%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: YetterPrimxxx1.dvi %%Pages: 152 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips YetterPrimxxx1.dvi %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 1999.05.30:1854 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N 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1 1 1 0 bop 326 436 a Fv(Y)-6 b(etter-Drinfel'd)22 b(Hopf)f(algebras)g(o)n (v)n(er)h(groups)f(of)806 527 y(prime)g(order)748 683 y Fu(Y)l(orc)o(k)16 b(Sommerh\177)-24 b(auser)889 1382 y Ft(Abstract)360 1467 y Fs(W)m(e)20 b(pro)o(v)o(e)g(a)g(structure)g (theorem)g(for)g(Y)m(etter-Drinfel'd)g(Hopf)g(algebras)h(o)o(v)o(er)361 1512 y(groups)d(of)f(prime)i(order)e(that)h(are)f(non)o(trivial,)j(co)q (comm)o(utativ)o(e,)f(and)f(cosemi-)361 1558 y(simple:)f(Under)f (certain)h(assumptions)h(on)e(the)g(base)g(\014eld,)h(these)f(algebras) h(can)361 1604 y(b)q(e)10 b(decomp)q(osed)i(in)o(to)e(a)g(tensor)h(pro) q(duct)f(of)g(the)g(dual)h(group)g(ring)g(of)e(the)h(group)h(of)361 1649 y(prime)j(order)f(and)g(an)g(ordinary)h(group)f(ring)h(of)e(some)h (other)g(group.)g(This)g(tensor)361 1695 y(pro)q(duct)i(is)f(a)g (crossed)h(pro)q(duct)f(as)g(an)g(algebra)i(and)e(an)g(ordinary)i (tensor)e(pro)q(d-)361 1741 y(uct)e(as)g(a)g(coalgebra.)h(In)f (particular,)h(the)f(dimension)j(of)c(suc)o(h)i(a)e(Y)m (etter-Drinfel'd)361 1786 y(Hopf)k(algebra)h(is)f(divisible)j(b)o(y)d (the)f(prime)i(under)f(consideration.)i(W)m(e)e(also)h(\014nd)361 1832 y(explicit)g(examples)f(of)f(suc)o(h)g(Y)m(etter-Drinfel'd)h(Hopf) e(algebras)j(and)e(apply)h(these)361 1878 y(results)f(to)f(the)g (classi\014catio)q(n)j(program)e(for)e(semisimple)k(Hopf)d(algebras.)p eop %%Page: 2 2 2 1 bop 257 262 a Fr(Con)n(ten)n(ts)257 389 y Fq(In)o(tro)q(duction) 1142 b(7)257 516 y(1)39 b(Prelimi)o(nari)o(es)1042 b(11)320 601 y Fp(1.1)41 b(Y)m(etter-Drinfel'd)14 b(mo)q(dules)h(.)20 b(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g (.)h(.)f(.)g(.)g(.)45 b(11)320 687 y(1.2)c(Duals)13 b(of)h(Y)m (etter-Drinfel'd)f(mo)q(dules)g(.)20 b(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)45 b(13)320 772 y(1.3)c(The)14 b(dual)g(Hopf)f(algebra)23 b(.)d(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.) f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)45 b(14)320 858 y(1.4)c(T)m(ensor)14 b(pro)q(ducts)f(.)20 b(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f (.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)45 b(14)320 943 y(1.5)c(Grouplik)o(e)13 b(elemen)o(ts)21 b(.)g(.)f(.)g(.)g(.)h(.)f(.)g (.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.) g(.)45 b(16)320 1029 y(1.6)c(The)14 b(Radford)f(bipro)q(duct)i (construction)32 b(.)20 b(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h (.)f(.)g(.)g(.)45 b(17)320 1115 y(1.7)c(Mo)q(dules)14 b(o)o(v)o(er)g(Radford)f(bipro)q(ducts)20 b(.)g(.)g(.)g(.)h(.)f(.)g(.)h (.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)45 b(18)320 1200 y(1.8)c(The)14 b(Nic)o(hols-Zo)q(eller)g(theorem)h(.)20 b(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f (.)g(.)g(.)45 b(19)320 1286 y(1.9)c(Dual)13 b(mo)q(dules)21 b(.)g(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g (.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)45 b(20)320 1371 y(1.10)20 b(Dimo)q(dules)14 b(.)20 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g 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b(of)f(the)i(Radford) e(bipro)q(duct)20 b(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f (.)g(.)h(.)f(.)g(.)g(.)45 b(30)320 2097 y(2.4)c(Purely)14 b(unstable)h(mo)q(dules)h(.)k(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g (.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)45 b(32)320 2183 y(2.5)c(Stable)14 b(mo)q(dules)28 b(.)20 b(.)g(.)h(.)f(.)g(.)g(.)h (.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.) f(.)g(.)g(.)45 b(33)320 2268 y(2.6)c(The)14 b(link)n(age)f(principle)20 b(.)g(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g (.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)45 b(34)320 2354 y(2.7)c(The)14 b(left)g(action)36 b(.)20 b(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)45 b(35)963 2628 y(2)p eop %%Page: 3 3 3 2 bop 257 262 a Fq(3)39 b(Examples)1121 b(37)320 347 y Fp(3.1)41 b(Con)o(v)o(en)o(tions)18 b(.)i(.)h(.)f(.)g(.)h(.)f(.)g(.)g (.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.) h(.)f(.)g(.)g(.)45 b(37)320 433 y(3.2)c(The)14 b(framew)o(ork)28 b(.)20 b(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.) h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)45 b(37)320 518 y(3.3)c(Finite)14 b(rings)g(and)f(\014nite)h(groups)33 b(.)20 b(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.) f(.)g(.)g(.)45 b(42)320 604 y(3.4)c(The)14 b(case)h(of)f(cyclic)g (groups)d(.)20 b(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.) g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)45 b(45)320 689 y(3.5)c(The)14 b(ev)o(en)h(case)k(.)i(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)45 b(46)320 775 y(3.6)c(The)14 b(case)h(of)f(dimension)e(4)28 b(.)20 b(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.) f(.)g(.)h(.)f(.)g(.)g(.)45 b(47)257 902 y Fq(4)39 b(Isomorphisms)1031 b(49)320 987 y Fp(4.1)41 b(Con)o(v)o(en)o(tions)18 b(.)i(.)h(.)f(.)g(.) h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g (.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)45 b(49)320 1073 y(4.2)c(Homomorphism)o (s)30 b(.)20 b(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g (.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)45 b(49)320 1159 y(4.3)c(Finite)14 b(rings,)f(\014nite)h(groups,)g(and)g(homom)o (orphism)o(s)20 b(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)45 b(52)320 1244 y(4.4)c(Categories)14 b(and)g(functors)36 b(.)20 b(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.) f(.)g(.)h(.)f(.)g(.)g(.)45 b(54)320 1330 y(4.5)c(Classi\014cation)13 b(of)h(isomorphism)o(s)e(in)h(the)i(o)q(dd)e(case)19 b(.)h(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)45 b(55)320 1415 y(4.6)c(The)14 b(case)h(of)f(dimension)e Fo(p)853 1400 y Fn(2)913 1415 y Fp(.)21 b(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.) h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)45 b(58)320 1501 y(4.7)c(Construction)15 b(of)e(isomorphisms)e(in)i(the)i(ev)o(en)f (case)43 b(.)20 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)45 b(60)320 1586 y(4.8)c(Classi\014cation)13 b(of)h(isomorphism)o(s)e(in)h (the)i(ev)o(en)f(case)38 b(.)20 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)45 b(62)320 1672 y(4.9)c(Isomorphisms)12 b(in)h(the)i(case)f(of)g (dimension)e(4)22 b(.)e(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g (.)45 b(64)257 1799 y Fq(5)39 b(Constructions)1028 b(66)320 1885 y Fp(5.1)41 b(Con)o(v)o(en)o(tions)18 b(.)i(.)h(.)f(.)g(.)h(.)f(.) g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f (.)g(.)h(.)f(.)g(.)g(.)45 b(66)320 1970 y(5.2)c(The)14 b(Radford)f(bipro)q(duct)f(.)20 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h (.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)45 b(67)320 2056 y(5.3)c(In)o(tegrals)14 b(of)f(the)i(Radford)e(bipro)q (duct)33 b(.)20 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f (.)g(.)g(.)45 b(68)320 2141 y(5.4)c(Hopf)14 b(algebra)f(extensions)34 b(.)20 b(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.) f(.)g(.)h(.)f(.)g(.)g(.)45 b(69)320 2227 y(5.5)c(The)14 b(second)h(construction)24 b(.)c(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.) g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)45 b(71)320 2312 y(5.6)c(The)14 b(t)o(wisted)h(dual)28 b(.)20 b(.)h(.)f(.)g(.)g(.)h (.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.) f(.)g(.)g(.)45 b(71)320 2398 y(5.7)c(The)14 b(adjoin)o(t)f(actions)j(.) 21 b(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.) g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)45 b(73)320 2484 y(5.8)c(The)14 b(coadjoin)o(t)f(actions)42 b(.)20 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.) h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)45 b(75)963 2628 y(3)p eop %%Page: 4 4 4 3 bop 320 262 a Fp(5.9)41 b(Structure)16 b(elemen)o(ts)e(of)f(the)h (second)h(construction)24 b(.)c(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.) 45 b(76)320 347 y(5.10)20 b(In)o(tegrals)14 b(of)f(the)i(second)g (construction)e(.)20 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h (.)f(.)g(.)g(.)45 b(81)320 433 y(5.11)20 b(Extensions)15 b(and)f(the)g(second)h(construction)38 b(.)20 b(.)g(.)h(.)f(.)g(.)g(.)h (.)f(.)g(.)h(.)f(.)g(.)g(.)45 b(83)320 518 y(5.12)20 b(Normal)12 b(bases)29 b(.)21 b(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h (.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.) 45 b(84)257 645 y Fq(6)39 b(Comm)o(utativ)o(e)13 b(Y)l(etter-Drinf)o (el)o('d)g(Hopf)i(algebras)384 b(88)320 731 y Fp(6.1)41 b(Con)o(v)o(en)o(tions)18 b(.)i(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g (.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.) g(.)45 b(88)320 816 y(6.2)c(Corresp)q(ondences)36 b(.)20 b(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)45 b(88)320 902 y(6.3)c(The)14 b(orbit)g(of)f(the)i(dual)i(.)j(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h (.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)45 b(89)320 987 y(6.4)c(In)o(v)n(arian)o(t)13 b(orbits)23 b(.)d(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h (.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)45 b(90)320 1073 y(6.5)c(Decomp)q(osition)12 b(of)i(tensor)g(pro)q(ducts)j(.)j(.)g (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)45 b(90)320 1159 y(6.6)c(Pro)q(ducts)15 b(of)f(c)o(haracters)27 b(.)20 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.) h(.)f(.)g(.)h(.)f(.)g(.)g(.)45 b(92)320 1244 y(6.7)c(Mo)q(dules)14 b(and)g(their)g(duals)25 b(.)20 b(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f (.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)45 b(95)320 1330 y(6.8)c(The)14 b(in)o(v)n(arian)o(t)f(c)o(haracter)40 b(.)20 b(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.) f(.)g(.)h(.)f(.)g(.)g(.)45 b(97)257 1457 y Fq(7)39 b(Co)q(comm)o (utativ)o(e)13 b(Y)l(etter-Drinfe)o(l'd)f(Hopf)k(algebras)313 b(100)320 1542 y Fp(7.1)41 b(Con)o(v)o(en)o(tions)18 b(.)i(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f (.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)k(100)320 1628 y(7.2)41 b(Co)q(comm)o(utativ)o(e)11 b(Y)m(etter-Drinfel'd)j(Hopf) f(algebras)36 b(.)20 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)k(100)320 1713 y(7.3)41 b(The)14 b(quotien)o(t)37 b(.)21 b(.)f(.)g(.)h(.)f(.)g(.) g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g (.)h(.)f(.)g(.)g(.)k(101)320 1799 y(7.4)41 b(The)14 b(quotien)o(t)g (group)i(.)21 b(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h (.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)k(102)320 1885 y(7.5)41 b(The)14 b(como)q(dule)f(algebra)g(structure)k(.)k(.)f(.) g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)k(103) 320 1970 y(7.6)41 b(The)14 b(compatibilit)o(y)d(condition)23 b(.)d(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h (.)f(.)g(.)g(.)k(104)320 2056 y(7.7)41 b(The)14 b(structure)i(theorem)e (in)f(the)i(o)q(dd)f(case)24 b(.)d(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g (.)h(.)f(.)g(.)g(.)k(107)320 2141 y(7.8)41 b(The)14 b(structure)i (theorem)e(in)f(the)i(ev)o(en)f(case)e(.)21 b(.)f(.)g(.)h(.)f(.)g(.)g (.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)k(108)320 2227 y(7.9)41 b(The)14 b(divisibilit)o(y)e(theorem)28 b(.)20 b(.)h(.)f(.)g(.)h(.)f(.) g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)k(109) 320 2312 y(7.10)c(Y)m(etter-Drinfel'd)14 b(Hopf)f(algebras)h(of)f (dimension)g Fo(p)1244 2297 y Fn(2)1301 2312 y Fp(.)20 b(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)k(109)963 2628 y(4)p eop %%Page: 5 5 5 4 bop 257 262 a Fq(8)39 b(Semisimp)o(le)13 b(Hopf)i(algebras)f(of)h (dimension)e Fo(p)1177 246 y Fn(3)1618 262 y Fq(111)320 341 y Fp(8.1)41 b(Con)o(v)o(en)o(tions)18 b(.)i(.)h(.)f(.)g(.)h(.)f(.)g (.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.) g(.)h(.)f(.)g(.)g(.)k(111)320 421 y(8.2)41 b(Grouplik)o(e)13 b(elemen)o(ts)21 b(.)g(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f (.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)k(111)320 501 y(8.3)41 b(The)14 b(bipro)q(duct)h(structure)28 b(.)20 b(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g (.)h(.)f(.)g(.)g(.)k(112)320 581 y(8.4)41 b(Comm)o(utativ)o(it)o(y)25 b(.)20 b(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.) h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)k(113)320 661 y(8.5)41 b(Basis)14 b(description)31 b(.)20 b(.)h(.)f(.)g(.)g(.)h (.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.) f(.)g(.)g(.)k(114)320 741 y(8.6)41 b(Isomorphisms)12 b(in)h(the)i(o)q(dd)e(case)26 b(.)20 b(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h (.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)k(115)320 821 y(8.7)41 b(Isomorphisms)12 b(of)h(Radford)g(bipro)q(ducts)29 b(.)20 b(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)k (116)320 901 y(8.8)41 b(Classi\014cation)31 b(.)21 b(.)f(.)g(.)h(.)f(.) g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h(.)f (.)g(.)h(.)f(.)g(.)g(.)k(120)320 981 y(8.9)41 b(The)14 b(case)h(of)f(dimension)e(8)28 b(.)20 b(.)h(.)f(.)g(.)h(.)f(.)g(.)g(.)h 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y(k)o(ey)d(step)h(in)f(the)g(pro)q(of)g(of)f(the)h(structure)i (theorem)e(in)f(the)i(next)f(section.)257 1315 y(In)j(Section)f(7,)g(w) o(e)g(dualize)g(the)h(situation)f(and)g(consider)h(non)o(trivial)e(Y)m (etter-Drinfel'd)257 1365 y(Hopf)g(algebras)f(o)o(v)o(er)h(groups)g(of) f(prime)g(order)i(that)e(are)i(co)q(comm)o(utativ)o(e)c(and)j(cosemi-) 257 1415 y(simple.)c(F)m(rom)g(the)i(previous)g(section,)g(w)o(e)f(kno) o(w)g(that)h(there)h(exists)f(an)f(in)o(v)n(arian)o(t,)f(coin-)257 1465 y(v)n(arian)o(t)17 b(grouplik)o(e)g(elemen)o(t)g(of)g(order)h Fo(p)g Fp(that)g(induces)g(action)f(and)h(coaction)f(on)h(the)257 1514 y(other)c(grouplik)o(e)f(elemen)o(ts.)f(W)m(e)h(then)h(pass)g(to)f (a)g(Y)m(etter-Drinfel'd)g(Hopf)g(algebra)g(quo-)257 1564 y(tien)o(t)h(in)e(whic)o(h)h(this)g(grouplik)o(e)f(elemen)o(t)h (is)g(equal)f(to)h(the)h(unit.)e(No)o(w)h(action)f(and)h(coac-)257 1614 y(tion)g(in)g(this)h(quotien)o(t)f(are)h(trivial,)e(and)h 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Fo(b)466 1713 y Fn(\(1\))510 1706 y Fp(\)\()p Fo(a)564 1713 y Fn(\(2\))609 1689 y(\(1\))654 1706 y Fo(b)672 1713 y Fn(\(2\))716 1689 y(\(1\))772 1706 y Fm(!)i Fp(\()p Fo(a)863 1689 y Fl(0)863 1717 y Fn(\(1\))917 1706 y Fm(\012)f Fo(b)977 1689 y Fl(0)977 1717 y Fn(\(1\))1021 1706 y Fp(\)\))g Fm(\012)f Fp(\()p Fo(a)1142 1713 y Fn(\(2\))1187 1689 y(\(2\))1241 1706 y Fm(\012)g Fo(b)1300 1713 y Fn(\(2\))1344 1689 y(\(2\))1389 1706 y Fp(\)\()p Fo(a)1443 1689 y Fl(0)1443 1717 y Fn(\(2\))1497 1706 y Fm(\012)h Fo(b)1557 1689 y Fl(0)1557 1717 y Fn(\(2\))1601 1706 y Fp(\))271 1780 y(=)i Fo(a)337 1787 y Fn(\(1\))381 1780 y Fp(\()p Fo(a)419 1787 y Fn(\(2\))464 1763 y(\(1\))508 1780 y Fo(b)526 1787 y Fn(\(2\))571 1763 y(\(1\))627 1780 y Fm(!)f Fo(a)702 1763 y Fl(0)702 1791 y Fn(\(1\))746 1780 y Fp(\))f Fm(\012)f Fo(b)831 1787 y Fn(\(1\))876 1780 y Fp(\()p Fo(a)914 1787 y Fn(\(2\))958 1763 y(\(2\))1003 1780 y Fo(b)1021 1787 y Fn(\(2\))1065 1763 y(\(2\))1121 1780 y Fm(!)i Fo(b)1192 1763 y Fl(0)1192 1791 y Fn(\(1\))1237 1780 y Fp(\))e Fm(\012)h Fo(a)1326 1787 y Fn(\(2\))1370 1763 y(\(3\))1415 1780 y Fo(a)1437 1763 y Fl(0)1437 1791 y Fn(\(2\))1484 1780 y Fm(\012)r Fo(b)1536 1787 y Fn(\(2\))1581 1763 y(\(3\))1625 1780 y Fo(b)1643 1763 y Fl(0)1643 1791 y Fn(\(2\))257 1864 y Fp(and)k(on)g(the)g(other)h(hand:)296 1949 y(\001)331 1955 y Fj(A)p Fl(\012)p Fj(B)410 1949 y Fp(\()q(\()p Fo(a)9 b Fm(\012)g Fo(b)p Fp(\)\()p Fo(a)587 1932 y Fl(0)608 1949 y Fm(\012)h Fo(b)668 1932 y Fl(0)679 1949 y Fp(\)\))i(=)g(\()p Fo(aa)827 1932 y Fl(0)839 1949 y Fp(\))855 1956 y Fn(\(1\))908 1949 y Fm(\012)e Fp(\()p Fo(bb)1002 1932 y Fl(0)1013 1949 y Fp(\))1029 1956 y Fn(\(1\))1083 1949 y Fm(\012)g Fp(\()p Fo(aa)1185 1932 y Fl(0)1197 1949 y Fp(\))1213 1956 y Fn(\(2\))1266 1949 y Fm(\012)g Fp(\()p Fo(bb)1360 1932 y Fl(0)1371 1949 y Fp(\))1387 1956 y Fn(\(2\))438 2019 y Fp(=)i Fo(a)504 2026 y Fn(\(1\))548 2019 y Fp(\()p Fo(a)586 2026 y Fn(\(2\))631 2002 y(\(1\))687 2019 y Fm(!)f Fo(a)762 2002 y Fl(0)762 2031 y Fn(\(1\))807 2019 y Fp(\))e Fm(\012)h Fo(b)892 2026 y Fn(\(1\))936 2019 y Fp(\()p Fo(b)970 2026 y Fn(\(2\))1014 2002 y(\(1\))1071 2019 y Fm(!)h Fo(b)1142 2002 y Fl(0)1142 2031 y Fn(\(1\))1186 2019 y Fp(\))e Fm(\012)h Fo(a)1275 2026 y Fn(\(2\))1319 2002 y(\(2\))1364 2019 y Fo(a)1386 2002 y Fl(0)1386 2031 y Fn(\(2\))1440 2019 y Fm(\012)f Fo(b)1499 2026 y Fn(\(2\))1544 2002 y(\(2\))1588 2019 y Fo(b)1606 2002 y Fl(0)1606 2031 y Fn(\(2\))257 2104 y Fp(Under)15 b(our)f(assumptions,)f(b)q(oth)h(expressions)h(are)f (equal.)257 2187 y(The)h(an)o(tip)q(o)q(de)g Fo(S)541 2193 y Fj(A)p Fl(\012)p Fj(B)634 2187 y Fp(:=)e Fo(S)716 2193 y Fj(A)753 2187 y Fm(\012)d Fo(S)820 2193 y Fj(B)864 2187 y Fp(is)k(a)g(tensor)i(pro)q(duct)g(of)e(t)o(w)o(o)g(linear)g(and) h(colinear)257 2236 y(maps,)c(and)i(therefore)h(is)f(itself)g(linear)f (and)g(colinear.)h(It)f(is)h(easy)g(to)g(see)h(that)f(it)f(really)h(is) 257 2286 y(an)h(an)o(tip)q(o)q(de.)257 2369 y(T)m(o)9 b(pro)o(v)o(e)h(that)g(the)h(\014rst)g(assertion)f(implies)e(the)j (second,)f(note)g(that,)g(if)f(b)q(oth)h(expressions)257 2419 y(calculated)k(ab)q(o)o(v)o(e)g(are)g(equal,)f(w)o(e)h(get)h(b)o (y)e(applying)g Fo(\017)1136 2425 y Fj(A)1172 2419 y Fm(\012)c Fp(id)1248 2425 y Fj(B)1284 2419 y Fm(\012)e Fp(id)1357 2425 y Fj(A)1391 2419 y Fm(\012)p Fo(\017)1440 2425 y Fj(B)1483 2419 y Fp(that:)688 2503 y Fo(b)p Fp(\()p Fo(a)744 2486 y Fn(\(1\))800 2503 y Fm(!)k Fo(b)871 2486 y Fl(0)883 2503 y Fp(\))e Fm(\012)h Fo(a)972 2486 y Fn(\(2\))1016 2503 y Fo(a)1038 2486 y Fl(0)1061 2503 y Fp(=)i Fo(bb)1141 2486 y Fl(0)1162 2503 y Fm(\012)d Fo(aa)1247 2486 y Fl(0)953 2628 y Fp(15)p eop %%Page: 16 16 16 15 bop 257 262 a Fp(F)m(or)15 b Fo(b)e Fp(=)h(1)431 268 y Fj(B)474 262 y Fp(and)h Fo(a)578 246 y Fl(0)603 262 y Fp(=)f(1)670 268 y Fj(A)697 262 y Fp(,)g(this)h(yields)g Fo(\033)947 268 y Fj(A;B)1010 262 y Fp(\()p Fo(a)10 b Fm(\012)h Fo(b)1119 246 y Fl(0)1130 262 y Fp(\))j(=)f Fo(b)1223 246 y Fl(0)1245 262 y Fm(\012)d Fo(a)p Fp(.)k(On)i(the)f (other)h(hand,)257 311 y(b)o(y)e(applying)f(id)519 317 y Fj(A)553 311 y Fm(\012)p Fo(\017)602 317 y Fj(B)640 311 y Fm(\012)d Fo(\017)699 317 y Fj(A)735 311 y Fm(\012)f Fp(id)811 317 y Fj(B)853 311 y Fp(to)14 b(these)h(expressions,)g(w)o(e) f(get:)690 390 y Fo(a)p Fp(\()p Fo(b)746 373 y Fn(\(1\))802 390 y Fm(!)d Fo(a)877 373 y Fl(0)889 390 y Fp(\))e Fm(\012)h Fo(b)974 373 y Fn(\(2\))1018 390 y Fo(b)1036 373 y Fl(0)1059 390 y Fp(=)i Fo(aa)1147 373 y Fl(0)1168 390 y Fm(\012)d Fo(bb)1245 373 y Fl(0)257 468 y Fp(F)m(or)14 b Fo(a)d Fp(=)h(1)430 474 y Fj(A)471 468 y Fp(and)h Fo(b)569 453 y Fl(0)592 468 y Fp(=)f(1)657 474 y Fj(B)686 468 y Fp(,)h(this)h (yields)g Fo(\033)933 474 y Fj(B)q(;A)996 468 y Fp(\()p Fo(b)9 b Fm(\012)g Fo(a)1102 453 y Fl(0)1114 468 y Fp(\))j(=)g Fo(a)1208 453 y Fl(0)1228 468 y Fm(\012)e Fo(b)p Fp(.)j Fi(2)257 578 y Fp(The)k(topic)f(of)f(tensor)i(pro)q(ducts)h(of)d(Hopf)h (algebras)g(in)g(quasisymmetric)e(categories)j(is)257 627 y(treated)e(in)f(greater)h(detail)e(in)h([60)o(],)e(Sec.)j(4.)257 757 y Fq(1.5)48 b Fp(As)20 b(for)g(an)o(y)f(coalgebra,)g(w)o(e)h(call)f (an)h(elemen)o(t)f Fo(g)i Fp(of)e(a)g(Y)m(etter-Drinfel'd)h(Hopf)257 807 y(algebra)c Fo(A)h Fp(a)f(grouplik)o(e)g(elemen)o(t)g(if)g(\001)905 813 y Fj(A)931 807 y Fp(\()p Fo(g)q Fp(\))h(=)f Fo(g)c Fm(\012)g Fo(g)17 b Fp(and)g Fo(\017)1263 813 y Fj(A)1290 807 y Fp(\()p Fo(g)q Fp(\))f(=)g(1)1428 813 y Fj(K)1460 807 y Fp(.)g(In)h(con)o(trast)257 857 y(to)d(ordinary)f(Hopf)g (algebras,)g(the)h(grouplik)o(e)f(elemen)o(ts)g(usually)g(do)g(not)h (form)e(a)h(group,)257 907 y(b)q(ecause)22 b(the)e(pro)q(duct)g(of)f(t) o(w)o(o)g(grouplik)o(e)g(elemen)o(ts)g(in)g(general)h(is)g(not)f(a)g (grouplik)o(e)257 957 y(elemen)o(t.)13 b(Ho)o(w)o(ev)o(er,)h(one)g(has) g(the)h(follo)o(wing)c(substitute:)257 1048 y Fq(Prop)q(osition)h(1)21 b Fp(Supp)q(ose)15 b(that)e Fo(A)h Fp(is)f(a)h(Y)m(etter-Drinfel'd)f (Hopf)g(algebra)g(o)o(v)o(er)h(a)f(Hopf)257 1098 y(algebra)h Fo(H)s Fp(.)f(Denote)h(b)o(y)g Fo(G)p Fp(\()p Fo(A)p Fp(\))g(the)g(set)h(of)f(grouplik)o(e)f(elemen)o(ts.)g(Denote)h(b)o(y) 525 1177 y Fo(G)558 1183 y Fj(I)577 1177 y Fp(\()p Fo(A)p Fp(\))e(:=)f Fm(f)p Fo(g)i Fm(2)e Fo(G)p Fp(\()p Fo(A)p Fp(\))h Fm(j)f(8)e Fo(h)i Fm(2)h Fo(H)i Fp(:)d Fo(h)g Fm(!)g Fo(g)i Fp(=)f Fo(\017)1292 1183 y Fj(H)1323 1177 y Fp(\()p Fo(h)p Fp(\))p Fo(g)q Fm(g)257 1255 y Fp(the)j(set)g(of)e(in) o(v)n(arian)o(t)f(grouplik)o(e)h(elemen)o(ts)h(and)g(b)o(y)617 1334 y Fo(G)650 1340 y Fj(C)677 1334 y Fp(\()p Fo(A)p Fp(\))e(:=)f Fm(f)p Fo(g)i Fm(2)e Fo(G)p Fp(\()p Fo(A)p Fp(\))h Fm(j)f Fo(\016)1049 1340 y Fj(A)1076 1334 y Fp(\()p Fo(g)q Fp(\))h(=)g(1)1206 1340 y Fj(H)1247 1334 y Fm(\012)d Fo(g)q Fm(g)257 1412 y Fp(the)15 b(set)g(of)e(coin)o(v)n(arian)o(t)f (grouplik)o(e)h(elemen)o(ts.)h(Then)g(w)o(e)g(ha)o(v)o(e:)308 1515 y(1.)20 b(Ev)o(ery)15 b(grouplik)o(e)e(elemen)o(t)g(is)h(in)o(v)o (ertible.)308 1593 y(2.)20 b Fo(G)394 1599 y Fj(I)413 1593 y Fp(\()p Fo(A)p Fp(\))14 b(and)g Fo(G)604 1599 y Fj(C)631 1593 y Fp(\()p Fo(A)p Fp(\))h(are)f(subgroups)h(of)e(the)h (group)g(of)f(units.)308 1671 y(3.)20 b Fo(G)394 1677 y Fj(I)413 1671 y Fp(\()p Fo(A)p Fp(\))14 b(acts)h(on)e Fo(G)p Fp(\()p Fo(A)p Fp(\))h(via)f(righ)o(t)h(m)o(ultipli)o(cation.) 308 1749 y(4.)20 b Fo(G)394 1755 y Fj(C)422 1749 y Fp(\()p Fo(A)p Fp(\))14 b(acts)h(on)e Fo(G)p Fp(\()p Fo(A)p Fp(\))h(via)f(left) h(m)o(ultiplicati)o(on.)257 1841 y Fq(Pro)q(of.)36 b Fp(As)20 b(for)f(ordinary)f(Hopf)h(algebras,)f Fo(S)1041 1847 y Fj(A)1069 1841 y Fp(\()p Fo(g)q Fp(\))i(is)e(an)h(in)o(v)o(erse) h(of)f(the)g(grouplik)o(e)257 1890 y(elemen)o(t)14 b Fo(g)q Fp(.)f(If)h Fo(g)h Fp(and)f Fo(g)636 1875 y Fl(0)662 1890 y Fp(are)g(grouplik)o(e)f(elemen)o(ts,)g(w)o(e)h(ha)o(v)o(e:)640 1969 y(\001)675 1975 y Fj(A)702 1969 y Fp(\()p Fo(g)q(g)760 1952 y Fl(0)772 1969 y Fp(\))e(=)g Fo(g)q Fp(\()p Fo(g)902 1952 y Fn(\(1\))958 1969 y Fm(!)f Fo(g)1032 1952 y Fl(0)1044 1969 y Fp(\))f Fm(\012)f Fp(\()p Fo(g)1148 1952 y Fn(\(2\))1205 1969 y Fm(!)i Fo(g)1279 1952 y Fl(0)1291 1969 y Fp(\))257 2047 y(Therefore,)21 b Fo(g)q(g)505 2032 y Fl(0)537 2047 y Fp(is)f(grouplik)o(e)f(if)h Fo(g)h Fp(is)f(coin)o(v)n(arian)o(t)e(or) i Fo(g)1202 2032 y Fl(0)1234 2047 y Fp(is)g(in)o(v)n(arian)o(t.)e(This) i(pro)o(v)o(es)257 2097 y(the)e(third)f(and)f(the)i(fourth)f(statemen)o (t.)f(The)h(pro)q(duct)h(of)e(t)o(w)o(o)h(in)o(v)n(arian)o(t)e(elemen)o (ts)i(is)257 2147 y(again)e(in)o(v)n(arian)o(t,)f(therefore)j Fo(G)767 2153 y Fj(I)786 2147 y Fp(\()p Fo(A)p Fp(\))f(is)g(a)f(m)o (ultiplicativ)o(ely)d(closed)17 b(set.)f(Similarly)l(,)d(the)257 2197 y(pro)q(duct)k(of)d(t)o(w)o(o)h(coin)o(v)n(arian)o(t)f(elemen)o (ts)h(is)g(again)f(coin)o(v)n(arian)o(t,)g(therefore)i Fo(G)1519 2203 y Fj(C)1547 2197 y Fp(\()p Fo(A)p Fp(\))g(is)f(a)257 2247 y(m)o(ultiplicativ)o(ely)f(closed)19 b(set.)f(T)m(o)f(pro)o(v)o(e) h(that)f(they)i(are)f(subgroups,)g(w)o(e)g(m)o(ust)e(pro)o(v)o(e)257 2296 y(that)e(they)h(con)o(tain)e(in)o(v)o(erses.)i(If)e Fo(g)i Fp(is)f(a)g(grouplik)o(e)f(elemen)o(t,)f(w)o(e)j(ha)o(v)o(e:)627 2375 y(\001)662 2381 y Fj(A)688 2375 y Fp(\()p Fo(S)729 2381 y Fj(A)757 2375 y Fp(\()p Fo(g)q Fp(\)\))d(=)g Fo(S)907 2381 y Fj(A)935 2375 y Fp(\()p Fo(g)972 2358 y Fn(\(1\))1028 2375 y Fm(!)f Fo(g)q Fp(\))f Fm(\012)f Fo(S)1194 2381 y Fj(A)1222 2375 y Fp(\()p Fo(g)1259 2358 y Fn(\(2\))1304 2375 y Fp(\))257 2453 y(Therefore)15 b(w)o(e)f(see)h(that,)f(if)f Fo(g)i Fp(is)e(in)o(v)n(arian)o(t)f(or)i(coin)o(v)n(arian)o(t,)e(then)i (the)g(in)o(v)o(erse)h Fo(S)1567 2459 y Fj(A)1594 2453 y Fp(\()p Fo(g)q Fp(\))g(is)257 2503 y(again)e(grouplik)o(e.)g(It)h(is) f(also)h(in)o(v)n(arian)o(t)e(or)i(coin)o(v)n(arian)o(t.)e Fi(2)953 2628 y Fp(16)p eop %%Page: 17 17 17 16 bop 257 262 a Fp(As)16 b(w)o(e)f(ha)o(v)o(e)g(seen)h(in)f(P)o (aragraph)f(1.2,)g(the)i(dual)e(of)g(a)h(\014nite-dimensional)e(left)i (Y)m(etter-)257 311 y(Drinfel'd)20 b(Hopf)h(algebra)f(o)o(v)o(er)h Fo(H)j 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Fp(is)d(comm)o(utativ)o(e)e(and)i (co)q(comm)o(utativ)o(e,)d(the)k(Y)m(etter-Drinfel'd)f(condition)257 1959 y(reads:)699 2009 y Fo(\016)717 2015 y Fj(V)746 2009 y Fp(\()p Fo(h)f Fm(!)f Fo(v)q Fp(\))h(=)g Fo(v)965 1992 y Fn(\(1\))1019 2009 y Fm(\012)e Fp(\()p Fo(h)i Fm(!)f Fo(v)1187 1992 y Fn(\(2\))1232 2009 y Fp(\))257 2082 y(A)k(mo)q(dule)f(that)h(is)g(sim)o(ultaneously)e(a)i(como)q(dule) f(suc)o(h)h(that)g(this)h(compatibilit)n(y)c(con-)257 2131 y(dition)17 b(is)g(satis\014ed)g(is)g(called)g(a)g(dimo)q(dule)e (\(cf.)i([39)o(]\).)g(In)g(particular,)f(if)h Fo(G)f Fp(is)h(a)g(\014nite)257 2181 y(ab)q(elian)12 b(group)g(and)h Fo(H)h Fp(=)e Fo(K)s Fp([)p Fo(G)p Fp(])f(is)h(its)h(group)f(ring,)g (then)h(an)f Fo(H)s Fp(-como)q(dule)f(is)h(the)h(same)257 2231 y(as)j(a)f Fo(G)p Fp(-graded)g(v)o(ector)i(space,)f(where)h(the)f (homogeneous)e(comp)q(onen)o(t)h(of)g(degree)i Fo(g)g Fp(is)257 2281 y(giv)o(en)d(b)o(y)700 2331 y Fo(V)724 2337 y Fj(g)755 2331 y Fp(=)e Fm(f)p Fo(v)h Fm(2)e Fo(V)21 b Fm(j)11 b Fo(\016)978 2337 y Fj(V)1007 2331 y Fp(\()p Fo(v)q Fp(\))i(=)f Fo(c)1135 2337 y Fj(g)1163 2331 y Fm(\012)e Fo(v)q Fm(g)257 2403 y Fp(where)18 b Fo(c)398 2409 y Fj(g)432 2403 y Fm(2)d Fo(K)s Fp([)p Fo(G)p Fp(])g(denotes)i (the)g(canonical)e(basis)h(elemen)o(t)g(of)f(the)i(group)f(ring)f (corre-)257 2453 y(sp)q(onding)g(to)f Fo(g)g Fm(2)e Fo(G)p Fp(.)i(The)g(dimo)q(dule)f(condition)h(ab)q(o)o(v)o(e)g(then)h(sa)o(ys) g(precisely)g(that)g(the)257 2503 y(homogeneous)e(comp)q(onen)o(ts)h (are)g(submo)q(dules.)953 2628 y(21)p eop %%Page: 22 22 22 21 bop 257 262 a Fp(No)o(w)11 b(supp)q(ose)h(that)f Fo(G)g Fp(=)h Fk(Z)708 268 y Fj(p)724 262 y Fp(,)e(where)i Fo(p)f Fp(is)g(not)f(necessarily)i(prime.)e(W)m(e)g(denote)i(b)o(y)e Fo(C)k Fp(the)257 311 y(set)j(of)f(grouplik)o(e)f(elemen)o(ts)h(of)g Fo(H)s Fp(,)f Fo(C)j Fp(:=)e Fo(G)p Fp(\()p Fo(H)s Fp(\).)f(These)j (are)e(precisely)h(the)g(canonical)257 361 y(basis)d(elemen)o(ts)g(of)f (the)i(group)f(ring,)f(i.)f(e.,)i(w)o(e)g(ha)o(v)o(e:)820 452 y Fo(C)g Fp(=)e Fm(f)p Fo(c)947 458 y Fj(i)972 452 y Fm(j)f Fo(i)g Fm(2)h Fk(Z)1090 458 y Fj(p)1107 452 y Fm(g)257 544 y Fp(Supp)q(ose)19 b(that)e Fo(K)j Fp(con)o(tains)d(a)g (primitiv)o(e)e Fo(p)p Fp(-th)i(ro)q(ot)h(of)e(unit)o(y)h Fo(\020)s Fp(,)g(whic)o(h)g(is)g(only)f(p)q(os-)257 594 y(sible)g(if)f(the)i(c)o(haracteristic)g(of)f Fo(K)j Fp(is)d(relativ)o(ely)f(prime)g(to)h Fo(p)p Fp(.)f(Then)i(w)o(e)f(ha)o (v)o(e)g(a)g(group)257 643 y(homomorphism)d Fo(\015)21 b Fp(:)c Fk(Z)656 649 y Fj(p)690 643 y Fm(!)h Fo(K)788 628 y Fl(\002)833 643 y Fp(that)g(maps)f(1)g(to)h Fo(\020)s Fp(.)f(If)1232 633 y(^)1228 643 y Fk(Z)1259 649 y Fj(p)1293 643 y Fp(denotes)i(the)f(c)o(haracter)257 693 y(group)13 b(of)g Fk(Z)451 699 y Fj(p)468 693 y Fp(,)f(i.)g(e.,)h(the)g(group)g (of)g(all)f(group)g(homomorphism)o(s)e(from)i Fk(Z)1427 699 y Fj(p)1456 693 y Fp(to)h Fo(K)1544 678 y Fl(\002)1586 693 y Fp(under)257 749 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Fj(p)884 737 y Fo(;)c(M)e Fp(\))11 b Fm(!)g Fo(H)1066 720 y Fn(2)1084 737 y Fp(\()p Fk(Z)1131 743 y Fj(p)1147 737 y Fo(;)c(M)e Fp(\))p Fo(;)i(q)12 b Fm(7!)f Fo(iq)257 837 y Fq(Pro)q(of.)36 b Fp(The)16 b(last)f(t)o(w)o(o)f(assertions)i(are)g(pro)o(v)o(ed)f(in)g (m)o(uc)o(h)f(greater)i(generalit)o(y)f(in)f([43)o(],)257 887 y(Chap.)19 b(IV,)g(Thm.)e(7.1,)h(p.)h(122,)f(resp.)i(Exerc.)g(7.6,) e(p.)g(124)h(and)g(in)g([22)o(],)f(Chap.)h(VI,)257 937 y(Prop.)c(7.1,)e(p.)h(201,)g(resp.)h(Exerc.)h(7.4,)d(p.)h(201.)g(T)m(o) g(pro)o(v)o(e)h(the)g(\014rst)g(assertion,)g(observ)o(e)257 986 y(that,)h(b)o(y)h(replacing)f Fo(q)h Fp(with)g Fo(q)12 b Fm(\000)f Fo(q)q Fp(\(0)p Fo(;)c Fp(0\),)15 b(w)o(e)i(can)f(assume)g (that)h Fo(q)g Fp(is)g(normalized.)d(By)257 1036 y(the)g(standard)f (description)g(of)g(extensions)g(via)f(co)q(cycles)j(\(cf.)d([6],)f (Chap.)h(IV,)h Fm(x)g Fp(3,)f(p.)g(91,)257 1086 y([43)o(],)h(Chap.)g (IV,)g Fm(x)h Fp(4,)f(p.)g(111\),)g(w)o(e)g(can)h(de\014ne)h(a)e(group) h(structure)h(on)f(the)g(set)g Fo(M)g Fm(\002)9 b Fk(Z)1673 1092 y Fj(p)257 1136 y Fp(b)o(y:)656 1186 y(\()p Fo(x;)e(i)p Fp(\)\()p Fo(y)q(;)g(j)r Fp(\))13 b(:=)e(\()p Fo(x)e Fp(+)g Fo(y)j Fp(+)d Fo(q)q Fp(\()p Fo(i;)e(j)r Fp(\))p Fo(;)g(i)j Fp(+)f Fo(j)r Fp(\))257 1260 y(where,)14 b(according)g(to)f (the)h(ab)q(o)o(v)o(e)f(lemma,)c(the)14 b(unit)f(elemen)o(t)g(is)g(\(0) p Fo(;)7 b Fp(0\))13 b(and)g(the)h(in)o(v)o(erse)257 1310 y(of)g(\()p Fo(x;)7 b(i)p Fp(\))13 b(is)h(\()p Fm(\000)p Fo(x)9 b Fm(\000)h Fo(q)q Fp(\()p Fo(i;)d Fm(\000)p Fo(i)p Fp(\))p Fo(;)g Fm(\000)p Fo(i)p Fp(\).)14 b(The)g(group)g(homomorphism) 747 1401 y Fo(M)i Fm(!)11 b Fo(M)j Fm(\002)9 b Fk(Z)982 1407 y Fj(p)998 1401 y Fo(;)e(x)k Fm(7!)g Fp(\()p Fo(x;)c Fp(0\))257 1493 y(iden)o(ti\014es)16 b Fo(M)j Fp(with)14 b(a)h(cen)o(tral)g(subgroup)g(of)f Fo(M)h Fm(\002)10 b Fk(Z)1118 1499 y Fj(p)1149 1493 y Fp(whose)15 b(corresp)q(onding)h (quotien)o(t)257 1543 y(is)e(isomorphic)e(to)h Fk(Z)590 1549 y Fj(p)606 1543 y 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b(get)h(a)e(left)h (and)f(a)h(righ)o(t)f(action)h(of)918 1510 y(^)909 1521 y Fo(C)i Fp(on)e(the)g(set)h(of)e(irreducible)i(c)o(haracters.)g(W)m(e) 257 1571 y(will)g(establish)i(a)f(link)n(age)f(principle)i(that)f(will) f(yield)h(a)g(one-to-one)g(corresp)q(ondence)257 1620 y(b)q(et)o(w)o(een)e(the)g(orbits)e(of)g(the)h(righ)o(t)f(action)h(of) 1021 1610 y(^)1012 1620 y Fo(C)i Fp(in)d(the)h(set)h(of)e(irreducible)h (c)o(haracters)257 1670 y(of)h Fo(B)j Fp(and)d(the)h(orbits)f(of)g(cen) o(trally)g(primitiv)o(e)e(idemp)q(oten)o(ts)i(of)f Fo(A)i Fp(under)g(the)g(action)257 1720 y(of)c Fo(C)s Fp(.)f(The)h(results)i (of)d(this)h(section)h(can)f(also)g(b)q(e)g(understo)q(o)q(d)i(from)c (the)j(p)q(oin)o(t)e(of)h(view)257 1770 y(of)h(the)h(theory)g(of)f (prime)g(ideals)g(in)g(crossed)i(pro)q(ducts)g(dev)o(elop)q(ed)f(b)o(y) f(M.)g(Lorenz)i(and)257 1820 y(D.)e(S.)f(P)o(assman)g(\(cf.)h([40)o(],) f([64)o(],)g(Chap.)h(4\).)f(W)m(e)h(shall)f(indicate)h(the)h (connections)g(with)257 1870 y(this)e(theory)h(where)g(they)f(o)q (ccur.)257 1955 y(Throughout)e(this)g(section,)g(w)o(e)g(will)e (constan)o(tly)i(use)h(the)f(con)o(v)o(en)o(tion)f(that)h(indices)g (tak)o(e)257 2005 y(v)n(alues)i(b)q(et)o(w)o(een)i(0)e(and)g Fo(p)9 b Fm(\000)h Fp(1)k(and)g(are)g(reduced)i(mo)q(dulo)c Fo(p)i Fp(if)g(they)g(do)g(not)h(lie)e(within)257 2055 y(this)j(range.)f(In)h(notation,)e(w)o(e)i(shall)f(not)g(distinguish)g (b)q(et)o(w)o(een)i(an)f(in)o(teger)g Fo(i)e Fm(2)g Fk(Z)p Fp(and)257 2105 y(its)g(equiv)n(alence)g(class)h(in)e Fk(Z)711 2111 y Fj(p)738 2105 y Fp(:=)f Fk(Z)-13 b Fo(=)o(p)p Fk(Z)d Fp(.)257 2240 y Fq(2.2)48 b Fp(W)m(e)14 b(b)q(egin)h(b)o(y)f (asso)q(ciating)g(to)h(ev)o(ery)g(simple)e Fo(B)r Fp(-mo)q(dule)h(a)g (subset)i(of)e(the)h(set)g Fo(E)257 2290 y Fp(of)k(cen)o(trally)f (primitiv)o(e)f(idemp)q(oten)o(ts)h(of)g Fo(A)p Fp(.)h(This)f(subset)j (essen)o(tially)d(c)o(haracterizes)257 2340 y(whic)o(h)f(simple)f Fo(A)p 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461 y Fq(De\014nition)33 b Fp(Supp)q(ose)15 b(that)f Fo(V)23 b Fp(is)14 b(a)f(simple)g Fo(B)r Fp(-mo)q(dule)g(with)g(c)o(haracter)i Fo(\037)1502 467 y Fj(V)1531 461 y Fp(.)308 580 y(1.)20 b(De\014ne:)605 629 y Fo(\024)p Fp(\()p Fo(V)10 b Fp(\))i(:=)f Fm(f)p Fo(e)g Fm(2)h Fo(E)h Fm(j)e(9)p Fo(v)j Fm(2)d Fo(V)g Fm(n)r(f)p Fp(0)p Fm(g)g Fp(:)g(\()p Fo(e)f Fm(\012)g Fp(1)1279 635 y Fj(H)1310 629 y Fp(\))p Fo(v)j Fp(=)f Fo(v)q Fm(g)308 721 y Fp(2.)20 b(De\014ne:)15 b Fo(\024)526 706 y Fl(\003)545 721 y Fp(\()p Fo(V)9 b Fp(\))14 b(to)g(b)q(e)h(the)g (isotrop)o(y)f(group)g(of)f(the)i(canonical)f(righ)o(t)f(action)h(of) 1666 710 y(^)1657 721 y Fo(C)361 770 y Fp(on)g(the)g(set)h(of)e (irreducible)i(c)o(haracters:)688 887 y Fo(\024)712 870 y Fl(\003)731 887 y Fp(\()p Fo(V)10 b Fp(\))h(:=)h Fm(f)p Fo(\015)i Fm(2)968 876 y Fp(^)959 887 y Fo(C)g Fm(j)d Fo(\037)1052 893 y Fj(V)1081 887 y Fp(\()p Fo(\017)1114 893 y Fj(A)1150 887 y Fm(\012)f Fo(\015)r Fp(\))i(=)g Fo(\037)1313 893 y Fj(V)1342 887 y 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Fj(A)793 1446 y Fm(\012)g Fo(c)p Fp(\))p Fo(v)j Fp(=)f(\(1)983 1452 y Fj(A)1019 1446 y Fm(\012)d Fo(c)p Fp(\)\()p Fo(e)h Fm(\012)g Fp(1)1202 1452 y Fj(H)1233 1446 y Fp(\))p Fo(v)j Fp(=)f(\(1)1363 1452 y Fj(A)1399 1446 y Fm(\012)e Fo(c)p Fp(\))p Fo(v)257 1537 y Fp(This)j(implies)e(that)i Fo(c)f Fm(!)f Fo(e)h Fm(2)f Fo(\024)p Fp(\()p Fo(V)e Fp(\),)k(and)g(therefore)h Fo(\024)p Fp(\()p Fo(V)c Fp(\))j(is)g(a)g(disjoin)o(t)f(union)g(of)g (orbits)257 1587 y(of)f Fo(C)s Fp(.)g(Supp)q(ose)i(that)e Fo(O)i Fp(is)e(one)h(of)f(these)i(orbits)f(and)f(consider)i(for)e(ev)o (ery)i(elemen)o(t)e Fo(e)g Fm(2)h Fo(O)257 1636 y Fp(the)j(space)701 1686 y Fo(V)725 1692 y Fj(e)755 1686 y Fp(:=)c Fm(f)p Fo(v)i Fm(2)e Fo(V)21 b Fm(j)12 b Fp(\()p Fo(e)d Fm(\012)h Fp(1)1079 1692 y Fj(H)1110 1686 y Fp(\))p Fo(v)j Fp(=)f Fo(v)q Fm(g)257 1761 y Fp(whic)o(h)19 b(is)g(nonzero)g(b)o(y)g (de\014nition.)e Fo(V)874 1767 y Fj(e)911 1761 y Fp(is)i(ob)o(viously)e (an)i Fo(A)p Fp(-submo)q(dule)e(of)i Fo(V)9 b Fp(,)18 b(and)h(a)257 1811 y(calculation)13 b(similar)f(to)h(the)i(one)f(ab)q (o)o(v)o(e)g(sho)o(ws)g(that)861 1902 y Fo(V)894 1885 y Fl(0)917 1902 y Fp(:=)975 1863 y Fg(X)973 1952 y Fj(e)p Fl(2)p Fj(O)1044 1902 y Fo(V)1068 1908 y Fj(e)257 2035 y Fp(is)d(in)o(v)n(arian)o(t)f(with)g(resp)q(ect)j(to)e Fo(C)s Fp(.)f(This)h(means)f(that)h Fo(V)1138 2020 y Fl(0)1160 2035 y Fp(is)g(a)g Fo(B)r Fp(-submo)q(dule)f(of)h Fo(V)e Fp(,)i(whic)o(h)257 2085 y(b)o(y)j(simplicit)o(y)d(implies)h (that)i Fo(V)769 2070 y Fl(0)792 2085 y Fp(=)e Fo(V)d Fp(.)k(Therefore,)i(w)o(e)f(ha)o(v)o(e)g Fo(\024)p Fp(\()p Fo(V)9 b Fp(\))j(=)g Fo(O)q Fp(.)h Fi(2)257 2204 y Fp(The)e(ab)q(o)o(v) o(e)f(prop)q(osition)g(is)g(a)f(sp)q(ecial)i(case)g(of)f([13)o(],)f (Exerc.)i(18.10,)d(p.)i(288,)f(where)i(mostly)257 2253 y(the)h(corresp)q(onding)f(simple)f(mo)q(dules)f(are)j(emphasized)e(o)o (v)o(er)h(cen)o(trally)f(primitiv)o(e)f(idem-)257 2303 y(p)q(oten)o(ts)18 b(\(cf.)e(also)g([12)o(],)f(Prop.)i(\(11.16\),)e(p.) 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Fp(Supp)q(ose)22 b(that)f Fo(e)i Fm(2)g Fo(E)f Fp(is)f(a)f(cen)o(trally)h(primitiv)o(e)e(idemp)q (oten)o(t)h(of)g Fo(A)p Fp(.)257 558 y(Denote)15 b(b)o(y)e Fo(C)487 564 y Fj(e)519 558 y Fp(the)h(isotrop)o(y)g(group)f(of)h Fo(e)p Fp(,)f(i.)g(e.,)h Fo(C)1082 564 y Fj(e)1111 558 y Fp(:=)d Fm(f)p Fo(c)1205 564 y Fj(i)1230 558 y Fm(2)g Fo(C)j Fm(j)e Fo(c)1355 564 y Fj(i)1380 558 y Fm(!)f Fo(e)h Fp(=)g Fo(e)p Fm(g)308 673 y Fp(1.)20 b Fo(e)14 b Fp(is)g(called)g(purely)g(unstable)g(if)f Fo(C)916 679 y Fj(e)945 673 y Fp(=)f Fm(f)p Fp(1)p Fm(g)p Fp(.)308 755 y(2.)20 b Fo(e)14 b Fp(is)g(called)g(stable)g(if)f Fo(C)742 761 y Fj(e)771 755 y Fp(=)f Fo(C)s Fp(.)257 869 y(A)g(simple)f Fo(A)p Fp(-mo)q(dule,)f(or)i(its)g(c)o(haracter,)h (will)d(b)q(e)j(called)f(purely)g(unstable,)g(resp.)h(stable,)257 919 y(if)g(the)g(corresp)q(onding)i(cen)o(trally)e(primitiv)o(e)e (idemp)q(oten)o(t)h(has)h(this)g(prop)q(ert)o(y)m(.)h(If)e Fo(O)h Fm(\032)e Fo(E)257 969 y Fp(is)g(an)f(orbit)g(of)g Fo(C)s Fp(,)g(it)g(will)f(b)q(e)j(called)e(purely)h(unstable,)f(resp.)h (stable,)g(if)f(its)g(elemen)o(ts)h(ha)o(v)o(e)257 1018 y(this)k(prop)q(ert)o(y)m(.)f(A)g(simple)e Fo(B)r Fp(-mo)q(dule)i(will) e(b)q(e)j(called)f(purely)g(unstable,)g(resp.)h(stable)f(if)257 1068 y Fo(\024)p Fp(\()p Fo(V)c Fp(\))k(is)g(purely)g(unstable,)g (resp.)g(stable.)257 1185 y(The)e(ab)q(o)o(v)o(e)g(de\014nition)f (should)g(b)q(e)i(compared)d(with)i([12)o(],)e(p.)i(269,)e([10)o(],)h (Def.)f(2.7,)h(p.)g(356,)257 1234 y([73)o(],)i(Def.)h(3.7,)e(p.)h(278,) g(and)h(p.)f(286.)257 1318 y(No)o(w)k(supp)q(ose)i(that)e Fo(O)640 1324 y Fn(1)658 1318 y Fo(;)7 b(:)g(:)g(:)12 b(;)7 b(O)790 1324 y Fj(m)838 1318 y Fp(are)17 b(the)h(orbits)f(of)g Fo(E)i Fp(with)e(resp)q(ect)i(to)e(the)h(action)257 1368 y(of)c Fo(C)s Fp(.)f(Then)h(it)g(is)f(easy)i(to)e(see)i(that)812 1487 y Fo(B)f Fp(=)917 1435 y Fj(m)900 1448 y Fg(M)905 1536 y Fj(i)p Fn(=1)970 1487 y Fo(AO)1033 1493 y Fj(i)1056 1487 y Fm(\012)9 b Fo(H)257 1611 y Fp(is)19 b(a)g(decomp)q(osition)f (of)g Fo(B)k Fp(in)o(to)c(t)o(w)o(o-sided)h(ideals.)f(Ho)o(w)o(ev)o (er,)h(these)h(ideals)f(are)h(not)257 1661 y(alw)o(a)o(ys)13 b(simple.)f(But)j(they)f(are)g(simple)f(if)g(the)h(orbit)g(is)g(purely) g(unstable:)257 1759 y Fq(Prop)q(osition)e(1)21 b Fp(Supp)q(ose)f(that) f Fo(O)h Fp(is)f(a)g(purely)g(unstable)h(orbit)e(of)h Fo(C)j Fp(in)c Fo(E)r Fp(.)h(Then)257 1808 y Fo(AO)11 b Fm(\012)e Fo(H)17 b Fp(is)d(a)f(simple)f(ideal)i(of)f Fo(B)r Fp(.)257 1906 y Fq(Pro)q(of.)36 b Fp(\(1\))21 b(Denote)10 b(b)o(y)g Fo(K)727 1891 y Fc(Z)749 1895 y Ff(p)775 1906 y Fp(the)h(algebra)e(of)g(functions)h(from)e Fk(Z)1326 1912 y Fj(p)1352 1906 y Fp(to)i(the)g(base)g(\014eld)g Fo(K)s Fp(,)257 1956 y(with)g(p)q(oin)o(t)o(wise)f(addition)g(and)h(m)o (ultiplicatio)o(n.)d(W)m(e)i(denote)i(the)g(canonical)e(basis)h(v)o (ector)257 2006 y(corresp)q(onding)15 b(to)f Fo(i)e Fm(2)g Fk(Z)671 2012 y Fj(p)702 2006 y Fp(b)o(y)h Fo(e)778 2012 y Fj(i)793 2006 y Fp(,)g(i.)g(e.,)h Fo(e)930 2012 y 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Fp(\)\))257 2403 y(is)i(an)g(algebra)f(isomorphism)e(\(cf.)j([57)o(],)e(Cor.)i (9.4.3,)d(p.)j(162,)f(note)h(that)g(the)g(setup)h(there)257 2453 y(is)i(sligh)o(tly)f(di\013eren)o(t\),)i(since)g(it)f(maps)f(the)h (elemen)o(ts)g Fo(e)1166 2459 y Fj(i)1191 2453 y Fm(\012)11 b Fo(c)1252 2459 y Fj(i)p Fl(\000)p Fj(j)1324 2453 y Fp(to)16 b(the)g(matrix)f(units.)257 2503 y(Therefore,)g(the)g(smash)e (pro)q(duct)h Fo(K)847 2488 y Fc(Z)870 2492 y Ff(p)896 2503 y Fm(\012)9 b Fo(H)17 b Fp(is)d(simple.)953 2628 y(30)p eop %%Page: 31 31 31 30 bop 257 262 a Fp(\(2\))21 b(F)m(or)10 b(all)g Fo(e)i Fm(2)f Fo(O)q Fp(,)f(the)h(tensor)g(pro)q(duct)h(algebra)e Fo(Ae)s Fm(\012)s Fo(K)1192 246 y Fc(Z)1214 250 y Ff(p)1241 262 y Fp(is)h(therefore)h(an)e Fo(H)s Fp(-mo)q(dule)257 311 y(algebra)15 b(if)g(it)g(is)h(endo)o(w)o(ed)g(with)f(the)h(action)f (on)g(the)h(second)h(tensor)f(factor.)g(It)f(can)h(b)q(e)257 361 y(v)o(eri\014ed)f(directly)f(that)g(the)g(bijection)543 480 y Fo(f)j Fp(:)11 b Fo(Ae)e Fm(\012)h 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966 y Fp(Although,)f(for)h(stable)g(orbits)g Fo(O)q Fp(,)f(the)i (ideals)e Fo(AO)d Fm(\012)e Fo(H)16 b Fp(are)d(not)g(simple)f(themselv) o(es,)g(w)o(e)257 1016 y(can)i(precisely)h(determine)f(their)g(decomp)q (osition)f(in)o(to)g(simple)f(ideals:)257 1109 y Fq(Prop)q(osition)g(2) 21 b Fp(Supp)q(ose)c(that)f Fo(O)f Fp(=)g Fm(f)p Fo(e)p Fm(g)h Fp(is)f(a)h(stable)g(orbit)f(of)h Fo(C)s Fp(.)e(Denote)j(the)f (cor-)257 1159 y(resp)q(onding)i(t)o(w)o(o-sided)f(ideal)g(of)g Fo(A)g Fp(b)o(y)g Fo(I)k Fp(:=)c Fo(Ae)h Fp(and)f(supp)q(ose)i(that)e Fo(W)24 b Fp(is)17 b(a)g(simple)257 1209 y Fo(I)s Fp(-mo)q(dule)e(with) h(corresp)q(onding)h(represen)o(tation)h Fo(\032)1102 1215 y Fj(W)1155 1209 y Fp(:)d Fo(I)k Fm(!)14 b Fp(End)1350 1215 y Fj(K)1382 1209 y Fp(\()p Fo(W)6 b Fp(\).)16 b(Then)g(there)257 1258 y(exists)f(a)f(group)f(homomorphism)843 1338 y Fo(u)867 1344 y Fj(e)897 1338 y Fp(:)e Fo(C)j Fm(!)d Fo(U)5 b Fp(\()p Fo(I)s Fp(\))257 1418 y(from)12 b Fo(C)17 b Fp(to)d(the)g (group)g(of)f(units)h 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Fp(is)c(in)o(v)n(arian) o(t)f(with)h(resp)q(ect)257 1820 y(to)j(the)h(action)e(of)g Fo(C)k Fp(on)d Fo(A)p Fp(.)f(By)h(the)h(Sk)o(olem-No)q(ether)e (theorem,)g(the)h(sequence)768 1917 y Fo(K)806 1900 y Fl(\002)863 1887 y Fj(\023)846 1917 y Fe(\032)d Fo(U)5 b Fp(\()p Fo(I)s Fp(\))1005 1887 y Fn(ad)1002 1917 y Fe(\020)11 b Fp(Aut\()p Fo(I)s Fp(\))257 1997 y(is)16 b(exact,)h(where)g Fo(\023)p Fp(\()p Fo(\025)p Fp(\))f(=)g Fo(\025e)h Fp(and)f(ad)o(\()p Fo(a)p Fp(\)\()p Fo(a)960 1982 y Fl(0)972 1997 y Fp(\))g(=)g Fo(aa)1096 1982 y Fl(0)1107 1997 y Fo(a)1129 1982 y Fl(\000)p Fn(1)1174 1997 y Fp(.)g(W)m(e)f(regard)i(these)h(groups)e(as)257 2047 y(trivial)d(\(nonab)q(elian\))h Fo(C)s Fp(-mo)q(dules.)e(F)m(rom)g (Prop)q(osition)i(1.13.1,)d(w)o(e)k(then)f(get)h(an)f(exact)257 2097 y(sequence)i(of)e(p)q(oin)o(ted)g(sets)499 2194 y Fo(H)s(om)p Fp(\()p Fo(C)q(;)7 b(K)697 2177 y Fl(\002)725 2194 y Fp(\))761 2163 y Fj(\023)773 2167 y Fh(\003)753 2194 y Fe(\032)k Fo(H)s(om)p Fp(\()p Fo(C)q(;)c(U)e 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(can)f(b)q(e)g(v)o(eri\014ed)h(directly)f(that)g(the)g(map)604 2503 y Fo(f)i Fp(:)11 b Fo(I)i Fm(\012)c Fo(H)15 b Fm(!)c Fo(I)h Fm(\012)e Fo(H)q(;)d(a)h Fm(\012)i Fo(c)1055 2509 y Fj(i)1080 2503 y Fm(7!)h Fo(au)1179 2509 y Fj(e)1197 2503 y Fp(\()p Fo(c)1231 2509 y Fj(i)1245 2503 y Fp(\))e Fm(\012)g Fo(c)1329 2509 y Fj(i)953 2628 y Fp(31)p eop %%Page: 32 32 32 31 bop 257 262 a Fp(is)13 b(an)f(algebra)g(isomorphism)d(from)h(the) j(smash)f(pro)q(duct)h(to)f(the)h(ordinary)f(tensor)h(pro)q(d-)257 311 y(uct.)j(Since)g(all)f(irreducible)h(represen)o(tations)i(of)d(the) h(ordinary)g(tensor)g(pro)q(duct)h(are)f(of)257 361 y(the)e(form)d Fo(\032)446 367 y Fj(W)491 361 y Fm(\012)c Fo(\015)14 b Fp(:)d Fo(I)f Fm(\012)d Fo(H)15 b Fm(!)c Fp(End)832 367 y Fj(K)864 361 y Fp(\()p Fo(W)6 b Fp(\))13 b(for)g(some)e Fo(\015)k Fm(2)1203 351 y Fp(^)1194 361 y Fo(C)s Fp(,)d(the)h (irreducible)g(represen-)257 411 y(tations)h(of)f(the)i(smash)e(pro)q (duct)i(are)f(of)f(the)i(form)d(stated)i(ab)q(o)o(v)o(e.)g Fi(2)257 525 y Fp(The)h(follo)o(wing)c(statemen)o(t)j(is)g(an)f(ob)o (vious)g(consequence)k(of)c(the)h(ab)q(o)o(v)o(e)g(pro)q(ofs:)257 621 y Fq(Corollary)35 b Fo(\024)17 b Fp(is)g(surjectiv)o(e,)g(i.)f(e.,) g(for)g(all)g Fo(C)s Fp(-orbits)g Fo(O)i Fp(in)e Fo(E)j Fp(there)f(is)e(a)h(simple)e Fo(B)r Fp(-)257 671 y(mo)q(dule)e Fo(V)23 b Fp(suc)o(h)15 b(that)f Fo(\024)p Fp(\()p Fo(V)9 b Fp(\))j(=)g Fo(O)q Fp(.)257 785 y(The)21 b(assertions)g(in)f(this)g (paragraph)g(should)g(b)q(e)g(compared)g(with)g([64)o(],)f(Thm.)f (14.7,)257 835 y(p.)c(138.)e(F)m(urther)j(references)i(will)12 b(b)q(e)j(giv)o(en)e(b)q(elo)o(w.)257 967 y Fq(2.4)48 b Fp(Our)11 b(next)f(aim)e(is)i(to)g(describ)q(e)h(in)f(greater)h (detail)e(those)i(simple)d Fo(B)r Fp(-mo)q(dules)i(whic)o(h)257 1017 y(are)15 b(purely)f(unstable.)g(Recall)f(that)h(the)g(function)g Fo(\025)1106 1023 y Fj(H)1149 1017 y Fp(:)d Fo(H)j Fm(!)d Fo(K)18 b Fp(de\014ned)c(b)o(y:)779 1137 y Fo(\025)803 1143 y Fj(H)834 1137 y Fp(\()p Fo(c)868 1143 y 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Fp(.)308 2495 y(4.)20 b(dim)5 b Fo(V)21 b Fp(=)12 b Fo(p)7 b Fp(dim)e Fo(W)669 2501 y Fj(e)953 2628 y Fp(32)p eop %%Page: 33 33 33 32 bop 257 262 a Fq(Pro)q(of.)36 b Fp(Supp)q(ose)19 b(that)f Fo(e)h Fm(2)f Fo(O)q Fp(.)f(It)h(is)g(clear)g(from)e(the)i (description)h(of)e(the)i(induced)257 311 y(mo)q(dule)d Fo(B)e Fm(\012)484 317 y Fj(A)523 311 y Fo(W)562 317 y Fj(e)597 311 y Fp(ab)q(o)o(v)o(e)j(that)g(ev)o(ery)h(cen)o(trally)f (primitiv)o(e)e(idemp)q(oten)o(t)h Fo(e)1520 296 y Fl(0)1549 311 y Fm(2)h Fo(E)7 b Fm(n)t Fo(O)257 361 y Fp(v)n(anishes)k(on)e Fo(B)t Fm(\012)537 367 y Fj(A)566 361 y Fo(W)605 367 y Fj(e)623 361 y Fp(.)h(Therefore,)h Fo(B)t Fm(\012)908 367 y Fj(A)936 361 y Fo(W)975 367 y Fj(e)1003 361 y Fp(is)f(a)g(p)q(o)o (w)o(er)g(of)g(the)g(unique)g(simple)f(mo)q(dule)257 411 y(corresp)q(onding)17 b(to)f(the)g(simple)e(ideal)h Fo(AO)d Fm(\012)e Fo(H)s Fp(.)16 b(Since)g(dim)5 b Fo(AO)11 b Fm(\012)g Fo(H)18 b Fp(=)d Fo(p)1491 396 y Fn(2)1516 411 y Fp(dim)5 b Fo(Ae)15 b Fp(=)257 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Fm(6)p Fp(=)h(0)h Fi(2)257 1632 y Fp(Note)j(that,)e(since)i(the)g(isotrop)o(y) f(group)g Fo(\024)941 1617 y Fl(\003)960 1632 y Fp(\()p Fo(V)9 b Fp(\))15 b(is)g(trivial,)e(the)j(orbit)f(of)f Fo(\037)1469 1638 y Fj(V)1513 1632 y Fp(under)i(the)257 1682 y(righ)o(t)h(action)f(of)547 1671 y(^)538 1682 y Fo(C)j Fp(consists)f(of)e Fo(p)h Fp(c)o(haracters)h(of)e(nonisomorphic) f(simple)g Fo(B)r Fp(-mo)q(dules)257 1732 y(whose)g(restrictions)g(to)f Fo(A)g Fp(are)g(isomorphic.)257 1866 y Fq(2.6)48 b Fp(It)17 b(no)o(w)f(turns)h(out)f(that)h(there)g(is)g(a)f(connection)h(b)q(et)o (w)o(een)h(the)f(action)f(of)g Fo(C)j Fp(on)257 1916 y(the)f(set)f Fo(E)h Fp(of)e(cen)o(trally)h(primitiv)o(e)d(idemp)q (oten)o(ts)i(of)g Fo(A)h Fp(and)f(the)h(action)f(of)1532 1905 y(^)1522 1916 y Fo(C)k Fp(on)c(the)257 1966 y(irreducible)i(c)o (haracters)h(of)e Fo(B)r Fp(:)g(Tw)o(o)g(simple)f Fo(B)r Fp(-mo)q(dules)g(ha)o(v)o(e)h(the)h(same)f(restriction)257 2015 y(to)12 b Fo(A)h Fp(if)e(and)h(only)g(if)f(their)i(c)o(haracters)h (are)e(link)o(ed)g(via)f(the)i(canonical)f(righ)o(t)f(action)h(of)1666 2005 y(^)1657 2015 y Fo(C)257 2065 y Fp(on)i(Ch\()p Fo(B)r Fp(\):)257 2163 y Fq(Theorem)36 b Fp(Supp)q(ose)16 b(that)f Fo(V)25 b Fp(and)15 b Fo(V)900 2148 y Fl(0)927 2163 y Fp(are)h(simple)d Fo(B)r Fp(-mo)q(dules)i(with)g(c)o(haracters)i Fo(\037)1661 2169 y Fj(V)257 2213 y Fp(resp.)e Fo(\037)383 2219 y Fj(V)410 2211 y Fh(0)423 2213 y Fp(.)e(Then)i(the)f(follo)o (wing)d(assertions)k(are)g(equiv)n(alen)o(t:)308 2329 y(1.)20 b Fo(\024)p Fp(\()p Fo(V)10 b Fp(\))h(=)h Fo(\024)p Fp(\()p Fo(V)580 2314 y Fl(0)591 2329 y Fp(\))308 2411 y(2.)20 b(The)14 b(restrictions)i(of)d Fo(V)23 b Fp(and)14 b Fo(V)870 2396 y Fl(0)896 2411 y Fp(to)g Fo(A)g Fp(are)g(isomorphic.) 308 2493 y(3.)20 b(There)15 b(exists)g Fo(\015)f Fm(2)680 2482 y Fp(^)670 2493 y Fo(C)j Fp(suc)o(h)d(that)g Fo(\037)926 2499 y Fj(V)953 2491 y Fh(0)978 2493 y Fp(=)e Fo(\037)1048 2499 y Fj(V)1077 2493 y Fp(\()p Fo(\017)1110 2499 y Fj(A)1146 2493 y Fm(\012)d Fo(\015)r Fp(\).)953 2628 y(34)p eop %%Page: 35 35 35 34 bop 257 262 a Fq(Pro)q(of.)36 b Fp(W)m(e)10 b(distinguish)g(t)o (w)o(o)g(cases:)i(Supp)q(ose)f(\014rst)g(that)g Fo(V)20 b Fp(is)10 b(purely)h(unstable.)f(Then)257 311 y(the)15 b(restriction)f(of)f Fo(V)23 b Fp(to)13 b Fo(A)h Fp(is)g(a)f(direct)h (sum)f(of)g Fo(p)g Fp(nonisomorphic)f(simple)g Fo(A)p Fp(-mo)q(dules,)257 361 y(eac)o(h)k(o)q(ccurring)f(with)f(m)o (ultiplicit)o(y)e(one,)i(that)h(corresp)q(ond)h(to)e(the)i(elemen)o(ts) e(of)g Fo(\024)p Fp(\()p Fo(V)c Fp(\).)257 411 y(Therefore,)k(the)e (\014rst)i(t)o(w)o(o)d(assertions)j(are)f(equiv)n(alen)o(t.)e(But)i (since)g(according)f(to)g(Prop)q(o-)257 461 y(sition)17 b(2.4,)e(the)j Fo(B)r Fp(-mo)q(dule)e Fo(V)26 b Fp(can)18 b(b)q(e)f(reco)o(v)o(ered)i(from)c(ev)o(ery)j(simple)d Fo(A)p Fp(-submo)q(dule)257 511 y(via)i(induction,)f Fo(V)27 b Fp(and)17 b Fo(V)696 496 y Fl(0)725 511 y Fp(are)g (isomorphic)f(as)h Fo(A)p 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Fo(A)p Fp(.)h(Therefore,)g(the)h(\014rst)f(t)o(w)o (o)f(assertions)i(are)f(equiv)n(alen)o(t.)f(According)h(to)f(Prop)q(o-) 257 892 y(sition)f(2.3.2,)e(there)j(are,)f(up)g(to)g(isomorphism,)d Fo(p)j Fp(simple)e Fo(B)r Fp(-mo)q(dules)i(that)g(ha)o(v)o(e)g(the)257 942 y(same)e(restriction)h(to)f Fo(A)h Fp(as)f Fo(V)10 b Fp(.)15 b(F)m(or)g Fo(\015)i Fm(2)940 931 y Fp(^)930 942 y Fo(C)s Fp(,)e Fo(\037)1016 948 y Fj(V)1045 942 y Fp(\()p Fo(\017)1078 948 y Fj(A)1115 942 y Fm(\012)10 b Fo(\015)r Fp(\))17 b(is)e(an)g(irreducible)h(c)o(haracter)257 991 y(of)g Fo(B)j Fp(that)e(has)f(the)h(same)f(restriction)h(to)g Fo(A)f Fp(as)h Fo(\037)1089 997 y Fj(V)1118 991 y Fp(.)f(According)h (to)f(Prop)q(osition)g(2.5,)257 1041 y(these)e(c)o(haracters)g(are)f (all)e(distinct,)h(and)g(therefore)i(exhaust)f(all)e(irreducible)i(c)o (haracters)257 1091 y(of)i Fo(B)i Fp(that)e(ha)o(v)o(e)g(the)h(same)e (restriction)i(to)f Fo(A)g Fp(as)g Fo(\037)1096 1097 y Fj(V)1125 1091 y 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Fo(x)1195 2432 y Fj(st)1225 2426 y Fp(\))606 2488 y(=)j Fo(c)668 2494 y Fj(b)696 2488 y Fm(!)f Fp(\(\()p Fo(e)800 2494 y Fj(u)831 2488 y Fm(\012)f Fo(x)897 2494 y Fj(s)914 2488 y Fp(\)\()p Fo(e)965 2494 y Fj(v)995 2488 y Fm(\012)f Fo(x)1060 2494 y Fj(t)1075 2488 y Fp(\)\))953 2628 y(39)p eop %%Page: 40 40 40 39 bop 257 262 a Fp(Similarly)m(,)11 b(since)k Fo(z)h Fp(is)f(a)f(1-co)q(cycle,)h(w)o(e)f(ha)o(v)o(e)h Fo(z)1025 268 y Fj(st)1055 262 y Fp(\()p Fo(u)p Fp(\))e(=)g Fo(z)1188 268 y Fj(s)1206 262 y Fp(\()p Fo(u)p Fp(\))p Fo(z)1281 268 y Fj(t)1296 262 y Fp(\()p Fo(s)1331 246 y Fl(\000)p Fn(1)1376 262 y Fo(:u)p Fp(\).)g(Therefore)j(w)o(e)257 311 y(ha)o(v)o(e:)454 403 y Fo(\016)472 409 y Fj(A)499 403 y Fp(\()p Fo(e)534 409 y Fj(u)565 403 y Fm(\012)10 b Fo(x)631 409 y Fj(s)648 403 y Fp(\))p Fo(\016)682 409 y Fj(A)710 403 y Fp(\()p Fo(e)745 409 y Fj(v)775 403 y Fm(\012)f Fo(x)840 409 y Fj(t)854 403 y Fp(\))j(=)g Fo(c)944 410 y Fj(z)960 414 y Ff(s)976 410 y Fn(\()p Fj(u)p Fn(\))1023 403 y Fo(c)1041 410 y Fj(z)1057 414 y Ff(t)1071 410 y Fn(\()p Fj(v)q Fn(\))1126 403 y Fm(\012)d Fp(\()p Fo(e)1202 409 y Fj(u)1234 403 y Fm(\012)g Fo(x)1299 409 y Fj(s)1317 403 y Fp(\)\()p Fo(e)1368 409 y Fj(v)1397 403 y Fm(\012)h Fo(x)1463 409 y Fj(t)1477 403 y Fp(\))660 465 y(=)i Fo(c)722 472 y Fj(z)738 476 y Ff(s)754 472 y Fn(\()p Fj(u)p Fn(\))p Fj(z)816 476 y Ff(t)829 472 y Fn(\()p Fj(v)q Fn(\))884 465 y Fm(\012)d Fo(\016)943 471 y Fj(u;s:v)1018 465 y Fo(\033)1042 471 y Fj(u)1064 465 y Fp(\()p Fo(s;)e(t)p Fp(\)\()p Fo(e)1184 471 y Fj(u)1215 465 y Fm(\012)j Fo(x)1281 471 y Fj(st)1311 465 y Fp(\))660 527 y(=)i Fo(c)722 535 y Fj(z)738 539 y Ff(s)754 535 y Fn(\()p Fj(u)p Fn(\))p Fj(z)816 539 y Ff(t)829 535 y Fn(\()p Fj(s)858 527 y Fh(\000)p Fd(1)896 535 y Fj(:u)p Fn(\))950 527 y Fm(\012)d Fo(\016)1009 533 y Fj(u;s:v)1085 527 y Fo(\033)1109 533 y Fj(u)1130 527 y Fp(\()p Fo(s;)e(t)p Fp(\)\()p Fo(e)1250 533 y Fj(u)1282 527 y Fm(\012)i Fo(x)1347 533 y Fj(st)1377 527 y Fp(\))660 594 y(=)j Fo(c)722 601 y Fj(z)738 605 y Ff(st)765 601 y Fn(\()p Fj(u)p Fn(\))822 594 y Fm(\012)e Fo(\016)882 600 y Fj(u;s:v)957 594 y Fo(\033)981 600 y Fj(u)1002 594 y Fp(\()p Fo(s;)d(t)p Fp(\)\()p Fo(e)1122 600 y Fj(u)1154 594 y Fm(\012)i Fo(x)1219 600 y Fj(st)1249 594 y Fp(\))660 656 y(=)j Fo(\016)722 662 y Fj(A)749 656 y Fp(\(\()p Fo(e)800 662 y Fj(u)832 656 y Fm(\012)d Fo(x)897 662 y Fj(s)915 656 y Fp(\)\()p Fo(e)966 662 y Fj(v)995 656 y Fm(\012)h Fo(x)1061 662 y Fj(t)1075 656 y Fp(\)\))257 764 y(\(4\))21 b(W)m(e)g(no)o(w)f(v)o (erify)g(the)i(Y)m(etter-Drinfel'd)e(bialgebra)g(axiom,)e(i.)i(e.,)g (the)i(fact)e(that)257 814 y(\001)292 820 y Fj(A)331 814 y Fp(:)11 b Fo(A)g Fm(!)g Fo(A)486 808 y Fp(^)480 814 y Fm(\012)q Fo(A)i Fp(is)g(an)g(algebra)g(homom)o(orphism)o(.)d (This)j(is)g(the)h(step)g(that)f(dep)q(ends)i(on)257 864 y(the)g(compatibilit)o(y)c(condition:)286 955 y(\001)321 961 y Fj(A)348 955 y Fp(\()p Fo(e)383 961 y Fj(u)414 955 y Fm(\012)f Fo(x)480 961 y Fj(s)497 955 y Fp(\)\001)548 961 y Fj(A)575 955 y Fp(\()p Fo(e)610 961 y Fj(v)639 955 y Fm(\012)g Fo(x)705 961 y Fj(t)719 955 y Fp(\))298 1030 y(=)370 991 y Fg(X)342 1080 y Fj(u)362 1071 y Fh(0)372 1080 y Fj(;v)400 1071 y Fh(0)411 1080 y Fl(2)p Fj(P)459 1030 y Fp([\()p Fo(e)506 1036 y Fj(u)526 1028 y Fh(0)548 1030 y Fm(\012)g Fo(x)614 1036 y Fj(s)631 1030 y Fp(\))g Fm(\012)f Fp(\()p Fo(e)733 1038 y Fj(u)753 1030 y Fh(0)r(\000)p Fd(1)803 1038 y Fj(u)834 1030 y Fm(\012)h Fo(x)900 1036 y Fj(s)917 1030 y Fp(\)][\()p Fo(e)992 1036 y Fj(v)1010 1028 y Fh(0)1032 1030 y Fm(\012)f Fo(x)1097 1036 y Fj(t)1112 1030 y Fp(\))g Fm(\012)h Fp(\()p Fo(e)1214 1038 y Fj(v)1232 1030 y Fh(0)r(\000)p Fd(1)1282 1038 y Fj(v)1311 1030 y Fm(\012)f Fo(x)1376 1036 y Fj(t)1391 1030 y Fp(\)])298 1150 y(=)370 1110 y Fg(X)342 1199 y Fj(u)362 1191 y Fh(0)372 1199 y Fj(;v)400 1191 y Fh(0)411 1199 y Fl(2)p Fj(P)459 1150 y Fp(\()p Fo(\015)496 1156 y Fj(t)511 1150 y Fp(\()p Fo(v)548 1133 y Fl(0)561 1150 y Fp(\)\)\()p Fo(z)628 1156 y Fj(s)646 1150 y Fp(\()p Fo(u)686 1133 y Fl(0)r(\000)p Fn(1)742 1150 y Fo(u)p Fp(\)\)\()p Fo(e)833 1156 y Fj(u)853 1148 y Fh(0)876 1150 y Fm(\012)h Fo(x)942 1156 y Fj(s)959 1150 y Fp(\)\()p Fo(e)1010 1156 y Fj(v)1028 1148 y Fh(0)1051 1150 y Fm(\012)f Fo(x)1116 1156 y Fj(t)1131 1150 y Fp(\))g Fm(\012)h Fp(\()p Fo(e)1233 1157 y Fj(u)1253 1149 y Fh(0)r(\000)p Fd(1)1303 1157 y Fj(u)1334 1150 y Fm(\012)f Fo(x)1399 1156 y Fj(s)1416 1150 y Fp(\)\()p Fo(e)1467 1157 y Fj(v)1485 1149 y Fh(0)s(\000)p Fd(1)1536 1157 y Fj(v)1565 1150 y Fm(\012)g Fo(x)1630 1156 y Fj(t)1644 1150 y Fp(\))298 1269 y(=)370 1230 y Fg(X)342 1319 y Fj(u)362 1311 y Fh(0)372 1319 y Fj(;v)400 1311 y Fh(0)411 1319 y Fl(2)p Fj(P)459 1269 y Fp(\()p Fo(\015)496 1275 y Fj(t)511 1269 y Fp(\()p Fo(v)548 1252 y Fl(0)561 1269 y Fp(\)\)\()p Fo(z)628 1275 y Fj(s)646 1269 y Fp(\()p Fo(u)686 1252 y Fl(0)r(\000)p Fn(1)742 1269 y Fo(u)p Fp(\)\))p Fo(\016)816 1275 y Fj(u)836 1267 y Fh(0)848 1275 y Fj(;s:v)902 1267 y Fh(0)914 1269 y Fo(\016)932 1277 y Fj(u)952 1269 y Fh(0)r(\000)p Fd(1)1002 1277 y Fj(u;s:)p Fn(\()p Fj(v)1089 1269 y Fh(0)q(\000)p Fd(1)1138 1277 y Fj(v)q Fn(\))1170 1269 y Fo(\033)1194 1275 y Fj(u)1214 1267 y Fh(0)1227 1269 y Fp(\()p Fo(s;)e(t)p Fp(\))p Fo(\033)1336 1277 y Fj(u)1356 1269 y Fh(0)r(\000)p Fd(1)1405 1277 y Fj(u)1427 1269 y Fp(\()p Fo(s;)g(t)p Fp(\))1093 1380 y(\()p Fo(e)1128 1386 y Fj(u)1148 1378 y Fh(0)1171 1380 y Fm(\012)i Fo(x)1236 1386 y Fj(st)1266 1380 y Fp(\))g Fm(\012)h Fp(\()p Fo(e)1368 1388 y Fj(u)1388 1380 y Fh(0)r(\000)p Fd(1)1438 1388 y Fj(u)1469 1380 y Fm(\012)f Fo(x)1534 1386 y Fj(st)1564 1380 y Fp(\))298 1455 y(=)j Fo(\016)360 1461 y Fj(u;s:v)451 1416 y Fg(X)442 1505 y Fj(u)462 1497 y Fh(0)473 1505 y Fl(2)p Fj(P)521 1455 y Fp(\()p Fo(\015)558 1461 y Fj(t)573 1455 y Fp(\()p Fo(s)608 1438 y Fl(\000)p Fn(1)653 1455 y Fo(:u)689 1438 y Fl(0)700 1455 y Fp(\)\)\()p Fo(z)767 1461 y Fj(s)785 1455 y Fp(\()p Fo(u)825 1438 y Fl(0)r(\000)p Fn(1)881 1455 y Fo(u)p Fp(\)\))p Fo(\033)961 1461 y Fj(u)981 1453 y Fh(0)994 1455 y Fp(\()p Fo(s;)7 b(t)p Fp(\))p Fo(\033)1103 1463 y Fj(u)1123 1455 y Fh(0)r(\000)p Fd(1)1173 1463 y Fj(u)1194 1455 y Fp(\()p Fo(s;)g(t)p Fp(\))1093 1563 y(\()p Fo(e)1128 1569 y Fj(u)1148 1561 y Fh(0)1171 1563 y Fm(\012)i Fo(x)1236 1569 y Fj(st)1266 1563 y Fp(\))g Fm(\012)h Fp(\()p Fo(e)1368 1571 y Fj(u)1388 1562 y Fh(0)r(\000)p Fd(1)1438 1571 y Fj(u)1469 1563 y Fm(\012)f Fo(x)1534 1569 y Fj(st)1564 1563 y Fp(\))298 1638 y(=)j Fo(\016)360 1644 y Fj(u;s:v)451 1598 y Fg(X)442 1687 y Fj(u)462 1679 y Fh(0)473 1687 y Fl(2)p Fj(P)527 1638 y Fo(\033)551 1644 y Fj(u)573 1638 y Fp(\()p Fo(s;)7 b(t)p Fp(\)\()p Fo(e)693 1644 y Fj(u)713 1636 y Fh(0)736 1638 y Fm(\012)i Fo(x)801 1644 y Fj(st)831 1638 y Fp(\))h Fm(\012)f Fp(\()p Fo(e)933 1646 y Fj(u)953 1637 y Fh(0)r(\000)p Fd(1)1003 1646 y Fj(u)1034 1638 y Fm(\012)h Fo(x)1100 1644 y Fj(st)1130 1638 y Fp(\))298 1745 y(=)i(\001)377 1751 y Fj(A)403 1745 y Fp(\(\()p Fo(e)454 1751 y Fj(u)486 1745 y Fm(\012)d Fo(x)551 1751 y Fj(s)569 1745 y Fp(\)\()p Fo(e)620 1751 y Fj(v)649 1745 y Fm(\012)h Fo(x)715 1751 y Fj(t)729 1745 y Fp(\)\))257 1853 y(\001)292 1859 y Fj(A)333 1853 y Fp(also)j(preserv)o(es)j(the)f(unit.)257 1936 y(\(5\))21 b(W)m(e)16 b(no)o(w)g(pro)o(v)o(e)g(that)h Fo(\017)718 1942 y Fj(A)761 1936 y Fp(is)f(an)g(algebra)g(homom)o(orphism)o(.)d (Inserting)k Fo(u)e Fp(=)h Fo(v)h Fp(=)f(1)257 1986 y(in)o(to)e(the)g (compatibilit)o(y)d(condition,)h(w)o(e)j(get:)446 2077 y Fo(\033)470 2083 y Fn(1)488 2077 y Fp(\()p Fo(s;)7 b(t)p Fp(\))12 b(=)g(\(\()p Fo(s:\015)713 2083 y Fj(t)728 2077 y Fp(\)\(1\)\)\()p Fo(z)848 2083 y Fj(s)867 2077 y Fp(\(1\)\))p Fo(\033)960 2083 y Fn(1)978 2077 y Fp(\()p Fo(s;)7 b(t)p Fp(\))p Fo(\033)1087 2083 y Fn(1)1106 2077 y Fp(\()p Fo(s;)g(t)p Fp(\))k(=)h Fo(\033)1270 2083 y Fn(1)1288 2077 y Fp(\()p Fo(s;)7 b(t)p Fp(\))p Fo(\033)1397 2083 y Fn(1)1416 2077 y Fp(\()p Fo(s;)g(t)p Fp(\))257 2169 y(and)14 b(therefore)h Fo(\033)536 2175 y Fn(1)555 2169 y Fp(\()p Fo(s;)7 b(t)p Fp(\))k(=)h(1.)h(This)h(implies:)354 2260 y Fo(\017)371 2266 y Fj(A)398 2260 y Fp(\(\()p Fo(e)449 2266 y Fj(u)480 2260 y Fm(\012)c Fo(x)546 2266 y Fj(s)563 2260 y Fp(\)\()p Fo(e)614 2266 y Fj(v)644 2260 y Fm(\012)f Fo(x)709 2266 y Fj(t)724 2260 y Fp(\)\))j(=)f Fo(\017)828 2266 y Fj(A)855 2260 y Fp(\()p Fo(\016)889 2266 y Fj(u;s:v)965 2260 y Fo(\033)989 2266 y Fj(u)1010 2260 y Fp(\()p Fo(s;)c(t)p Fp(\))p Fo(e)1114 2266 y Fj(u)1145 2260 y Fm(\012)j Fo(x)1211 2266 y Fj(st)1241 2260 y Fp(\))i(=)f Fo(\016)1330 2266 y Fj(u;s:v)1406 2260 y Fo(\033)1430 2266 y Fj(u)1451 2260 y Fp(\()p Fo(s;)c(t)p Fp(\))p Fo(\016)1554 2266 y Fj(u)p Fn(1)768 2322 y Fp(=)k Fo(\016)829 2328 y Fj(u)p Fn(1)868 2322 y Fo(\016)886 2328 y Fj(v)q Fn(1)934 2322 y Fp(=)h Fo(\017)995 2328 y Fj(A)1022 2322 y Fp(\()p Fo(e)1057 2328 y Fj(u)1088 2322 y Fm(\012)e Fo(x)1154 2328 y Fj(s)1171 2322 y Fp(\))p Fo(\017)1204 2328 y Fj(A)1231 2322 y Fp(\()p Fo(e)1266 2328 y Fj(v)1296 2322 y Fm(\012)f Fo(x)1361 2328 y Fj(t)1376 2322 y Fp(\))257 2414 y(It)14 b(is)g(easy)g(to)g(see)h(that)f Fo(\017)660 2420 y Fj(A)701 2414 y Fp(preserv)o(es)i(the)f(unit.)953 2628 y(40)p eop %%Page: 41 41 41 40 bop 257 262 a Fp(\(6\))21 b(W)m(e)11 b(no)o(w)f(pro)q(ceed)j(to)e (pro)o(v)o(e)g(that)g(the)h(an)o(tip)q(o)q(de)f(is)g Fo(H)s Fp(-linear)f(and)h(colinear.)f(Since)i Fo(\015)257 311 y Fp(is)i(a)g(1-co)q(cycle,)g(w)o(e)g(kno)o(w)f(from)f(Lemma)f (1.13)i(that)g Fo(\015)1136 319 y Fj(s)1152 311 y Fh(\000)p Fd(1)k Fp(=)12 b(\()p Fo(s)1284 296 y Fl(\000)p Fn(1)1329 311 y Fo(:\015)1362 317 y Fj(s)1380 311 y Fp(\))1396 296 y Fl(\000)p Fn(1)1454 311 y Fp(for)i(all)e Fo(s)g Fm(2)f Fo(G)p Fp(,)257 361 y(i.)i(e.,)h(w)o(e)g(ha)o(v)o(e)f Fo(\015)527 369 y Fj(s)543 361 y Fh(\000)p Fd(1)5 b Fp(\()p Fo(u)p Fp(\))12 b(=)g Fo(\015)717 367 y Fj(s)735 361 y Fp(\()p Fo(s:u)806 346 y Fl(\000)p Fn(1)850 361 y Fp(\).)i(This)g (implies:)307 440 y Fo(c)325 446 y Fj(b)354 440 y Fm(!)d Fo(S)432 446 y Fj(A)459 440 y Fp(\()p Fo(e)494 446 y Fj(u)526 440 y Fm(\012)e Fo(x)591 446 y Fj(s)609 440 y Fp(\))i(=)h Fo(\015)701 448 y Fj(s)717 440 y Fh(\000)p Fd(1)5 b Fp(\()p Fo(s)793 423 y Fl(\000)p Fn(1)838 440 y Fo(:u)874 423 y Fl(\000)p Fn(1)918 440 y Fp(\)\()p Fo(b)p Fp(\))k Fo(S)1018 446 y Fj(A)1046 440 y Fp(\()p Fo(e)1081 446 y Fj(u)1112 440 y Fm(\012)h Fo(x)1178 446 y Fj(s)1195 440 y Fp(\))i(=)g Fo(\015)1288 446 y Fj(s)1306 440 y Fp(\()p Fo(u)p Fp(\)\()p Fo(b)p Fp(\))d Fo(S)1446 446 y Fj(A)1474 440 y Fp(\()p Fo(e)1509 446 y Fj(u)1540 440 y Fm(\012)h Fo(x)1606 446 y Fj(s)1623 440 y Fp(\))636 503 y(=)i Fo(S)705 509 y Fj(A)733 503 y Fp(\()p Fo(c)767 509 y Fj(b)795 503 y Fm(!)f Fo(e)867 509 y Fj(u)898 503 y Fm(\012)f Fo(x)964 509 y Fj(s)981 503 y Fp(\))257 596 y(Since)16 b Fo(z)h Fp(is)d(also)h(a)f(1-co)q(cycle,)h(w)o(e)g(kno)o(w) g(from)e(Lemma)f(1.13)h(that)i Fo(z)1373 604 y Fj(s)1389 596 y Fh(\000)p Fd(1)j Fp(=)c(\()p Fo(s)1524 581 y Fl(\000)p Fn(1)1569 596 y Fo(:z)1600 602 y Fj(s)1617 596 y Fp(\))1633 581 y Fl(\000)p Fn(1)1678 596 y Fp(,)257 646 y(i.)f(e.,)h(that)g Fo(z)459 654 y Fj(s)475 645 y Fh(\000)p Fd(1)515 646 y Fp(\()p Fo(u)p Fp(\))e(=)g Fo(z)646 652 y Fj(s)664 646 y Fp(\()p Fo(s:u)735 631 y Fl(\000)p Fn(1)779 646 y Fp(\).)i(This)f(implies:)333 725 y Fo(\016)351 731 y Fj(A)379 725 y Fp(\()p Fo(S)420 731 y Fj(A)447 725 y Fp(\()p Fo(e)482 731 y Fj(u)514 725 y Fm(\012)c Fo(x)579 731 y Fj(s)597 725 y Fp(\)\))j(=)f Fo(c)702 733 y Fj(z)718 743 y Ff(s)732 737 y Fh(\000)p Fd(1)773 733 y Fn(\()p Fj(s)802 725 y Fh(\000)p Fd(1)840 733 y Fj(:u)870 725 y Fh(\000)p Fd(1)908 733 y Fn(\))933 725 y Fm(\012)e Fo(S)999 731 y Fj(A)1027 725 y Fp(\()p Fo(e)1062 731 y Fj(u)1093 725 y Fm(\012)h Fo(x)1159 731 y Fj(s)1176 725 y Fp(\))i(=)f Fo(c)1265 732 y Fj(z)1281 736 y Ff(s)1297 732 y Fn(\()p Fj(u)p Fn(\))1354 725 y Fm(\012)f Fo(S)1421 731 y Fj(A)1448 725 y Fp(\()p Fo(e)1483 731 y Fj(u)1515 725 y Fm(\012)f Fo(x)1580 731 y Fj(s)1598 725 y Fp(\))641 794 y(=)i(\(id)735 800 y Fj(H)773 794 y Fm(\012)p Fo(S)830 800 y Fj(A)858 794 y Fp(\))p Fo(\016)892 800 y Fj(A)920 794 y Fp(\()p Fo(e)955 800 y Fj(u)986 794 y Fm(\012)f Fo(x)1052 800 y Fj(s)1069 794 y Fp(\))257 888 y(\(7\))21 b(A)o(t)14 b(last,)g(w)o(e)g(sho)o(w)h(that)f Fo(S)762 894 y Fj(A)804 888 y Fp(really)f(is)h(an)g(an)o(tip)q(o)q(de)h(for)e Fo(A)p Fp(.)h(W)m(e)g(b)q(egin)g(b)o(y)g(pro)o(ving)257 938 y(that)g(it)g(is)g(a)f(righ)o(t)h(an)o(tip)q(o)q(de)f(for)h Fo(A)p Fp(:)404 1023 y Fo(\026)429 1029 y Fj(A)465 1023 y Fm(\016)9 b Fp(\(id)546 1029 y Fj(A)580 1023 y Fm(\012)p Fo(S)637 1029 y Fj(A)665 1023 y Fp(\))g Fm(\016)g Fp(\001)755 1029 y Fj(A)782 1023 y Fp(\()p Fo(e)817 1029 y Fj(u)848 1023 y Fm(\012)h Fo(x)914 1029 y Fj(s)931 1023 y Fp(\))i(=)1005 983 y Fg(X)1003 1073 y Fj(v)q Fl(2)p Fj(P)1068 1023 y Fp(\()p Fo(e)1103 1029 y Fj(v)1133 1023 y Fm(\012)d Fo(x)1198 1029 y Fj(s)1216 1023 y Fp(\))p Fo(S)1257 1029 y Fj(A)1284 1023 y Fp(\()p Fo(e)1319 1031 y Fj(v)1337 1022 y Fh(\000)p Fd(1)1376 1031 y Fj(u)1407 1023 y Fm(\012)g Fo(x)1472 1029 y Fj(s)1490 1023 y Fp(\))676 1139 y(=)723 1100 y Fg(X)720 1189 y Fj(v)q Fl(2)p Fj(P)793 1139 y Fo(\033)818 1121 y Fl(\000)p Fn(1)817 1154 y Fj(u)837 1145 y Fh(\000)p Fd(1)875 1154 y Fj(v)894 1139 y Fp(\()p Fo(s;)e(s)967 1122 y Fl(\000)p Fn(1)1012 1139 y Fp(\)\()p Fo(e)1063 1145 y Fj(v)1093 1139 y Fm(\012)j Fo(x)1159 1145 y Fj(s)1176 1139 y Fp(\)\()p Fo(e)1227 1147 y Fj(s)1243 1139 y Fh(\000)p Fd(1)1282 1147 y Fj(:)p Fn(\()p Fj(u)1325 1139 y Fh(\000)p Fd(1)1363 1147 y Fj(v)q Fn(\))1405 1139 y Fm(\012)g Fo(x)1471 1147 y Fj(s)1487 1139 y Fh(\000)p Fd(1)t Fp(\))676 1255 y(=)723 1216 y Fg(X)720 1305 y Fj(v)q Fl(2)p Fj(P)793 1255 y Fo(\033)818 1237 y Fl(\000)p Fn(1)817 1270 y Fj(u)837 1261 y Fh(\000)p Fd(1)875 1270 y Fj(v)894 1255 y Fp(\()p Fo(s;)d(s)967 1238 y Fl(\000)p Fn(1)1012 1255 y Fp(\))p Fo(\033)1052 1261 y Fj(v)1072 1255 y Fp(\()p Fo(s;)g(s)1145 1238 y Fl(\000)p Fn(1)1190 1255 y Fp(\))p Fo(\016)1224 1263 y Fj(v)q(;u)1272 1255 y Fh(\000)p Fd(1)1311 1263 y Fj(v)1330 1255 y Fp(\()p Fo(e)1365 1261 y Fj(v)1395 1255 y Fm(\012)i Fo(x)1460 1261 y Fn(1)1479 1255 y Fp(\))676 1371 y(=)j Fo(\016)738 1377 y Fj(u)p Fn(1)787 1332 y Fg(X)784 1421 y Fj(v)q Fl(2)p Fj(P)856 1371 y Fo(e)875 1377 y Fj(v)905 1371 y Fm(\012)d Fo(x)970 1377 y Fn(1)1000 1371 y Fp(=)j Fo(\017)1061 1377 y Fj(A)1088 1371 y Fp(\()p Fo(e)1123 1377 y Fj(u)1154 1371 y Fm(\012)e Fo(x)1220 1377 y Fj(s)1237 1371 y Fp(\)1)1274 1377 y Fj(A)257 1504 y Fp(Next,)k(w)o(e)g(pro)o(v)o(e)g(that)g Fo(S)658 1510 y Fj(A)700 1504 y Fp(is)f(a)h(left)g(an)o(tip)q(o)q(de)f(for)h Fo(A)p Fp(.)f(F)m(or)h(this,)f(w)o(e)h(observ)o(e)h(\014rst)g(that,)257 1554 y(b)o(y)f(Lemma)d(1.13,)h(w)o(e)i(ha)o(v)o(e:)749 1633 y Fo(\033)q Fp(\()p Fo(s)809 1616 y Fl(\000)p Fn(1)854 1633 y Fo(;)7 b(s)p Fp(\))12 b(=)f Fo(s)982 1616 y Fl(\000)p Fn(1)1027 1633 y Fo(:\033)q Fp(\()p Fo(s;)c(s)1137 1616 y Fl(\000)p Fn(1)1182 1633 y Fp(\))257 1712 y(This)18 b(implies)f(that)h Fo(\033)620 1718 y Fj(u)641 1712 y Fp(\()p Fo(s;)7 b(s)714 1697 y Fl(\000)p Fn(1)759 1712 y Fp(\))19 b(=)g Fo(\033)869 1720 y Fj(s)885 1712 y Fh(\000)p Fd(1)923 1720 y Fj(:u)954 1712 y Fp(\()p Fo(s)989 1697 y Fl(\000)p Fn(1)1035 1712 y Fo(;)7 b(s)p Fp(\).)17 b(Therefore,)i(the) g(an)o(tip)q(o)q(de)f(is)g(also)257 1762 y(giv)o(en)c(b)o(y)f(the)i (form)o(ula:)558 1841 y Fo(S)583 1847 y Fj(A)611 1841 y Fp(\()p Fo(e)646 1847 y Fj(u)677 1841 y Fm(\012)9 b Fo(x)742 1847 y Fj(s)760 1841 y Fp(\))j(=)f Fo(\033)856 1824 y Fl(\000)p Fn(1)855 1856 y Fj(s)871 1847 y Fh(\000)p Fd(1)910 1856 y Fj(:u)940 1847 y Fh(\000)p Fd(1)980 1841 y Fp(\()p Fo(s)1015 1824 y Fl(\000)p Fn(1)1060 1841 y Fo(;)c(s)p Fp(\))p Fo(e)1133 1849 y Fj(s)1149 1841 y Fh(\000)p Fd(1)1188 1849 y Fj(:u)1218 1841 y Fh(\000)p Fd(1)13 b Fm(\012)d Fo(x)1333 1849 y Fj(s)1349 1841 y Fh(\000)p Fd(1)257 1921 y Fp(This)k(implies:)283 2006 y Fo(\026)308 2012 y Fj(A)344 2006 y Fm(\016)9 b Fp(\()p Fo(S)415 2012 y Fj(A)452 2006 y Fm(\012)h Fp(id)528 2012 y Fj(A)555 2006 y Fp(\))g Fm(\016)f Fp(\001)646 2012 y Fj(A)672 2006 y Fp(\()p Fo(e)707 2012 y Fj(u)739 2006 y Fm(\012)g Fo(x)804 2012 y Fj(s)822 2006 y Fp(\))i(=)896 1966 y Fg(X)893 2055 y Fj(v)q Fl(2)p Fj(P)966 2006 y Fo(S)991 2012 y Fj(A)1018 2006 y Fp(\()p Fo(e)1053 2012 y Fj(v)1083 2006 y Fm(\012)e Fo(x)1148 2012 y Fj(s)1166 2006 y Fp(\)\()p Fo(e)1217 2014 y Fj(v)1235 2005 y Fh(\000)p Fd(1)1274 2014 y Fj(u)1304 2006 y Fm(\012)h Fo(x)1370 2012 y Fj(s)1387 2006 y Fp(\))402 2122 y(=)449 2082 y Fg(X)446 2172 y Fj(v)q Fl(2)p Fj(P)518 2122 y Fo(\033)543 2104 y Fl(\000)p Fn(1)542 2136 y Fj(s)558 2128 y Fh(\000)p Fd(1)596 2136 y Fj(:v)624 2128 y Fh(\000)p Fd(1)665 2122 y Fp(\()p Fo(s)700 2105 y Fl(\000)p Fn(1)745 2122 y Fo(;)d(s)p Fp(\)\()p Fo(e)834 2130 y Fj(s)850 2121 y Fh(\000)p Fd(1)889 2130 y Fj(:v)917 2121 y Fh(\000)p Fd(1)966 2122 y Fm(\012)j Fo(x)1032 2130 y Fj(s)1048 2121 y Fh(\000)p Fd(1)1088 2122 y Fp(\)\()p Fo(e)1139 2130 y Fj(v)1157 2121 y Fh(\000)p Fd(1)1196 2130 y Fj(u)1227 2122 y Fm(\012)f Fo(x)1292 2128 y Fj(s)1310 2122 y Fp(\))402 2238 y(=)449 2199 y Fg(X)446 2288 y Fj(v)q Fl(2)p Fj(P)518 2238 y Fo(\033)543 2220 y Fl(\000)p Fn(1)542 2253 y Fj(s)558 2244 y Fh(\000)p Fd(1)596 2253 y Fj(:v)624 2244 y Fh(\000)p Fd(1)665 2238 y Fp(\()p Fo(s)700 2221 y Fl(\000)p Fn(1)745 2238 y Fo(;)e(s)p Fp(\))p Fo(\033)823 2246 y Fj(s)839 2238 y Fh(\000)p Fd(1)877 2246 y Fj(:v)905 2238 y Fh(\000)p Fd(1)t Fp(\()p Fo(s)980 2221 y Fl(\000)p Fn(1)1025 2238 y Fo(;)g(s)p Fp(\))p Fo(\016)1097 2246 y Fj(s)1113 2238 y Fh(\000)p Fd(1)1152 2246 y Fj(:v)1180 2238 y Fh(\000)p Fd(1)1218 2246 y Fj(;s)1244 2238 y Fh(\000)p Fd(1)1283 2246 y Fj(:)p Fn(\()p Fj(v)1324 2238 y Fh(\000)p Fd(1)1362 2246 y Fj(u)p Fn(\))1397 2238 y Fp(\()p Fo(e)1432 2246 y Fj(s)1448 2238 y Fh(\000)p Fd(1)1486 2246 y Fj(:v)1514 2238 y Fh(\000)p Fd(1)14 b Fm(\012)9 b Fo(x)1629 2244 y Fn(1)1648 2238 y Fp(\))402 2354 y(=)j Fo(\016)464 2360 y Fj(u)p Fn(1)512 2315 y Fg(X)509 2404 y Fj(v)q Fl(2)p Fj(P)582 2354 y Fo(e)601 2362 y Fj(s)617 2354 y Fh(\000)p Fd(1)656 2362 y Fj(:v)684 2354 y Fh(\000)p Fd(1)h Fm(\012)d Fo(x)799 2360 y Fn(1)829 2354 y Fp(=)h Fo(\017)889 2360 y Fj(A)916 2354 y Fp(\()p Fo(e)951 2360 y Fj(u)983 2354 y Fm(\012)e Fo(x)1048 2360 y Fj(s)1066 2354 y Fp(\)1)1103 2360 y Fj(A)257 2503 y Fp(This)14 b(completes)g(the)g(pro)q(of)g(of)f(the)i (prop)q(osition.)d Fi(2)953 2628 y Fp(41)p eop %%Page: 42 42 42 41 bop 257 262 a Fp(Note)15 b(that)f Fo(A)g Fp(is)g(semisimple)d(if) j(the)g(c)o(haracteristic)i(of)d(the)i(base)f(\014eld)g(do)q(es)h(not)f (divide)257 311 y(the)g(order)g(of)f Fo(G)p Fp(,)g(since)h(a)f(crossed) i(pro)q(duct)f(of)f(a)g(semisimple)e(group)i(ring)g(with)g(a)g(semi-) 257 361 y(simple)g(algebra)h(is)g(again)f(semisimple)f(\(cf.)i([57)o (],)f(Thm.)f(7.4.2,)g(p.)i(116,)f([64)o(],)h(Thm.)e(4.4,)257 411 y(p.)j(31\).)g Fo(A)g Fp(is)h(cosemisimple)d(if)h(the)i(c)o (haracteristic)h(of)e(the)h(base)g(\014eld)f(do)q(es)i(not)e(divide)257 461 y(the)h(order)g(of)f Fo(P)6 b Fp(,)14 b(since)i Fo(K)689 446 y Fj(P)732 461 y Fp(is)g(cosemisimple)d(in)h(this)i(case,)g (whereas)g Fo(K)s Fp([)p Fo(G)p Fp(])e(is)h(alw)o(a)o(ys)257 511 y(cosemisimple.)257 646 y Fq(3.3)48 b Fp(In)12 b(the)g(second)h (stage,)e(w)o(e)h(shall)f(construct)j(a)d(Y)m(etter-Drinfel'd)g(Hopf)h (algebra)f(for)257 696 y(an)o(y)k(group)g(homomorphism)c(from)j(a)h (\014nite)g(group)h(to)f(the)h(group)f(of)g(units)g(of)g(a)g(\014nite) 257 746 y(ring.)h(Supp)q(ose)h(that)f Fo(G)g Fp(is)g(a)h(\014nite)f (group)g(and)g(that)h Fo(R)f Fp(is)g(a)g(\014nite)h(ring.)e(W)m(e)h (assume)257 795 y(that)e(w)o(e)g(are)h(giv)o(en)e(a)h(group)f (homomorphism)847 887 y Fo(\027)h Fp(:)d Fo(G)g Fm(!)g Fo(U)5 b Fp(\()p Fo(R)p Fp(\))257 978 y(from)15 b Fo(G)h Fp(to)g(the)h(m)o(ultiplicati)o(v)o(e)d(group)i Fo(U)5 b Fp(\()p Fo(R)p Fp(\))16 b(of)g(units)g(of)g Fo(R)p Fp(.)g(W)m(e)f(use)j Fo(\027)g Fp(to)e(turn)h(the)257 1028 y(additiv)o(e)f(group)g(of)f Fo(R)h Fp(in)o(to)g(a)g(left)g Fo(G)p Fp(-mo)q(dule)e(in)i(t)o(w)o(o)f(w)o(a)o(ys.)h(First,)g Fo(R)g Fp(b)q(ecomes)g(a)g Fo(G)p Fp(-)257 1078 y(mo)q(dule)d(via:)660 1127 y Fo(G)c Fm(\002)g Fo(R)j Fm(!)f Fo(R;)c Fp(\()p Fo(s;)g(u)p Fp(\))j Fm(7!)h Fo(s)f Fm(\001)f Fo(u)i Fp(:=)g Fo(\027)s Fp(\()p Fo(s)p Fp(\))p Fo(u)257 1202 y Fp(W)m(e)17 b(denote)g Fo(R)g Fp(b)o(y)578 1208 y Fj(G)606 1202 y Fo(R)f Fp(if)g(it)h(is)f(regarded)i(as)f(a)f(left)h Fo(G)p Fp(-mo)q(dule)e(in)h(this)h(w)o(a)o(y)m(.)e(Second,)257 1252 y Fo(R)f Fp(b)q(ecomes)g(a)g(left)f Fo(G)p Fp(-mo)q(dule)f(via:) 640 1343 y Fo(G)d Fm(\002)g Fo(R)j Fm(!)f Fo(R;)c Fp(\()p Fo(s;)g(u)p Fp(\))j Fm(7!)h Fo(s)c(:)g(u)12 b Fp(:=)f Fo(u\027)s Fp(\()p Fo(s)1247 1326 y Fl(\000)p Fn(1)1291 1343 y Fp(\))257 1435 y(W)m(e)j(denote)h Fo(R)e Fp(b)o(y)h Fo(R)598 1441 y Fj(G)639 1435 y Fp(if)f(it)h(is)g(regarded)g(as)g(a)g (left)g Fo(G)p Fp(-mo)q(dule)e(in)h(this)h(w)o(a)o(y)m(.)257 1520 y(W)m(e)g(no)o(w)f(assume)h(that)g(the)g(follo)o(wing)d (additional)i(structure)j(elemen)o(ts)d(are)i(giv)o(en:)308 1639 y(1.)20 b(Tw)o(o)14 b(1-co)q(cycles)g Fo(\013;)7 b(\014)14 b Fm(2)d Fo(Z)801 1624 y Fn(1)820 1639 y Fp(\()p Fo(G;)894 1645 y Fj(G)922 1639 y Fo(R)p Fp(\),)i(i.)g(e.,)h(mappings)e (from)g Fo(G)h Fp(to)h Fo(R)g Fp(satisfying)559 1730 y Fo(\013)p Fp(\()p Fo(st)p Fp(\))d(=)h Fo(\013)p Fp(\()p Fo(s)p Fp(\))e(+)f Fo(\027)s Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(t)p Fp(\))83 b Fo(\014)r Fp(\()p Fo(st)p Fp(\))13 b(=)f Fo(\014)r Fp(\()p Fo(s)p Fp(\))f(+)e Fo(\027)s Fp(\()p Fo(s)p Fp(\))p Fo(\014)r Fp(\()p Fo(t)p Fp(\))361 1822 y(for)14 b(all)e Fo(s;)7 b(t)12 b Fm(2)f Fo(G)p Fp(.)308 1905 y(2.)20 b(A)14 b(normalized)e(2-co)q(cycle)j Fo(q)e Fm(2)e Fo(Z)898 1890 y Fn(2)917 1905 y Fp(\()p Fo(G;)991 1911 y Fj(G)1019 1905 y Fo(R)p Fp(\).)308 1988 y(3.)20 b(Tw)o(o)11 b(c)o(haracters)i Fo(\037;)7 b(\021)12 b Fm(2)773 1977 y Fp(^)764 1988 y Fo(R)f Fp(of)g(the)g(additiv)o(e)g (group)g(of)f Fo(R)p Fp(,)h(i.)f(e.,)h(mappings)e(from)h Fo(R)361 2037 y Fp(to)k Fo(K)450 2022 y Fl(\002)492 2037 y Fp(satisfying)618 2129 y Fo(\037)p Fp(\()p Fo(u)9 b Fp(+)h Fo(v)q Fp(\))i(=)g Fo(\037)p Fp(\()p Fo(u)p Fp(\))p Fo(\037)p Fp(\()p Fo(v)q Fp(\))84 b Fo(\021)q Fp(\()p Fo(u)9 b Fp(+)g Fo(v)q Fp(\))k(=)e Fo(\021)q Fp(\()p Fo(u)p Fp(\))p Fo(\021)q Fp(\()p Fo(v)q Fp(\))361 2220 y(for)j(all)e Fo(u;)7 b(v)13 b Fm(2)e Fo(R)p Fp(.)257 2339 y(These)18 b(structure)g(elemen)o(ts)d(can)h(b)q(e)h(c)o(hosen)g (freely)m(,)e(except)i(that)f(w)o(e)g(supp)q(ose)i(that)e Fo(\037)257 2389 y Fp(satis\014es)f(the)g(condition:)811 2439 y Fo(\037)p Fp(\()p Fo(uv)q(w)q Fp(\))d(=)g Fo(\037)p Fp(\()p Fo(v)q(uw)q Fp(\))953 2628 y(42)p eop %%Page: 43 43 43 42 bop 257 262 a Fp(for)14 b(all)f Fo(u;)7 b(v)q(;)g(w)13 b Fm(2)e Fo(R)p Fp(.)j(F)m(or)g(example,)e(this)j(condition)e(is)h (satis\014ed)h(if)f Fo(R)g Fp(is)g(comm)o(utativ)o(e.)257 311 y(It)g(should)g(b)q(e)h(observ)o(ed)g(that)e(this)h(condition)g (implies)378 391 y Fo(\037)p Fp(\()p Fo(u)444 397 y Fn(1)472 391 y Fm(\001)9 b Fo(:)e(:)g(:)g Fm(\001)i Fo(u)595 397 y Fj(i)609 391 y Fo(u)633 397 y Fj(i)p Fn(+1)697 391 y Fm(\001)g Fo(:)e(:)g(:)h Fm(\001)h Fo(u)821 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Fj(R)1008 2414 y Fo(\033)1032 2420 y Fj(u)1053 2414 y Fp(\()p Fo(s;)7 b(t)p Fp(\))p Fo(e)1157 2420 y Fj(u)953 2628 y Fp(43)p eop %%Page: 44 44 44 43 bop 257 262 a Fp(for)14 b(functions)g Fo(\033)524 268 y Fj(u)557 262 y Fp(:)d Fo(G)e Fm(\002)g Fo(G)i Fm(!)g Fo(K)798 246 y Fl(\002)827 262 y Fp(.)i(W)m(e)h(de\014ne:)296 345 y(1.)20 b Fo(z)368 351 y Fj(s)386 345 y Fp(\()p Fo(u)p Fp(\))11 b(:=)h Fo(u\014)r Fp(\()p Fo(s)p Fp(\))296 411 y(2.)20 b(\()p Fo(\015)386 417 y Fj(s)404 411 y Fp(\()p Fo(u)p Fp(\)\)\()p Fo(v)q Fp(\))13 b(:=)e Fo(\037)p Fp(\()p Fo(uv)q(\013)p Fp(\()p Fo(s)p Fp(\)\))778 396 y Fn(2)296 477 y Fp(3.)20 b Fo(\033)373 483 y Fj(u)394 477 y Fp(\()p Fo(s;)7 b(t)p Fp(\))12 b(:=)f Fo(\021)q Fp(\()p Fo(uq)q Fp(\()p Fo(s;)c(t)p Fp(\)\))p Fo(\037)p Fp(\()p Fo(u)795 462 y Fn(2)814 477 y Fo(\027)s Fp(\()p Fo(s)p Fp(\))p Fo(\014)r Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(t)p Fp(\)\))257 560 y(W)m(e)k(no)o(w)g(ha)o(v)o(e)h(to)f(pro)o(v)o(e)g (that)h Fo(z)r Fp(,)f Fo(\015)j Fp(and)d 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Fn(2)651 1732 y Fo(\027)s Fp(\()p Fo(r)q(s)p Fp(\))p Fo(\014)r Fp(\()p Fo(r)q Fp(\))p Fo(\013)p Fp(\()p Fo(t)p Fp(\))j(+)f Fo(u)972 1715 y Fn(2)991 1732 y Fo(\027)s Fp(\()p Fo(r)q(s)p Fp(\))p Fo(\027)s Fp(\()p Fo(r)q Fp(\))p Fo(\014)r Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(t)p Fp(\)\))320 1800 y(=)j Fo(\033)388 1806 y Fj(u)409 1800 y Fp(\()p Fo(r)o(;)7 b(s)p Fp(\))p Fo(\021)q Fp(\()p Fo(uq)q Fp(\()p Fo(r)q(s;)g(t)p Fp(\)\))p Fo(\037)p Fp(\()p Fo(u)766 1782 y Fn(2)785 1800 y Fo(\027)s Fp(\()p Fo(r)q(s)p Fp(\)\()p Fo(\014)r Fp(\()p Fo(r)q Fp(\))j(+)f Fo(\027)s Fp(\()p Fo(r)q Fp(\))p Fo(\014)r Fp(\()p Fo(s)p Fp(\)\))p Fo(\013)p Fp(\()p Fo(t)p Fp(\)\))320 1862 y(=)j Fo(\033)388 1868 y Fj(u)409 1862 y Fp(\()p Fo(r)o(;)7 b(s)p Fp(\))p Fo(\033)521 1868 y Fj(u)542 1862 y Fp(\()p Fo(r)q(s;)g(t)p Fp(\))257 1953 y(It)14 b(no)o(w)g(follo)o(ws)e(from)g(an)i(easy)g (calculation)f(that)h Fo(\033)h Fp(is)f(a)f(2-co)q(cycle.)257 2020 y(F)m(rom)g(Lemma)f(1.13,)g(w)o(e)j(ha)o(v)o(e)f Fo(\013)p 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Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(t)p Fp(\)\))p Fo(\037)p Fp(\(2)p Fo(uv)q(\027)s Fp(\()p Fo(s)p Fp(\))p Fo(\014)r Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(t)p Fp(\)\))274 2379 y(=)12 b Fo(\033)342 2385 y Fj(u)363 2379 y Fp(\()p Fo(s;)7 b(t)p Fp(\))p Fo(\033)472 2385 y Fj(v)492 2379 y Fp(\()p Fo(s;)g(t)p Fp(\)\()p Fo(\015)614 2385 y Fj(t)629 2379 y Fp(\()p Fo(s)664 2361 y Fl(\000)p Fn(1)709 2379 y Fo(:u)p Fp(\)\)\()p Fo(v)q(\014)r Fp(\()p Fo(s)p Fp(\)\))953 2628 y(44)p eop %%Page: 45 45 45 44 bop 257 262 a Fp(\(5\))21 b(It)12 b(no)o(w)g(follo)o(ws)e(from)g (Prop)q(osition)i(3.2)f(that)h Fo(A)g Fp(b)q(ecomes)g(a)g(Y)m (etter-Drinfel'd)g(Hopf)257 311 y(algebra.)f(It)h(is)f(easy)h(to)f(see) i(that)e(the)i(structure)g(elemen)o(ts)f(giv)o(en)f(there)h(are)g (those)h(stated)257 361 y(ab)q(o)o(v)o(e,)i(except)i(for)e(the)h(an)o (tip)q(o)q(de.)f(The)h(an)o(tip)q(o)q(de)f(giv)o(en)g(in)g(Prop)q (osition)g(3.2)f(has)i(the)257 411 y(form)387 486 y Fo(S)412 492 y Fj(A)440 486 y Fp(\()p Fo(e)475 492 y Fj(u)506 486 y Fm(\012)10 b Fo(x)572 492 y Fj(s)589 486 y Fp(\))i(=)g Fo(\033)686 469 y Fl(\000)p Fn(1)685 497 y Fl(\000)p Fj(u)732 486 y Fp(\()p Fo(s;)7 b(s)805 469 y Fl(\000)p Fn(1)850 486 y Fp(\))p Fo(e)885 493 y Fl(\000)p Fj(u\027)r Fn(\()p Fj(s)p Fn(\))1003 486 y Fm(\012)i Fo(x)1068 494 y Fj(s)1084 486 y Fh(\000)p Fd(1)617 554 y Fp(=)j Fo(\021)q Fp(\()p Fo(uq)q Fp(\()p Fo(s;)7 b(s)816 537 y Fl(\000)p Fn(1)861 554 y Fp(\)\))p Fo(\037)p Fp(\()p Fm(\000)p Fo(u)991 537 y Fn(2)1010 554 y Fo(\027)s Fp(\()p Fo(s)p Fp(\))p Fo(\014)r Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)1223 537 y Fl(\000)p Fn(1)1269 554 y Fp(\)\))p Fo(e)1320 561 y Fl(\000)p Fj(u\027)r Fn(\()p Fj(s)p Fn(\))1438 554 y Fm(\012)i Fo(x)1503 562 y Fj(s)1519 553 y Fh(\000)p Fd(1)257 629 y Fp(But)17 b(since)g(w)o(e)g(ha)o(v)o(e)f Fo(\013)p Fp(\()p Fo(s)670 614 y Fl(\000)p Fn(1)715 629 y Fp(\))g(=)g Fm(\000)p Fo(\027)s Fp(\()p Fo(s)886 614 y Fl(\000)p Fn(1)930 629 y Fp(\))p Fo(\013)p Fp(\()p Fo(s)p 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1036 y Fp(\()p Fo(\013;)7 b(\014)r(;)g(q)q Fp(\).)257 1114 y(In)15 b(the)f(case)i(where)f Fo(R)f Fp(is)g(comm)o(utativ)n(e,)e(the)i(de\014nition)g(of)g Fo(\033)h Fp(can)g(b)q(e)f(rewritten)i(using)257 1164 y(the)f(cup)f(pro)q(duct.)h(The)f(m)o(ultiplicatio)o(n)d(map)745 1240 y Fo(R)e Fm(\012)818 1246 y Fc(Z)849 1240 y Fo(R)i Fm(!)g Fo(R;)c(u)i Fm(\012)g Fo(v)k Fm(7!)e Fo(uv)257 1315 y Fp(is)17 b(then)h Fo(G)p Fp(-equiv)n(arian)o(t,)d(pro)o(vided)i (that)g(w)o(e)g(consider)h Fo(R)f Fp(as)g(a)g Fo(G)p Fp(-mo)q(dule)e(in)i(a)f(third)257 1365 y(w)o(a)o(y)e(via)f(the)h(mo)q (dule)f(structure)720 1441 y Fo(G)c Fm(\002)h Fo(R)h Fm(!)g Fo(R;)c Fp(\()p Fo(s;)g(u)p Fp(\))k Fm(7!)g Fo(\027)s Fp(\()p Fo(s)p Fp(\))1184 1423 y Fn(2)1203 1441 y Fo(u)257 1516 y Fp(The)23 b(cup)g(pro)q(duct)g Fo(\014)h Fm([)d Fo(\013)k Fm(2)g Fo(Z)834 1501 y Fn(2)853 1516 y Fp(\()p Fo(G;)7 b(R)14 b Fm(\012)999 1522 y Fc(Z)1035 1516 y Fo(R)p Fp(\))22 b(then)h(is)f(mapp)q(ed)f(to)h(a)g(2-co)q(cycle)257 1566 y(in)e Fo(Z)343 1551 y Fn(2)362 1566 y Fp(\()p Fo(G;)7 b(R)p Fp(\))19 b(whic)o(h)h(w)o(e,)g(follo)o(wing)e(common)f(usage,)j (also)g(denote)h(b)o(y)f Fo(\014)c Fm([)d Fo(\013)p Fp(.)19 b(The)257 1616 y(de\014nition)14 b(of)f Fo(\033)i Fp(then)g(tak)o(es)f (the)h(form:)630 1691 y Fo(\033)654 1697 y Fj(u)676 1691 y Fp(\()p Fo(s;)7 b(t)p Fp(\))12 b(=)f Fo(\021)q Fp(\()p Fo(uq)q Fp(\()p Fo(s;)c(t)p Fp(\)\))p Fo(\037)p Fp(\()p Fo(u)1065 1674 y Fn(2)1084 1691 y Fp(\()p Fo(\014)12 b Fm([)d Fo(\013)p Fp(\)\()p Fo(s;)e(t)p Fp(\)\))257 1820 y Fq(3.4)48 b Fp(W)m(e)14 b(no)o(w)g(lo)q(ok)f(at)h(the)h(sp)q (ecial)f(case)h(of)f(the)h(preceding)g(construction)g(whic)o(h)f(will) 257 1870 y(b)q(e)21 b(the)h(most)d(imp)q(ortan)o(t)g(one)h(in)g(the)h (follo)o(wing,)d(b)q(ecause)k(the)f(structure)i(theorem)257 1919 y(men)o(tioned)14 b(ab)q(o)o(v)o(e)g(and)g(stated)h(in)f(P)o (aragraph)g(7.7)f(will)g(sa)o(y)h(that)h(all)e(Y)m(etter-Drinfel'd)257 1969 y(Hopf)e(algebras)h(of)f(a)g(certain)h(t)o(yp)q(e)g(are)g (necessarily)g(of)f(the)h(form)e(that)h(w)o(e)h(describ)q(e)h(no)o(w.) 257 2019 y(Supp)q(ose)f(that)f Fo(p)f Fp(is)h(an)f(o)q(dd)h(prime)e (and)i(that)f Fo(R)i Fp(=)g Fk(Z)1103 2025 y Fj(p)1119 2019 y Fp(,)e(the)i(\014nite)f(\014eld)f(with)h Fo(p)f Fp(elemen)o(ts.)257 2069 y(Supp)q(ose)17 b(that)e Fo(\020)i Fm(2)c Fo(K)19 b Fp(is)c(a)g Fo(p)p Fp(-th)g(ro)q(ot)g(of)g(unit)o(y,)f (whic)o(h)h(need)i(not)e(b)q(e)h(primitiv)o(e)d(here.)257 2119 y(Since)h Fo(p)e Fp(is)g(o)q(dd,)h(2)f(is)g(an)h(in)o(v)o(ertible) f(elemen)o(t)g(of)g Fk(Z)1072 2125 y Fj(p)1088 2119 y Fp(,)g(and)h(therefore)h(the)f(expression)h Fo(i=)p Fp(2)257 2169 y(for)g Fo(i)e Fm(2)f Fk(Z)417 2175 y Fj(p)447 2169 y Fp(mak)o(es)h(sense.)j(W)m(e)f(then)h(de\014ne)f(the)h(c)o(haracters) 577 2249 y Fo(\037)c Fp(:)g Fk(Z)668 2255 y Fj(p)696 2249 y Fm(!)g Fo(K)q(;)c(i)k Fm(7!)g Fo(\020)903 2232 y Fj(i=)p Fn(2)1034 2249 y Fo(\021)h Fp(:)f Fk(Z)1121 2255 y Fj(p)1149 2249 y Fm(!)g Fo(K)q(;)c(i)k Fm(7!)g Fo(\020)1356 2232 y Fj(i)257 2354 y Fp(No)o(w,)18 b(if)g Fo(G)g Fp(is)h(a)f(\014nite)h(group,)f Fo(\027)j Fp(:)e Fo(G)h Fm(!)e Fk(Z)1010 2339 y Fl(\002)1010 2364 y Fj(p)1054 2354 y Fp(is)g(a)h(group)f(homomorphi)o(sm)o(,)e Fo(\013;)7 b(\014)21 b Fm(2)257 2403 y Fo(Z)288 2388 y Fn(1)307 2403 y Fp(\()p Fo(G;)381 2409 y Fj(G)409 2403 y Fk(Z)440 2409 y Fj(p)456 2403 y Fp(\))15 b(are)f(t)o(w)o(o)g(1-co)q(cycles,)h (and)g Fo(q)e Fm(2)f Fo(Z)1026 2388 y Fn(2)1045 2403 y Fp(\()p Fo(G;)1119 2409 y Fj(G)1147 2403 y Fk(Z)1178 2409 y Fj(p)1194 2403 y Fp(\))i(is)g(a)g(normalized)f(2-co)q(cycle,)257 2453 y(w)o(e)19 b(can)g(construct)h(the)f(Y)m(etter-Drinfel'd)f(Hopf)h (algebra)f Fo(A)1260 2459 y Fj(G)1288 2453 y Fp(\()p Fo(\013;)7 b(\014)r(;)g(q)q Fp(\).)17 b(Its)i(structure)257 2503 y(elemen)o(ts)14 b(then)h(tak)o(e)f(the)g(follo)o(wing)d(form:)953 2628 y(45)p eop %%Page: 46 46 46 45 bop 308 262 a Fp(1.)20 b(Crossed)15 b(pro)q(duct)g(algebra)e (structure:)515 352 y(\()p Fo(e)550 358 y Fj(i)574 352 y Fm(\012)c Fo(x)639 358 y Fj(s)657 352 y Fp(\)\()p Fo(e)708 358 y Fj(j)735 352 y Fm(\012)g Fo(x)800 358 y Fj(t)815 352 y Fp(\))i(:=)h Fo(\016)916 359 y Fj(i\027)r Fn(\()p Fj(s)p Fn(\))p Fj(;j)1015 352 y Fo(\020)1036 335 y Fj(iq)q Fn(\()p Fj(s;t)p Fn(\))1131 352 y Fo(\020)1152 335 y Fj(i)1164 323 y Fd(2)1180 335 y Fj(\027)r Fn(\()p Fj(s)p Fn(\))p Fj(\014)q Fn(\()p Fj(s)p Fn(\))p Fj(\013)p Fn(\()p Fj(t)p Fn(\))p Fj(=)p Fn(2)1398 352 y Fo(e)1417 358 y Fj(i)1440 352 y Fm(\012)e Fo(x)1506 358 y Fj(st)361 447 y Fp(Unit:)j(1)489 453 y Fj(A)528 447 y Fp(:=)583 416 y Fg(P)627 426 y Fj(p)p Fl(\000)p Fn(1)627 459 y Fj(i)p Fn(=0)695 447 y Fo(e)714 453 y Fj(i)738 447 y Fm(\012)c Fo(x)803 453 y Fn(1)308 527 y Fp(2.)20 b(T)m(ensor)14 b(pro)q(duct)h(coalgebra)f(structure:)647 652 y(\001)682 658 y Fj(A)708 652 y Fp(\()p Fo(e)743 658 y Fj(i)767 652 y Fm(\012)9 b Fo(x)832 658 y Fj(s)850 652 y Fp(\))j(:=)933 598 y Fj(p)p Fl(\000)p Fn(1)933 613 y Fg(X)934 701 y Fj(j)r Fn(=0)993 652 y Fp(\()p Fo(e)1028 658 y Fj(j)1055 652 y Fm(\012)e Fo(x)1121 658 y Fj(s)1138 652 y Fp(\))f Fm(\012)h Fp(\()p Fo(e)1240 658 y Fj(i)p Fl(\000)p Fj(j)1305 652 y Fm(\012)f Fo(x)1370 658 y Fj(s)1388 652 y Fp(\))361 782 y(Counit:)k Fo(\017)528 788 y Fj(A)555 782 y Fp(\()p Fo(e)590 788 y Fj(i)613 782 y Fm(\012)d Fo(x)679 788 y Fj(s)696 782 y Fp(\))i(:=)f Fo(\016)797 788 y Fj(i)p Fn(0)308 863 y Fp(3.)20 b(Action:)14 b Fo(c)526 869 y Fj(i)551 863 y Fm(!)d Fp(\()p Fo(e)639 869 y Fj(j)666 863 y Fm(\012)f Fo(x)732 869 y Fj(s)749 863 y Fp(\))i(:=)f Fo(\020)853 847 y Fj(ij)r(\013)p Fn(\()p Fj(s)p Fn(\))946 863 y Fo(e)965 869 y Fj(j)992 863 y Fm(\012)e Fo(x)1057 869 y Fj(s)308 943 y Fp(4.)20 b(Coaction:)13 b Fo(\016)566 949 y Fj(A)593 943 y Fp(\()p Fo(e)628 949 y Fj(i)652 943 y Fm(\012)c Fo(x)717 949 y Fj(s)735 943 y Fp(\))j(:=)f Fo(c)836 950 y Fj(i\014)q Fn(\()p Fj(s)p Fn(\))921 943 y Fm(\012)e Fp(\()p Fo(e)997 949 y Fj(i)1021 943 y Fm(\012)g Fo(x)1086 949 y Fj(s)1104 943 y Fp(\))308 1034 y(5.)20 b(An)o(tip)q(o)q(de:)14 b Fo(S)580 1040 y Fj(A)607 1034 y Fp(\()p Fo(e)642 1040 y Fj(i)666 1034 y Fm(\012)9 b Fo(x)731 1040 y Fj(s)749 1034 y Fp(\))j(:=)f Fo(\020)853 1019 y Fj(iq)q Fn(\()p Fj(s;s)936 1007 y Fh(\000)p Fd(1)974 1019 y Fn(\))989 1034 y Fo(\020)1010 1019 y Fj(i)1022 1007 y Fd(2)1038 1019 y Fj(\014)q Fn(\()p Fj(s)p Fn(\))p Fj(\013)p Fn(\()p Fj(s)p Fn(\))p Fj(=)p Fn(2)1199 1034 y Fo(e)1218 1041 y Fl(\000)p Fj(i\027)r Fn(\()p Fj(s)p Fn(\))1327 1034 y Fm(\012)f Fo(x)1393 1042 y Fj(s)1409 1034 y Fh(\000)p Fd(1)257 1167 y Fq(3.5)48 b Fp(W)m(e)19 b(shall)g(no)o(w)g(giv)o(e)g(another)g(application)f(of)h(the)h(framew) o(ork)e(considered)j(in)257 1217 y(P)o(aragraph)f(3.2.)f(Supp)q(ose)i (that)g Fo(G)e Fp(is)h(a)g(\014nite)h(group)f(and)g(that)g(the)h(base)g (\014eld)f Fo(K)257 1267 y Fp(con)o(tains)f(a)g(primitiv)o(e)e(fourth)i (ro)q(ot)g(of)g(unit)o(y)g Fo(\023)p Fp(;)f(this)h(can,)g(of)g(course,) h(only)e(happ)q(en)257 1317 y(if)e Fo(K)j Fp(do)q(es)d(not)g(ha)o(v)o 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Fo(G)g Fm(!)g Fk(Z)1396 1522 y Fn(2)1427 1516 y Fp(are)i(1-co)q(cycles;)257 1566 y(since)i(the)f Fo(G)p Fp(-mo)q(dule)e(structure)k(is)d(trivial,)f (these)k(are)e(just)g(group)f(homomorphism)o(s.)257 1616 y(Supp)q(ose)c(that)f Fo(q)i Fm(2)e Fo(Z)606 1601 y Fn(2)625 1616 y Fp(\()p Fo(G;)c Fk(Z)723 1622 y Fn(4)738 1616 y Fp(\))k(is)g(a)g(normalized)e(2-co)q(cycle)j(of)e(the)i(trivial)d Fo(G)p Fp(-mo)q(dule)g Fk(Z)1674 1622 y Fn(4)257 1665 y Fp(suc)o(h)15 b(that)857 1715 y(^)-24 b Fo(\031)11 b Fm(\016)e Fo(q)j Fp(=)g Fo(\014)g Fm([)d Fo(\013)257 1786 y Fp(where)15 b(^)-23 b Fo(\031)12 b Fp(:)f Fk(Z)465 1792 y Fn(4)492 1786 y Fm(!)g Fk(Z)576 1792 y Fn(2)603 1786 y Fp(is)g(the)g(unique)g(surjectiv)o(e)i(group)d(homomorphism)o(.) d(Here)13 b(w)o(e)e(ha)o(v)o(e,)257 1836 y(as)17 b(in)g(P)o(aragraph)f (3.3,)g(used)h(the)h(isomorphism)13 b Fk(Z)1098 1842 y Fn(2)1125 1836 y Fm(\012)1157 1842 y Fc(Z)1190 1836 y Fk(Z)1221 1842 y Fn(2)1253 1825 y Fm(\030)1253 1838 y Fp(=)1301 1836 y Fk(Z)1332 1842 y Fn(2)1364 1836 y Fp(to)k(regard)g(the)h(cup)257 1886 y(pro)q(duct)d Fo(\014)d Fm([)d Fo(\013)i Fm(2)g Fo(Z)593 1871 y Fn(2)612 1886 y Fp(\()p Fo(G;)c Fk(Z)710 1892 y Fn(2)735 1886 y Fm(\012)767 1892 y Fc(Z)798 1886 y Fk(Z)829 1892 y Fn(2)844 1886 y Fp(\))14 b(as)g(an)g(elemen)o(t)f(of)h Fo(Z)1215 1871 y Fn(2)1233 1886 y Fp(\()p Fo(G;)7 b Fk(Z)1331 1892 y Fn(2)1347 1886 y Fp(\),)14 b(i.)e(e.,)i(w)o(e)g(ha)o(v)o(e:)634 1972 y Fo(\014)e Fm([)c Fo(\013)k Fp(:)f Fo(G)e Fm(\002)g Fo(G)i Fm(!)g Fk(Z)978 1978 y Fn(2)994 1972 y Fo(;)c Fp(\()p Fo(s;)g(t)p Fp(\))k Fm(7!)g Fo(\014)r Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(t)p Fp(\))257 2057 y(Using)f(these)i(data,)d(w)o(e)h(can)g(construct)i(group)d (homomorphism)o(s)e Fo(z)f Fp(:)12 b Fo(G)f Fm(!)g Fp(Hom)o(\()p Fk(Z)1581 2063 y Fn(2)1597 2057 y Fo(;)c Fk(Z)1646 2063 y Fn(2)1662 2057 y Fp(\))p Fo(;)257 2111 y(s)12 b Fm(7!)f Fo(z)360 2117 y Fj(s)392 2111 y Fp(and)j Fo(\015)7 b Fp(:)k Fo(G)h Fm(!)f Fp(Hom)n(\()p Fk(Z)755 2117 y Fn(2)771 2111 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Fo(\034)1184 2492 y Fj(f)1200 2496 y Ff(P)1225 2492 y Fn(\()p Fj(u)p Fn(\))1272 2485 y Fp(\()p Fo(st)p Fp(\))1338 2467 y Fl(\000)p Fn(1)1384 2485 y Fo(\034)1402 2492 y Fj(f)1418 2496 y Ff(P)1442 2492 y Fn(\()p Fj(u)p Fn(\))1489 2485 y Fp(\()p Fo(s)p Fp(\))953 2628 y(50)p eop %%Page: 51 51 51 50 bop 257 262 a Fq(Prop)q(osition)33 b Fp(Supp)q(ose)17 b(that)g(w)o(e)f(ha)o(v)o(e)g Fo(\034)973 268 y Fj(u)993 259 y Fh(0)1004 268 y Fj(v)1022 259 y Fh(0)1035 262 y Fp(\()p Fo(s)p Fp(\))g(=)g Fo(\034)1168 268 y Fj(u)1188 259 y Fh(0)1201 262 y Fp(\()p Fo(s)p Fp(\))p Fo(\034)1270 268 y Fj(v)1288 259 y Fh(0)1302 262 y Fp(\()p Fo(s)p Fp(\))h(for)f(all)f Fo(s)h Fm(2)f Fo(G)h Fp(and)257 311 y Fo(u)281 296 y Fl(0)293 311 y Fo(;)7 b(v)333 296 y Fl(0)356 311 y Fm(2)k Fo(P)428 296 y Fl(0)439 311 y Fp(.)j(Then)g(w)o (e)g(ha)o(v)o(e:)308 412 y(1.)20 b(F)m(or)14 b(all)e Fo(s;)7 b(t)12 b Fm(2)f Fo(G)i Fp(and)h(all)f Fo(u;)7 b(v)12 b Fm(2)f Fo(P)6 b Fp(,)13 b(the)i(compatibilit)o(y)10 b(condition)638 489 y 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Ff(G)1412 780 y Fn(\()p Fj(s)p Fn(\))361 846 y Fp(is)14 b(a)f(morphism)f(of)h(Y)m(etter-Drinfel'd)g (Hopf)h(algebras.)257 937 y Fq(Pro)q(of.)36 b Fp(First,)14 b(w)o(e)g(v)o(erify)g(the)g(compatibilit)o(y)d(condition:)331 1013 y Fo(\033)355 1019 y Fj(uv)394 1013 y Fp(\()p Fo(s;)c(t)p Fp(\))12 b(=)g Fo(\033)560 996 y Fl(0)559 1025 y Fj(f)575 1029 y Ff(P)599 1025 y Fn(\()p Fj(u)p Fn(\))p Fj(f)661 1029 y Ff(P)684 1025 y Fn(\()p Fj(v)q Fn(\))730 1013 y Fp(\()p Fo(f)766 1019 y Fj(G)794 1013 y Fp(\()p Fo(s)p Fp(\))p Fo(;)7 b(f)884 1019 y Fj(G)913 1013 y Fp(\()p Fo(t)p Fp(\)\))437 1084 y Fo(\034)455 1092 y Fj(f)471 1096 y Ff(P)496 1092 y Fn(\()p Fj(s)525 1084 y Fh(\000)p Fd(1)563 1092 y Fj(:u)p Fn(\))p Fj(f)622 1096 y Ff(P)645 1092 y Fn(\()p Fj(s)674 1084 y Fh(\000)p Fd(1)713 1092 y Fj(:v)q Fn(\))755 1084 y Fp(\()p Fo(t)p Fp(\))p Fo(\034)820 1091 y Fj(f)836 1095 y Ff(P)861 1091 y Fn(\()p Fj(u)p Fn(\))p Fj(f)923 1095 y Ff(P)946 1091 y Fn(\()p Fj(v)q Fn(\))992 1084 y Fp(\()p Fo(st)p Fp(\))1058 1067 y Fl(\000)p Fn(1)1103 1084 y Fo(\034)1121 1091 y Fj(f)1137 1095 y Ff(P)1162 1091 y Fn(\()p Fj(u)p Fn(\))p Fj(f)1224 1095 y Ff(P)1247 1091 y Fn(\()p Fj(v)q Fn(\))1293 1084 y Fp(\()p Fo(s)p Fp(\))384 1151 y(=)12 b(\(\()p Fo(f)480 1157 y Fj(G)509 1151 y Fp(\()p Fo(s)p Fp(\))p Fo(:\015)595 1134 y Fl(0)593 1162 y Fj(f)609 1166 y Ff(G)634 1162 y Fn(\()p Fj(t)p Fn(\))674 1151 y Fp(\)\()p Fo(f)726 1157 y Fj(P)755 1151 y Fp(\()p Fo(u)p Fp(\)\)\)\()p Fo(z)880 1134 y Fl(0)878 1162 y Fj(f)894 1166 y Ff(G)919 1162 y Fn(\()p Fj(s)p Fn(\))963 1151 y Fp(\()p Fo(f)999 1157 y Fj(P)1027 1151 y Fp(\()p Fo(v)q Fp(\)\)\))437 1221 y Fo(\033)462 1204 y Fl(0)461 1232 y Fj(f)477 1236 y Ff(P)501 1232 y Fn(\()p Fj(u)p Fn(\))549 1221 y Fp(\()p Fo(f)585 1227 y Fj(G)613 1221 y Fp(\()p Fo(s)p Fp(\))p Fo(;)7 b(f)703 1227 y Fj(G)732 1221 y Fp(\()p Fo(t)p Fp(\)\))p Fo(\033)820 1204 y Fl(0)819 1232 y Fj(f)835 1236 y Ff(P)859 1232 y Fn(\()p Fj(v)q Fn(\))904 1221 y Fp(\()p Fo(f)940 1227 y Fj(G)969 1221 y Fp(\()p 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y Fj(s)669 1359 y Fp(\()p Fo(v)q Fp(\)\))p Fo(\033)762 1365 y Fj(u)785 1359 y Fp(\()p Fo(s;)7 b(t)p Fp(\))p Fo(\033)894 1365 y Fj(v)913 1359 y Fp(\()p Fo(s;)g(t)p Fp(\))257 1465 y(W)m(e)17 b(no)o(w)f(pro)o(v)o(e)g(the)i(second)f (assertion.)g(Linearit)o(y)f(and)h(colinearit)o(y)e(of)h Fo(f)1474 1471 y Fj(A)1518 1465 y Fp(follo)o(w)f(di-)257 1515 y(rectly)f(from)c(our)j(de\014nition)f(of)g Fo(z)i Fp(and)e Fo(\015)r Fp(.)h(W)m(e)f(no)o(w)g(sho)o(w)h(that)f Fo(f)1296 1521 y Fj(A)1336 1515 y Fp(is)g(an)h(algebra)e(homo-)257 1564 y(morphism:)286 1641 y Fo(f)306 1647 y Fj(A)333 1641 y Fp(\()p Fo(e)368 1647 y Fj(u)400 1641 y Fm(\012)e Fo(x)465 1647 y Fj(s)483 1641 y Fp(\))p Fo(f)519 1647 y Fj(A)546 1641 y Fp(\()p Fo(e)581 1647 y Fj(v)611 1641 y Fm(\012)g Fo(x)676 1647 y Fj(t)690 1641 y Fp(\))j(=)g Fo(\034)780 1648 y Fj(f)796 1652 y Ff(P)820 1648 y Fn(\()p Fj(u)p Fn(\))868 1641 y Fp(\()p Fo(s)p Fp(\))p Fo(\034)937 1648 y Fj(f)953 1652 y Ff(P)978 1648 y Fn(\()p Fj(v)q Fn(\))1023 1641 y Fp(\()p Fo(t)p Fp(\)\()p Fo(e)1105 1624 y Fl(0)1105 1652 y Fj(f)1121 1656 y Ff(P)1146 1652 y Fn(\()p Fj(u)p Fn(\))1203 1641 y Fm(\012)d Fo(x)1268 1624 y Fl(0)1268 1652 y Fj(f)1284 1656 y Ff(G)1309 1652 y Fn(\()p Fj(s)p Fn(\))1352 1641 y Fp(\)\()p Fo(e)1403 1624 y Fl(0)1403 1652 y Fj(f)1419 1656 y Ff(P)1444 1652 y Fn(\()p Fj(v)q Fn(\))1499 1641 y Fm(\012)g Fo(x)1564 1624 y Fl(0)1564 1652 y Fj(f)1580 1656 y Ff(G)1604 1652 y Fn(\()p Fj(t)p Fn(\))1645 1641 y Fp(\))361 1711 y(=)j Fo(\034)423 1718 y Fj(f)439 1722 y Ff(P)463 1718 y Fn(\()p Fj(u)p Fn(\))511 1711 y Fp(\()p Fo(s)p Fp(\))p Fo(\034)580 1718 y Fj(f)596 1722 y Ff(P)620 1718 y Fn(\()p Fj(v)q Fn(\))666 1711 y Fp(\()p Fo(t)p Fp(\))p Fo(\016)731 1718 y Fj(f)747 1722 y Ff(P)772 1718 y Fn(\()p Fj(u)p Fn(\))p Fj(;f)844 1722 y Ff(G)868 1718 y Fn(\()p Fj(s)p Fn(\))p Fj(:f)936 1722 y Ff(P)959 1718 y Fn(\()p Fj(v)q Fn(\))1005 1711 y Fo(\033)1030 1694 y Fl(0)1029 1722 y Fj(f)1045 1726 y Ff(P)1069 1722 y Fn(\()p Fj(u)p Fn(\))1116 1711 y Fp(\()p Fo(f)1152 1717 y Fj(G)1181 1711 y Fp(\()p Fo(s)p Fp(\))p Fo(;)7 b(f)1271 1717 y Fj(G)1299 1711 y Fp(\()p Fo(t)p Fp(\)\))p Fo(e)1381 1694 y Fl(0)1381 1722 y Fj(f)1397 1726 y Ff(P)1422 1722 y Fn(\()p Fj(u)p Fn(\))1479 1711 y Fm(\012)i Fo(x)1544 1694 y Fl(0)1544 1722 y Fj(f)1560 1726 y Ff(G)1584 1722 y Fn(\()p Fj(st)p Fn(\))361 1781 y Fp(=)j Fo(\016)423 1787 y Fj(u;s:v)498 1781 y Fo(\033)522 1787 y Fj(u)543 1781 y Fp(\()p Fo(s;)7 b(t)p Fp(\))p Fo(\034)646 1788 y Fj(f)662 1792 y Ff(P)687 1788 y Fn(\()p Fj(u)p Fn(\))734 1781 y Fp(\()p Fo(st)p Fp(\))p Fo(e)819 1764 y Fl(0)819 1792 y Fj(f)835 1796 y Ff(P)860 1792 y Fn(\()p Fj(u)p Fn(\))917 1781 y Fm(\012)j Fo(x)983 1764 y Fl(0)983 1792 y Fj(f)999 1796 y Ff(G)1023 1792 y Fn(\()p Fj(st)p Fn(\))1091 1781 y Fp(=)h Fo(f)1154 1787 y Fj(A)1182 1781 y Fp(\(\()p Fo(e)1233 1787 y Fj(u)1264 1781 y Fm(\012)f Fo(x)1330 1787 y Fj(s)1347 1781 y Fp(\)\()p Fo(e)1398 1787 y Fj(v)1428 1781 y Fm(\012)f Fo(x)1493 1787 y Fj(t)1508 1781 y Fp(\)\))257 1858 y(Our)15 b(assumption)d(on)i Fo(\034)19 b Fp(also)13 b(assures)i(that)f Fo(f)991 1864 y Fj(A)1033 1858 y Fp(is)f(a)h(coalgebra)g(homom)o(orphism)o(:)366 1935 y(\()p Fo(f)402 1941 y Fj(A)439 1935 y Fm(\012)c Fo(f)501 1941 y Fj(A)528 1935 y Fp(\)\001)579 1941 y Fj(A)606 1935 y Fp(\()p Fo(e)641 1941 y Fj(u)672 1935 y Fm(\012)g Fo(x)738 1941 y Fj(s)755 1935 y Fp(\))394 2010 y(=)441 1970 y Fg(X)438 2059 y Fj(v)q Fl(2)p Fj(P)510 2010 y Fo(\034)528 2017 y Fj(f)544 2021 y Ff(P)569 2017 y Fn(\()p Fj(v)q Fn(\))614 2010 y Fp(\()p Fo(s)p Fp(\))p Fo(\034)683 2018 y Fj(f)699 2022 y Ff(P)724 2018 y Fn(\()p Fj(v)755 2009 y Fh(\000)p Fd(1)793 2018 y Fj(u)p Fn(\))828 2010 y Fp(\()p Fo(s)p Fp(\)\()p Fo(e)914 1993 y Fl(0)914 2021 y Fj(f)930 2025 y Ff(P)955 2021 y Fn(\()p Fj(v)q Fn(\))1010 2010 y Fm(\012)g Fo(x)1076 1993 y Fl(0)1076 2021 y Fj(f)1092 2025 y Ff(G)1116 2021 y Fn(\()p Fj(s)p Fn(\))1160 2010 y Fp(\))f Fm(\012)g Fp(\()p Fo(e)1261 1993 y Fl(0)1261 2021 y Fj(f)1277 2025 y Ff(P)1302 2021 y Fn(\()p Fj(v)1333 2013 y Fh(\000)p Fd(1)1371 2021 y Fj(u)p Fn(\))1415 2010 y Fm(\012)h Fo(x)1481 1993 y Fl(0)1481 2021 y Fj(f)1497 2025 y Ff(G)1521 2021 y Fn(\()p Fj(s)p Fn(\))1564 2010 y Fp(\))394 2117 y(=)i Fo(\034)456 2124 y Fj(f)472 2128 y Ff(P)496 2124 y Fn(\()p Fj(u)p Fn(\))544 2117 y Fp(\()p Fo(s)p Fp(\)\001)630 2123 y Fj(A)655 2115 y Fh(0)668 2117 y Fp(\()p Fo(e)703 2100 y Fl(0)703 2128 y Fj(f)719 2132 y Ff(P)744 2128 y Fn(\()p Fj(u)p Fn(\))801 2117 y Fm(\012)d Fo(x)866 2100 y Fl(0)866 2128 y Fj(f)882 2132 y Ff(G)906 2128 y Fn(\()p Fj(s)p Fn(\))950 2117 y Fp(\))j(=)f(\001)1056 2123 y Fj(A)1081 2115 y Fh(0)1094 2117 y Fp(\()p Fo(f)1130 2123 y Fj(A)1158 2117 y Fp(\()p Fo(e)1193 2123 y Fj(u)1224 2117 y Fm(\012)f Fo(x)1290 2123 y Fj(s)1307 2117 y Fp(\)\))257 2194 y(Since)16 b(w)o(e)f(ha)o(v)o (e)h Fo(\034)5 b Fp(\(1\))13 b(=)h(1,)h(i.)f(e.,)g Fo(\034)822 2200 y Fj(u)842 2192 y Fh(0)856 2194 y Fp(\(1\))f(=)h(1)h(for)g(all)f Fo(u)1152 2179 y Fl(0)1177 2194 y Fm(2)g Fo(P)1252 2179 y Fl(0)1263 2194 y Fp(,)g Fo(f)1309 2200 y Fj(A)1352 2194 y Fp(preserv)o(es)j(the)f(unit.)257 2244 y Fo(f)277 2250 y Fj(A)324 2244 y Fp(preserv)o(es)22 b(the)e(counit)f(since)h Fo(u)848 2229 y Fl(0)880 2244 y Fm(7!)g Fo(\034)960 2250 y Fj(u)980 2242 y Fh(0)993 2244 y Fp(\()p Fo(s)p Fp(\))g(is)f(a)g (group)g(homomorphism)o(,)d(whic)o(h)257 2293 y(implies)h(that)i Fo(\034)516 2299 y Fn(1)535 2293 y Fp(\()p Fo(s)p Fp(\))i(=)f(1)e(for)h (all)e Fo(s)k Fm(2)e Fo(G)p Fp(.)f Fo(f)999 2299 y Fj(A)1046 2293 y Fp(is)g(therefore)j(a)d(morphism)e(of)j(Y)m(etter-)257 2343 y(Drinfel'd)f(bialgebras.)g(Exactly)h(as)g(for)f(ordinary)h(Hopf)f (algebras,)g(it)h(can)g(b)q(e)g(sho)o(wn)257 2393 y(that)10 b(a)f(morphism)e(of)i(Y)m(etter-Drinfel'd)h(bialgebras)f(comm)o(utes)f (with)h(the)h(an)o(tip)q(o)q(de,)g(and)257 2443 y(therefore)17 b(is)e(a)f(morphism)f(of)h(Y)m(etter-Drinfel'd)h(Hopf)f(algebras)h (\(cf.)g([84)o(],)f(Lem.)g(4.0.4,)257 2493 y(p.)g(81\).)f Fi(2)953 2628 y Fp(51)p eop %%Page: 52 52 52 51 bop 257 262 a Fq(4.3)48 b Fp(In)9 b(P)o(aragraph)h(3.3,)e(w)o(e)h (ha)o(v)o(e)g(explained)g(ho)o(w)g(w)o(e)h(can)g(construct)h(Y)m (etter-Drinfel'd)257 311 y(Hopf)20 b(algebras)g(from)f(group)h (homomorphi)o(sm)o(s)e(to)i(the)h(additiv)o(e)f(group)g(of)f(a)h (\014nite)257 361 y(ring.)15 b(In)g(this)g(paragraph,)g(w)o(e)g(shall)f (\014nd)i(su\016cien)o(t)f(conditions)g(under)h(whic)o(h)g(Y)m(etter-) 257 411 y(Drinfel'd)d(Hopf)g(algebras)h(arising)f(from)g(this)g (construction)i(are)g(isomorphic.)257 497 y(Supp)q(ose)i(that)f Fo(G)547 481 y Fl(0)575 497 y Fp(is)g(a)f(\014nite)i(group)e(and)h (that)g Fo(R)g Fp(is)g(a)g(\014nite)g(ring.)f(W)m(e)h(assume)f(that)257 546 y(w)o(e)f(are)h(giv)o(en)e(a)h(group)f(homomorphism)836 638 y Fo(\027)860 621 y Fl(0)882 638 y Fp(:)e Fo(G)938 621 y Fl(0)961 638 y Fm(!)g Fo(U)5 b Fp(\()p Fo(R)p Fp(\))257 729 y(from)13 b Fo(G)389 714 y Fl(0)416 729 y Fp(to)h(the)i(m)o (ultiplicati)o(v)o(e)c(group)j Fo(U)5 b Fp(\()p Fo(R)p Fp(\))15 b(of)f(units)h(of)f Fo(R)p 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1923 y Fp(\()p Fo(f)t Fp(\()p Fo(s)p Fp(\)\)\))1271 1906 y Fn(2)1303 1923 y Fp(=)g Fo(\037)p Fp(\()p Fo(v)q(u\013)p Fp(\()p Fo(s)p Fp(\)\))1528 1906 y Fn(2)257 2009 y Fp(Finally)h Fo(\033)q Fp(,)g(if)g(written)i(in)e(the)i(form)d Fo(\033)q Fp(\()p Fo(s;)7 b(t)p Fp(\))12 b(=)1016 1978 y Fg(P)1060 2021 y Fj(u)p Fl(2)p Fj(P)1137 2009 y Fo(\033)1161 2015 y Fj(u)1182 2009 y Fp(\()p Fo(s;)7 b(t)p Fp(\))p Fo(e)1286 2015 y Fj(u)1308 2009 y Fp(,)14 b(is)g(determined)f(b)o(y:)271 2099 y Fo(\033)295 2105 y Fj(u)317 2099 y Fp(\()p Fo(s;)7 b(t)p Fp(\))k(=)h Fo(\033)482 2082 y Fl(0)481 2111 y Fj(f)497 2115 y Ff(P)521 2111 y Fn(\()p Fj(u)p Fn(\))569 2099 y Fp(\()p Fo(f)605 2105 y Fj(G)633 2099 y Fp(\()p Fo(s)p Fp(\))p Fo(;)7 b(f)723 2105 y Fj(G)751 2099 y Fp(\()p Fo(t)p Fp(\)\))p Fo(\034)832 2107 y Fj(f)848 2111 y Ff(P)873 2107 y Fn(\()p Fj(s)902 2099 y Fh(\000)p Fd(1)940 2107 y Fj(:u)p Fn(\))985 2099 y Fp(\()p Fo(t)p Fp(\))p Fo(\034)1050 2106 y Fj(f)1066 2110 y Ff(P)1090 2106 y Fn(\()p Fj(u)p 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Fm(2)f Fo(G)h Fp(:)f Fo(\027)s Fp(\()p Fo(s)p Fp(\))g(=)h Fo(\027)676 1435 y Fl(0)687 1450 y Fp(\()p Fo(f)t Fp(\()p Fo(s)p Fp(\)\))308 1526 y(2.)20 b Fm(8)p Fo(s)12 b Fm(2)f Fo(G)h Fp(:)f Fo(\013)p Fp(\()p Fo(s)p Fp(\))h(=)f Fo(x\013)706 1511 y Fl(0)718 1526 y Fp(\()p Fo(f)t Fp(\()p Fo(s)p Fp(\)\))308 1602 y(3.)20 b Fm(8)p Fo(s)12 b Fm(2)f Fo(G)h Fp(:)f Fo(\014)r Fp(\()p Fo(s)p Fp(\))i(=)e Fo(x\014)703 1587 y Fl(0)715 1602 y Fp(\()p Fo(f)t Fp(\()p Fo(s)p Fp(\)\))308 1677 y(4.)20 b Fm(8)p Fo(s;)7 b(t)12 b Fm(2)f Fo(G)g Fp(:)g Fo(q)q Fp(\()p Fo(s;)c(t)p Fp(\))12 b(:=)f Fo(x)p Fp(\()p Fo(q)787 1662 y Fl(0)799 1677 y Fp(\()p Fo(f)t Fp(\()p Fo(s)p Fp(\))p Fo(;)c(f)t Fp(\()p Fo(t)p Fp(\)\))k(+)f Fo(\027)s Fp(\()p Fo(s)p Fp(\))p Fo(w)q Fp(\()p Fo(t)p Fp(\))f Fm(\000)h Fo(w)q Fp(\()p Fo(st)p Fp(\))f(+)h Fo(w)q Fp(\()p Fo(s)p Fp(\)\))257 1774 y(If)15 b(\()p Fo(f)340 1759 y Fl(0)353 1774 y Fo(;)7 b(x)396 1759 y Fl(0)407 1774 y Fo(;)g(w)457 1759 y Fl(0)467 1774 y Fp(\))14 b(:)f(\()p Fo(G)571 1759 y Fl(0)583 1774 y Fo(;)7 b(\027)626 1759 y Fl(0)636 1774 y Fo(;)g(\013)682 1759 y Fl(0)693 1774 y Fo(;)g(\014)737 1759 y Fl(0)749 1774 y Fo(;)g(q)788 1759 y Fl(0)799 1774 y Fp(\))13 b Fm(!)g Fp(\()p Fo(G)932 1759 y Fl(00)953 1774 y Fo(;)7 b(\027)996 1759 y Fl(0)o(0)1016 1774 y Fo(;)g(\013)1062 1759 y Fl(0)o(0)1083 1774 y Fo(;)g(\014)1127 1759 y Fl(0)o(0)1148 1774 y Fo(;)g(q)1187 1759 y Fl(0)o(0)1208 1774 y Fp(\))15 b(is)g(another)g(morphism,)d(w)o(e)257 1824 y(de\014ne)j(the)g(comp)q(osition)d(as:)502 1898 y(\()p Fo(f)542 1880 y Fl(0)555 1898 y Fo(;)7 b(x)598 1880 y Fl(0)609 1898 y Fo(;)g(w)659 1880 y Fl(0)670 1898 y Fp(\))i Fm(\016)g Fp(\()p Fo(f)r(;)e(x;)g(w)q Fp(\))k(:=)g(\()p Fo(f)978 1880 y Fl(0)1000 1898 y Fm(\016)e Fo(f)r(;)e(xx)1119 1880 y Fl(0)1130 1898 y Fo(;)g(x)1173 1880 y Fl(0)q(\000)p Fn(1)1228 1898 y Fo(w)k Fp(+)e Fo(f)1334 1880 y Fn(1)1353 1898 y Fp(\()p Fo(w)1400 1880 y Fl(0)1412 1898 y Fp(\)\))257 1971 y(This)k(is)f(in)g(fact)g(a)g(morphism,)d(since)k(w)o(e)g(ob)o (viously)e(ha)o(v)o(e)h Fo(\027)s Fp(\()p Fo(s)p Fp(\))g(=)f Fo(\027)1345 1956 y Fl(0)o(0)1366 1971 y Fp(\()p Fo(f)1406 1956 y Fl(0)1418 1971 y Fp(\()p Fo(f)t Fp(\()p Fo(s)p Fp(\)\)\),)j Fo(\013)p Fp(\()p Fo(s)p Fp(\))e(=)257 2021 y Fo(xx)305 2006 y Fl(0)316 2021 y Fo(\013)343 2006 y Fl(00)364 2021 y Fp(\()p Fo(f)404 2006 y Fl(0)417 2021 y Fp(\()p Fo(f)t Fp(\()p Fo(s)p Fp(\)\)\),)j(and)f Fo(\014)r Fp(\()p Fo(s)p Fp(\))f(=)f Fo(xx)829 2006 y Fl(0)840 2021 y Fo(\014)865 2006 y Fl(0)877 2021 y Fp(\()p Fo(f)917 2006 y Fl(0)930 2021 y Fp(\()p Fo(f)t Fp(\()p Fo(s)p Fp(\)\)\),)j(but)f(also)290 2094 y Fo(q)q Fp(\()p Fo(s;)7 b(t)p Fp(\))12 b(:=)f Fo(xx)510 2077 y Fl(0)521 2094 y Fp(\()p Fo(q)557 2077 y Fl(00)579 2094 y Fp(\()p Fo(f)619 2077 y Fl(0)631 2094 y Fp(\()p Fo(f)t Fp(\()p Fo(s)p Fp(\)\))p Fo(;)c(f)781 2077 y Fl(0)795 2094 y Fp(\()p Fo(f)t Fp(\()p Fo(t)p Fp(\)\)\))k(+)e Fo(\027)s Fp(\()p Fo(s)p Fp(\))p Fo(w)1072 2077 y Fl(0)1084 2094 y Fp(\()p Fo(f)t Fp(\()p Fo(t)p Fp(\)\))h Fm(\000)g 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Fp(\))i Fm(\016)g Fp(\()p Fo(f)831 2471 y Fl(0)843 2488 y Fo(;)e(x)886 2471 y Fl(0)897 2488 y Fo(;)g(w)947 2471 y Fl(0)958 2488 y Fp(\)\))j Fm(\016)f Fp(\()p Fo(f)r(;)e(x;)g(w)q Fp(\))953 2628 y(54)p eop %%Page: 55 55 55 54 bop 257 262 a Fp(this)16 b(comp)q(osition)f(la)o(w)g(is)h(asso)q (ciativ)o(e.)f(The)i(iden)o(tit)o(y)e(homomorphism)d(of)j(the)i(ob)r (ject)257 311 y(\()p Fo(G;)7 b(\027;)g(\013;)g(\014)r(;)g(q)q Fp(\))12 b(is)i(\(id)596 317 y Fj(G)624 311 y Fo(;)7 b Fp(1)664 317 y Fj(R)690 311 y Fo(;)g Fp(0\).)257 395 y(No)o(w)14 b(consider)h(the)g(Hopf)e(algebra)h Fo(H)h Fp(:=)c Fo(K)s Fp([)p Fo(R)p Fp(],)i(i.)g(e.,)h(the)g(group)g(ring)g (of)g(the)g(additiv)o(e)257 445 y(group)j(of)g Fo(R)p Fp(.)f(F)m(or)h Fo(u)g Fm(2)g Fo(R)p Fp(,)f(w)o(e)h(denote)h(the)g (corresp)q(onding)g(canonical)f(basis)g(elemen)o(t)257 495 y(of)i Fo(K)s Fp([)p Fo(R)p Fp(])g(b)o(y)g Fo(c)504 501 y Fj(u)545 495 y Fp(and)g(the)h(corresp)q(onding)h(primitiv)o(e)c (idemp)q(oten)o(t)i(of)g Fo(K)1480 480 y 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Fp(is)g(an)g(o)q(dd)g(prime)f(and)h(that)g Fo(R)g Fp(=)g Fk(Z)1372 1628 y Fj(p)1388 1622 y Fp(.)f(W)m(e)h(shall)f(pro)o(v)o(e) 257 1672 y(no)o(w)22 b(that)g(isomorphisms)e(b)q(et)o(w)o(een)j(Y)m (etter-Drinfel'd)f(Hopf)g(algebras)g(of)f(the)i(t)o(yp)q(e)257 1721 y Fo(A)288 1727 y Fj(G)316 1721 y Fp(\()p Fo(\013;)7 b(\014)r(;)g(q)q Fp(\))15 b(o)o(v)o(er)h Fo(H)g Fp(:=)e Fo(K)s Fp([)p Fk(Z)754 1727 y Fj(p)770 1721 y Fp(])h(describ)q(ed)i(in) e(P)o(aragraph)g(3.4)f(are)i(necessarily)g(of)f(the)257 1771 y(form)d(presen)o(ted)k(in)e(the)g(previous)h(t)o(w)o(o)e (paragraphs.)257 1855 y(T)m(o)e(consider)h(t)o(w)o(o)g(suc)o(h)g (algebras,)f(w)o(e)g(supp)q(ose)i(that)f(w)o(e)f(are)h(giv)o(en)f(t)o (w)o(o)g(\014nite)h(groups)g Fo(G)257 1905 y Fp(and)21 b Fo(G)378 1890 y Fl(0)389 1905 y Fp(,)g(t)o(w)o(o)f(group)g (homomorphism)o(s)e Fo(\027)25 b Fp(:)e Fo(G)f Fm(!)h Fk(Z)1179 1890 y Fl(\002)1179 1915 y Fj(p)1225 1905 y Fp(and)e Fo(\027)1337 1890 y Fl(0)1371 1905 y Fp(:)h Fo(G)1438 1890 y Fl(0)1472 1905 y Fm(!)h Fk(Z)1568 1890 y Fl(\002)1568 1915 y Fj(p)1593 1905 y Fp(,)d(t)o(w)o(o)257 1954 y(co)q(cycles)13 b Fo(\013;)7 b(\014)13 b Fm(2)e Fo(Z)566 1939 y Fn(1)585 1954 y Fp(\()p Fo(G;)646 1960 y Fj(G)673 1954 y Fk(Z)704 1960 y Fj(p)720 1954 y Fp(\))g(as)g(w)o(ell) f(as)h(t)o(w)o(o)g(co)q(cycles)h Fo(\013)1183 1939 y Fl(0)1194 1954 y Fo(;)7 b(\014)1238 1939 y Fl(0)1262 1954 y Fm(2)k Fo(Z)1332 1939 y Fn(1)1351 1954 y Fp(\()p Fo(G)1400 1939 y Fl(0)1411 1954 y Fo(;)1423 1960 y Fj(G)1449 1952 y Fh(0)1462 1954 y Fk(Z)1493 1960 y Fj(p)1509 1954 y Fp(\),)f(and)h(t)o(w)o(o)257 2004 y(normalized)g(co)q(cycles)j Fo(q)f Fm(2)e Fo(Z)728 1989 y Fn(2)747 2004 y Fp(\()p Fo(G;)808 2010 y Fj(G)835 2004 y Fk(Z)866 2010 y Fj(p)882 2004 y Fp(\))p Fo(;)c(q)937 1989 y Fl(0)960 2004 y Fm(2)k Fo(Z)1030 1989 y Fn(2)1049 2004 y Fp(\()p Fo(G)1098 1989 y Fl(0)1109 2004 y Fo(;)1121 2010 y Fj(G)1147 2002 y Fh(0)1160 2004 y Fk(Z)1191 2010 y Fj(p)1207 2004 y Fp(\).)h(W)m(e)g 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Fq(Prop)q(osition)33 b Fp(Supp)q(ose)15 b(that)f Fo(\013)d Fm(6)p Fp(=)h(0)h(or)h Fo(\014)g Fm(6)p Fp(=)e(0.)g(Then)j(there)g(is)e(an)g(elemen)o(t)h Fo(k)e Fm(2)f Fk(Z)1653 531 y Fl(\002)1653 557 y Fj(p)1678 546 y Fp(,)257 596 y(a)18 b(group)f(isomorphism)e Fo(f)23 b Fp(:)17 b Fo(G)h Fm(!)f Fo(G)879 581 y Fl(0)890 596 y Fp(,)h(and)f(a)h(1-co)q(c)o(hain)f Fo(w)i Fp(:)e Fo(G)h Fm(!)1426 602 y Fj(G)1454 596 y Fk(Z)1485 602 y Fj(p)1518 596 y Fp(satisfying)257 646 y Fo(w)q Fp(\(1\))12 b(=)g(0)h(suc)o(h)i (that)308 765 y(1.)20 b Fm(8)p Fo(s)12 b Fm(2)f Fo(G)h Fp(:)f Fo(\027)s Fp(\()p Fo(s)p Fp(\))g(=)h Fo(\027)676 750 y Fl(0)687 765 y Fp(\()p Fo(f)t Fp(\()p Fo(s)p Fp(\)\))308 848 y(2.)20 b Fm(8)p Fo(s)12 b Fm(2)f Fo(G)h Fp(:)f Fo(\013)p Fp(\()p Fo(s)p Fp(\))h(=)f Fo(k)q(\013)705 833 y Fl(0)717 848 y Fp(\()p Fo(f)t Fp(\()p Fo(s)p Fp(\)\))308 931 y(3.)20 b Fm(8)p Fo(s)12 b Fm(2)f Fo(G)h Fp(:)f Fo(\014)r Fp(\()p Fo(s)p Fp(\))i(=)e Fo(k)q(\014)702 916 y Fl(0)715 931 y Fp(\()p Fo(f)t Fp(\()p 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b(In)e(the)h(algebras)e Fo(A)828 1330 y Fj(G)856 1324 y Fp(\()p Fo(\013;)7 b(\014)r(;)g(q)q Fp(\))19 b(resp.)h Fo(A)1153 1330 y Fj(G)1179 1322 y Fh(0)1192 1324 y Fp(\()p Fo(\013)1235 1309 y Fl(0)1246 1324 y Fo(;)7 b(\014)1290 1309 y Fl(0)1302 1324 y Fo(;)g(q)1341 1309 y Fl(0)1352 1324 y Fp(\),)19 b(w)o(e)g(consider)h(the)257 1374 y(elemen)o(ts)600 1455 y Fo(u)11 b Fp(:=)691 1401 y Fj(p)p Fl(\000)p Fn(1)691 1415 y Fg(X)692 1504 y Fj(j)r Fn(=0)757 1455 y Fo(\020)778 1438 y Fj(j)796 1455 y Fo(e)815 1461 y Fj(j)842 1455 y Fm(\012)f Fo(x)908 1461 y Fn(1)1009 1455 y Fo(u)1033 1438 y Fl(0)1056 1455 y Fp(:=)1112 1401 y Fj(p)p Fl(\000)p Fn(1)1111 1415 y Fg(X)1113 1504 y Fj(j)r Fn(=0)1178 1455 y Fo(\020)1199 1438 y Fj(j)1217 1455 y Fo(e)1236 1461 y Fj(j)1263 1455 y Fm(\012)g Fo(x)1329 1438 y Fl(0)1329 1465 y Fn(1)257 1571 y Fp(W)m(e)k(then)g(ha)o(v)o(e) 314 1694 y Fo(u)p Fp(\()p Fo(e)373 1700 y Fj(i)396 1694 y Fm(\012)c Fo(x)462 1700 y Fj(s)479 1694 y Fp(\))i(=)551 1640 y Fj(p)p Fl(\000)p Fn(1)551 1654 y Fg(X)552 1743 y Fj(j)r Fn(=0)618 1694 y Fo(\020)639 1677 y Fj(j)656 1694 y Fp(\()p Fo(e)691 1700 y Fj(j)719 1694 y Fm(\012)d Fo(x)784 1700 y Fn(1)802 1694 y Fp(\)\()p Fo(e)853 1700 y Fj(i)877 1694 y Fm(\012)h Fo(x)943 1700 y Fj(s)960 1694 y Fp(\))i(=)1032 1640 y Fj(p)p Fl(\000)p Fn(1)1032 1654 y Fg(X)1033 1743 y Fj(j)r Fn(=0)1098 1694 y Fo(\020)1119 1677 y Fj(j)1137 1694 y Fo(\016)1155 1701 y Fj(j)r(\027)r Fn(\(1\))p Fj(;i)1256 1694 y Fp(\()p Fo(e)1291 1700 y Fj(j)1318 1694 y Fm(\012)d Fo(x)1383 1700 y Fj(s)1401 1694 y Fp(\))j(=)f Fo(\020)1493 1677 y Fj(i)1507 1694 y Fo(e)1526 1700 y Fj(i)1550 1694 y Fm(\012)e Fo(x)1615 1700 y Fj(s)257 1835 y Fp(and)18 b(similarly)e Fo(u)541 1820 y Fl(0)552 1835 y Fp(\()p Fo(e)587 1841 y Fj(i)614 1835 y Fm(\012)c Fo(x)682 1820 y Fl(0)682 1846 y Fj(s)698 1838 y Fh(0)711 1835 y Fp(\))19 b(=)g Fo(\020)818 1820 y Fj(i)832 1835 y Fo(e)851 1841 y Fj(i)877 1835 y Fm(\012)13 b Fo(x)946 1820 y Fl(0)946 1846 y Fj(s)962 1838 y Fh(0)974 1835 y Fp(.)18 b(Since)h Fo(\036)p Fp(\()p Fo(e)1177 1841 y Fj(i)1203 1835 y Fm(\012)12 b Fo(x)1271 1841 y Fj(s)1289 1835 y Fp(\))19 b(=)g Fo(\020)1396 1820 y Fj(i\013)p Fn(\()p Fj(s)p Fn(\))1473 1835 y Fo(e)1492 1841 y Fj(i)1518 1835 y Fm(\012)13 b Fo(x)1587 1841 y Fj(s)1623 1835 y Fp(and)257 1885 y Fo( )q Fp(\()p Fo(e)320 1891 y Fj(i)344 1885 y Fm(\012)d Fo(x)410 1891 y Fj(s)427 1885 y Fp(\))i(=)g Fo(\020)520 1870 y Fj(i\014)q Fn(\()p Fj(s)p Fn(\))596 1885 y Fo(e)615 1891 y Fj(i)638 1885 y Fm(\012)e Fo(x)704 1891 y Fj(s)721 1885 y Fp(,)k(w)o(e)g(see)h(that:)479 1976 y Fo(\036)p Fp(\()p Fo(e)539 1982 y Fj(i)562 1976 y Fm(\012)9 b Fo(x)627 1982 y Fj(s)645 1976 y Fp(\))i(=)h Fo(u)740 1959 y Fj(\013)p Fn(\()p Fj(s)p Fn(\))805 1976 y Fo(e)824 1982 y Fj(i)848 1976 y Fm(\012)d Fo(x)913 1982 y Fj(s)1014 1976 y Fo( )q Fp(\()p Fo(e)1077 1982 y Fj(i)1101 1976 y Fm(\012)g Fo(x)1166 1982 y Fj(s)1184 1976 y Fp(\))i(=)h Fo(u)1279 1959 y Fj(\014)q Fn(\()p Fj(s)p Fn(\))1343 1976 y Fo(e)1362 1982 y Fj(i)1385 1976 y Fm(\012)e Fo(x)1451 1982 y Fj(s)257 2067 y Fp(Similarly)m(,)g(w)o(e)k (ha)o(v)o(e:)410 2159 y Fo(\036)435 2142 y Fl(0)447 2159 y Fp(\()p Fo(e)482 2165 y Fj(i)505 2159 y Fm(\012)c Fo(x)571 2142 y Fl(0)571 2169 y Fj(s)587 2161 y Fh(0)599 2159 y Fp(\))i(=)g Fo(u)695 2142 y Fl(0)q Fj(\013)728 2129 y Fh(0)739 2142 y Fn(\()p Fj(s)768 2129 y Fh(0)779 2142 y Fn(\))794 2159 y Fo(e)813 2165 y Fj(i)836 2159 y Fm(\012)e Fo(x)902 2142 y Fl(0)902 2169 y Fj(s)918 2161 y Fh(0)1014 2159 y Fo( )1042 2142 y Fl(0)1054 2159 y Fp(\()p Fo(e)1089 2165 y Fj(i)1112 2159 y Fm(\012)g Fo(x)1178 2142 y Fl(0)1178 2169 y Fj(s)1194 2161 y Fh(0)1207 2159 y Fp(\))h(=)h Fo(u)1302 2142 y Fl(0)q Fj(\014)1333 2129 y Fh(0)1345 2142 y Fn(\()p Fj(s)1374 2129 y Fh(0)1385 2142 y Fn(\))1400 2159 y Fo(e)1419 2165 y Fj(i)1442 2159 y Fm(\012)e Fo(x)1508 2142 y Fl(0)1508 2169 y Fj(s)1524 2161 y Fh(0)953 2628 y Fp(56)p eop %%Page: 57 57 57 56 bop 257 262 a Fp(\(2\))21 b(The)c(grouplik)o(e)e(elemen)o(ts)h (of)g(the)g(dual)g(group)g(ring)g Fo(K)1237 246 y Fc(Z)1259 250 y Ff(p)1292 262 y Fp(are)g(the)h(F)m(ourier)f(trans-)257 316 y(formed)f(elemen)o(ts)572 285 y Fg(P)615 295 y Fj(p)p Fl(\000)p Fn(1)615 328 y Fj(j)r Fn(=0)684 316 y Fo(\020)705 301 y Fj(ij)734 316 y Fo(e)753 322 y Fj(j)788 316 y Fp(of)g(the)i (idemp)q(oten)o(ts)f Fo(e)1168 322 y Fj(j)1186 316 y Fp(.)g(Since)g(the)h(coalgebra)f(struc-)257 365 y(ture)g(of)f Fo(A)426 371 y Fj(G)454 365 y Fp(\()p Fo(\013;)7 b(\014)r(;)g(q)q Fp(\))15 b(is)f(the)i(ordinary)f(tensor)h(pro)q(duct)g(coalgebra)f (structure,)i(the)e(ele-)257 420 y(men)o(ts)381 388 y Fg(P)425 399 y Fj(p)p Fl(\000)p Fn(1)425 432 y Fj(j)r Fn(=0)493 420 y Fo(\020)514 405 y Fj(ij)544 420 y Fo(e)563 426 y Fj(j)592 420 y Fm(\012)c Fo(x)659 426 y Fj(s)692 420 y Fp(form)k(a)h(basis)g(of)g Fo(A)1015 426 y Fj(G)1043 420 y Fp(\()p Fo(\013;)7 b(\014)r(;)g(q)q Fp(\))16 b(consisting)g(of)g (grouplik)o(e)f(ele-)257 481 y(men)o(ts.)h(Since)h(the)g(p)q(o)o(w)o (ers)g(of)f Fo(u)g Fp(are)h(giv)o(en)f(b)o(y)g(the)h(form)o(ula)d Fo(u)1304 466 y Fj(i)1334 481 y Fp(=)1382 450 y Fg(P)1426 461 y Fj(p)p Fl(\000)p Fn(1)1426 494 y Fj(j)r Fn(=0)1494 481 y Fo(\020)1515 466 y Fj(ij)1545 481 y Fo(e)1564 487 y Fj(j)1592 481 y Fm(\012)e Fo(x)1660 487 y Fn(1)1678 481 y Fp(,)257 531 y(these)k(elemen)o(ts)d(can)i(b)q(e)f(written)g(in)g (the)g(form:)731 596 y Fj(p)p Fl(\000)p Fn(1)731 610 y Fg(X)732 699 y Fj(j)r Fn(=0)798 650 y Fo(\020)819 633 y Fj(ij)849 650 y Fo(e)868 656 y Fj(j)895 650 y Fm(\012)9 b Fo(x)960 656 y Fj(s)989 650 y Fp(=)j Fo(u)1057 633 y Fj(i)1070 650 y Fp(\(1)e Fm(\012)f Fo(x)1182 656 y Fj(s)1200 650 y Fp(\))257 782 y Fo(A)288 788 y Fj(G)314 780 y Fh(0)328 782 y Fp(\()p Fo(\013)371 767 y Fl(0)382 782 y Fo(;)e(\014)426 767 y Fl(0)438 782 y Fo(;)g(q)477 767 y Fl(0)488 782 y Fp(\))21 b(has)g(a)f(similar)e(basis)j(consisting) g(of)f(grouplik)o(e)g(elemen)o(ts.)g(Since)h Fo(f)1662 788 y Fj(A)257 832 y Fp(tak)o(es)13 b(grouplik)o(e)f(elemen)o(ts)h(to)f (grouplik)o(e)g(elemen)o(ts,)g(there)i(exists,)f(for)f(ev)o(ery)i Fo(s)e Fm(2)f Fo(G)p Fp(,)h(an)257 881 y(elemen)o(t)i Fo(w)q Fp(\()p Fo(s)p Fp(\))e Fm(2)f Fk(Z)575 887 y Fj(p)605 881 y Fp(and)i(an)h(elemen)o(t)f Fo(f)t Fp(\()p Fo(s)p Fp(\))h Fm(2)d Fo(G)1057 866 y Fl(0)1082 881 y Fp(suc)o(h)j(that)702 974 y Fo(f)722 980 y Fj(A)750 974 y Fp(\(1)9 b Fm(\012)g Fo(x)861 980 y Fj(s)879 974 y Fp(\))i(=)h Fo(u)974 957 y Fl(0)r Fj(w)q Fn(\()p Fj(s)p Fn(\))1054 974 y Fp(\(1)d Fm(\012)h Fo(x)1166 957 y Fl(0)1166 985 y Fj(f)s Fn(\()p Fj(s)p Fn(\))1229 974 y Fp(\))257 1062 y(No)o(w)k(the)g(linearit)o(y)f (of)h Fo(f)655 1068 y Fj(A)696 1062 y Fp(o)o(v)o(er)g Fo(H)i Fp(implies:)330 1149 y Fo(f)350 1155 y Fj(A)377 1149 y Fp(\()p Fo(u)417 1132 y Fj(\013)p Fn(\()p Fj(s)p Fn(\))482 1149 y Fp(\))p Fo(u)522 1132 y Fl(0)r Fj(w)q Fn(\()p Fj(s)p Fn(\))603 1149 y Fp(\(1)9 b Fm(\012)g Fo(x)714 1132 y Fl(0)714 1161 y Fj(f)s Fn(\()p Fj(s)p Fn(\))777 1149 y Fp(\))j(=)f Fo(f)868 1155 y Fj(A)896 1149 y Fp(\()p Fo(u)936 1132 y Fj(\013)p Fn(\()p Fj(s)p Fn(\))1001 1149 y Fp(\(1)e Fm(\012)h Fo(x)1113 1155 y Fj(s)1130 1149 y Fp(\)\))i(=)g Fo(f)1238 1155 y Fj(A)1265 1149 y Fp(\()p Fo(\036)p Fp(\(1)d Fm(\012)h Fo(x)1418 1155 y Fj(s)1435 1149 y Fp(\)\))805 1226 y(=)h Fo(\036)873 1208 y Fl(0)885 1226 y Fp(\()p Fo(f)921 1232 y Fj(A)948 1226 y Fp(\(1)e Fm(\012)h Fo(x)1060 1232 y Fj(s)1077 1226 y Fp(\)\))i(=)g Fo(u)1189 1208 y Fl(0)r Fj(\013)1223 1196 y Fh(0)1233 1208 y Fn(\()p Fj(f)s Fn(\()p Fj(s)p Fn(\)\))1322 1226 y Fo(u)1346 1208 y Fl(0)r Fj(w)q Fn(\()p Fj(s)p Fn(\))1426 1226 y Fp(\(1)d Fm(\012)h Fo(x)1538 1208 y Fl(0)1538 1237 y Fj(f)s Fn(\()p Fj(s)p Fn(\))1601 1226 y Fp(\))257 1313 y(Since)15 b(grouplik)o(e)e(elemen)o(ts)g(are)i (in)o(v)o(ertible,)e(this)h(implies:)783 1401 y Fo(f)803 1407 y Fj(A)830 1401 y Fp(\()p Fo(u)870 1384 y Fj(\013)p Fn(\()p Fj(s)p Fn(\))935 1401 y Fp(\))e(=)g Fo(u)1031 1384 y Fl(0)q Fj(\013)1064 1371 y Fh(0)1075 1384 y Fn(\()p Fj(f)s Fn(\()p Fj(s)p Fn(\)\))257 1488 y Fp(Similarly)m(,)d(the)14 b(colinearit)o(y)e(of)h Fo(f)781 1494 y Fj(A)821 1488 y Fp(implies)e Fo(f)981 1494 y Fj(A)1008 1488 y Fp(\()p Fo(u)1048 1473 y Fj(\014)q Fn(\()p Fj(s)p Fn(\))1112 1488 y Fp(\))h(=)g Fo(u)1208 1473 y Fl(0)q Fj(\014)1239 1461 y Fh(0)1251 1473 y Fn(\()p Fj(f)s Fn(\()p Fj(s)p Fn(\)\))1340 1488 y Fp(.)g(No)o(w)h(supp)q(ose)h(that)257 1538 y Fo(\013)h Fm(6)p Fp(=)g(0.)g(Then)h(there)h(exists)g(an)e (elemen)o(t)h Fo(s)f Fm(2)f Fo(G)i Fp(suc)o(h)g(that)g Fo(\013)p Fp(\()p Fo(s)p Fp(\))f Fm(6)p Fp(=)g(0,)g(and,)h(since)g Fo(f)1662 1544 y Fj(A)257 1588 y Fp(is)e(injectiv)o(e,)e(w)o(e)i(also)f (ha)o(v)o(e)g Fo(\013)740 1573 y Fl(0)751 1588 y Fp(\()p Fo(f)t Fp(\()p Fo(s)p Fp(\)\))h Fm(6)p Fp(=)d(0.)i(Since)h Fo(\013)1096 1573 y Fl(0)1107 1588 y Fp(\()p Fo(f)t Fp(\()p Fo(s)p Fp(\)\))h(generates)g Fk(Z)1442 1594 y Fj(p)1458 1588 y Fp(,)e(there)i(exists)257 1638 y Fo(k)e Fm(2)f Fk(Z)363 1623 y Fl(\002)363 1648 y Fj(p)402 1638 y Fp(suc)o(h)j(that)g Fo(f)607 1644 y Fj(A)634 1638 y Fp(\()p Fo(u)674 1623 y Fj(k)694 1638 y Fp(\))e(=)f Fo(u)791 1623 y Fl(0)803 1638 y Fp(.)h(By)i(a)f(similar)e(reasoning,)h(this)i(also)e(holds)h(if) g Fo(\014)h Fm(6)p Fp(=)d(0.)257 1688 y(The)j(ab)q(o)o(v)o(e)e (equations)h(no)o(w)g(yield:)629 1775 y Fo(\013)p Fp(\()p Fo(s)p Fp(\))e(=)g Fo(k)q(\013)813 1758 y Fl(0)824 1775 y Fp(\()p Fo(f)t Fp(\()p Fo(s)p Fp(\)\))85 b Fo(\014)r Fp(\()p Fo(s)p Fp(\))13 b(=)f Fo(k)q(\014)1197 1758 y Fl(0)1209 1775 y Fp(\()p Fo(f)t Fp(\()p Fo(s)p Fp(\)\))257 1863 y(This)i(pro)o(v)o(es)g(the)h(second)g(and)f(the)g(third)g (assertion.)257 1944 y(\(3\))21 b(W)m(e)14 b(ha)o(v)o(e:)440 2073 y(\()p Fo(e)475 2079 y Fj(i)499 2073 y Fm(\012)9 b Fo(x)564 2079 y Fj(s)582 2073 y Fp(\))p Fo(u)i Fp(=)677 2019 y Fj(p)p Fl(\000)p Fn(1)677 2034 y Fg(X)678 2122 y Fj(j)r Fn(=0)744 2073 y Fo(\020)765 2056 y Fj(j)783 2073 y Fp(\()p Fo(e)818 2079 y Fj(i)841 2073 y Fm(\012)f Fo(x)907 2079 y Fj(s)924 2073 y Fp(\)\()p Fo(e)975 2079 y Fj(j)1003 2073 y Fm(\012)f Fo(x)1068 2079 y Fn(1)1086 2073 y Fp(\))j(=)1158 2019 y Fj(p)p Fl(\000)p Fn(1)1158 2034 y Fg(X)1159 2122 y Fj(j)r Fn(=0)1225 2073 y Fo(\020)1246 2056 y Fj(j)1263 2073 y Fo(\016)1281 2080 y Fj(i\027)r Fn(\()p Fj(s)p Fn(\))p Fj(;j)1381 2073 y Fo(e)1400 2079 y Fj(i)1424 2073 y Fm(\012)d Fo(x)1489 2079 y Fj(s)633 2188 y Fp(=)j Fo(\020)698 2170 y Fj(i\027)r Fn(\()p Fj(s)p Fn(\))772 2188 y Fo(e)791 2194 y Fj(i)815 2188 y Fm(\012)d Fo(x)880 2194 y Fj(s)909 2188 y Fp(=)j Fo(u)977 2170 y Fj(\027)r Fn(\()p Fj(s)p Fn(\))1039 2188 y Fp(\()p Fo(e)1074 2194 y Fj(i)1098 2188 y Fm(\012)d Fo(x)1163 2194 y Fj(s)1181 2188 y Fp(\))257 2284 y(By)21 b(a)e(similar)f (calculation,)h(w)o(e)h(ha)o(v)o(e)g(\()p Fo(e)944 2290 y Fj(i)971 2284 y Fm(\012)14 b Fo(x)1041 2269 y Fl(0)1041 2295 y Fj(s)1057 2287 y Fh(0)1070 2284 y Fp(\))p Fo(u)1110 2269 y Fl(0)1143 2284 y Fp(=)22 b Fo(u)1221 2269 y Fl(0)r Fj(\027)1252 2256 y Fh(0)1263 2269 y Fn(\()p Fj(s)1292 2256 y Fh(0)1302 2269 y Fn(\))1317 2284 y Fp(\()p Fo(e)1352 2290 y Fj(i)1380 2284 y Fm(\012)14 b Fo(x)1450 2269 y Fl(0)1450 2295 y Fj(s)1466 2287 y Fh(0)1478 2284 y Fp(\).)20 b(The)h(\014rst)257 2333 y(equation)12 b(implies)e(\()p Fo(e)599 2339 y Fj(i)618 2333 y Fm(\012)5 b Fo(x)679 2339 y Fj(s)697 2333 y Fp(\))p Fo(u)737 2318 y Fj(k)769 2333 y Fp(=)11 b Fo(u)836 2318 y Fj(k)q(\027)r Fn(\()p Fj(s)p Fn(\))917 2333 y Fp(\()p Fo(e)952 2339 y Fj(i)971 2333 y Fm(\012)5 b Fo(x)1032 2339 y Fj(s)1050 2333 y Fp(\).)11 b(Summing)e(o)o(v)o(er)j Fo(i)g Fp(and)f(applying)g Fo(f)1651 2339 y Fj(A)1678 2333 y Fp(,)257 2383 y(w)o(e)19 b(get)g Fo(u)421 2368 y Fl(0)r Fj(w)q Fn(\()p Fj(s)p Fn(\))501 2383 y Fp(\(1)13 b Fm(\012)f Fo(x)619 2368 y Fl(0)619 2397 y Fj(f)s Fn(\()p Fj(s)p Fn(\))682 2383 y Fp(\))p Fo(u)722 2368 y Fl(0)753 2383 y Fp(=)20 b Fo(u)829 2368 y Fl(0)r Fj(\027)r Fn(\()p Fj(s)p Fn(\))903 2383 y Fo(u)927 2368 y Fl(0)q Fj(w)q Fn(\()p Fj(s)p Fn(\))1007 2383 y Fp(\(1)12 b Fm(\012)h Fo(x)1125 2368 y Fl(0)1125 2397 y Fj(f)s Fn(\()p Fj(s)p Fn(\))1188 2383 y Fp(\).)18 b(Since)h(on)g(the)g(other)h(hand)257 2447 y Fo(u)281 2432 y Fl(0)r Fj(w)q Fn(\()p Fj(s)p Fn(\))361 2447 y Fp(\(1)12 b Fm(\012)g Fo(x)478 2432 y Fl(0)478 2460 y Fj(f)s Fn(\()p Fj(s)p Fn(\))540 2447 y Fp(\))p Fo(u)580 2432 y Fl(0)609 2447 y Fp(=)18 b Fo(u)683 2432 y Fl(0)r Fj(\027)714 2420 y Fh(0)724 2432 y Fn(\()p Fj(f)s Fn(\()p Fj(s)p Fn(\)\))813 2447 y Fo(u)837 2432 y Fl(0)r Fj(w)q Fn(\()p Fj(s)p Fn(\))917 2447 y Fp(\(1)12 b Fm(\012)g Fo(x)1034 2432 y Fl(0)1034 2460 y Fj(f)s Fn(\()p Fj(s)p Fn(\))1097 2447 y Fp(\),)17 b(w)o(e)g(see)i(that)e Fo(u)1394 2432 y Fl(0)r Fj(\027)r Fn(\()p Fj(s)p Fn(\))1485 2447 y Fp(=)h Fo(u)1559 2432 y Fl(0)r Fj(\027)1590 2420 y Fh(0)1600 2432 y Fn(\()p Fj(f)s Fn(\()p Fj(s)p Fn(\)\))257 2503 y Fp(and)c(therefore)h Fo(\027)s Fp(\()p Fo(s)p Fp(\))d(=)g Fo(\027)667 2488 y Fl(0)677 2503 y Fp(\()p Fo(f)t Fp(\()p Fo(s)p Fp(\)\).)k(This)d(pro)o(v)o(es)i(the)f(\014rst)h (assertion.)953 2628 y(57)p eop %%Page: 58 58 58 57 bop 257 262 a Fp(\(4\))21 b(By)14 b(in)o(v)o(erting)g(the)g (discrete)i(F)m(ourier)d(transform)g(ab)q(o)o(v)o(e,)g(w)o(e)h(get:)699 391 y Fo(e)718 397 y Fj(i)742 391 y Fm(\012)9 b Fo(x)807 397 y Fj(s)836 391 y Fp(=)885 362 y(1)p 885 381 21 2 v 885 419 a Fo(p)918 337 y Fj(p)p Fl(\000)p Fn(1)918 351 y Fg(X)919 440 y Fj(j)r Fn(=0)985 391 y Fo(\020)1006 373 y Fl(\000)p Fj(ij)1061 391 y Fo(u)1085 373 y Fj(j)1102 391 y Fp(\(1)g Fm(\012)h Fo(x)1214 397 y Fj(s)1231 391 y Fp(\))257 520 y(Therefore,)15 b(w)o(e)f(ha)o(v)o(e:)310 647 y Fo(f)330 653 y Fj(A)357 647 y Fp(\()p Fo(e)392 653 y Fj(i)416 647 y Fm(\012)9 b Fo(x)481 653 y Fj(s)499 647 y Fp(\))j(=)575 619 y(1)p 575 637 V 575 675 a Fo(p)608 593 y Fj(p)p Fl(\000)p Fn(1)608 607 y Fg(X)609 696 y Fj(j)r Fn(=0)675 647 y Fo(\020)696 630 y Fl(\000)p Fj(ij)751 647 y Fo(f)771 653 y Fj(A)799 647 y Fp(\()p Fo(u)839 630 y Fj(j)856 647 y Fp(\(1)d Fm(\012)h Fo(x)968 653 y Fj(s)985 647 y Fp(\)\))i(=)1078 619 y(1)p 1078 637 V 1078 675 a Fo(p)1111 593 y Fj(p)p Fl(\000)p Fn(1)1111 607 y 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1255 y Fn(\()p Fj(f)s Fn(\()p Fj(s)p Fn(\)\))p Fj(;k)q(j)889 1248 y Fo(\020)910 1231 y Fj(k)q(iq)957 1218 y Fh(0)968 1231 y Fn(\()p Fj(f)s Fn(\()p Fj(s)p Fn(\))p Fj(;f)s Fn(\()p Fj(t)p Fn(\)\))1124 1248 y Fo(\020)1145 1231 y Fj(k)1164 1218 y Fd(2)1180 1231 y Fj(i)1192 1218 y Fd(2)1208 1231 y Fj(\027)1227 1218 y Fh(0)1238 1231 y Fn(\()p Fj(f)s Fn(\()p Fj(s)p Fn(\)\))p Fj(\014)1345 1218 y Fh(0)1357 1231 y Fn(\()p Fj(f)s Fn(\()p Fj(s)p Fn(\)\))p Fj(\013)1466 1218 y Fh(0)1476 1231 y Fn(\()p Fj(f)s Fn(\()p Fj(t)p Fn(\)\))p Fj(=)p Fn(2)1351 1310 y Fp(\()p Fo(e)1386 1316 y Fj(k)q(i)1428 1310 y Fm(\012)e Fo(x)1493 1293 y Fl(0)1493 1322 y Fj(f)s Fn(\()p Fj(s)p Fn(\))p Fj(f)s Fn(\()p Fj(t)p Fn(\))1614 1310 y Fp(\))392 1389 y(=)i Fo(\020)456 1371 y Fj(k)q(i)p Fn(\()p Fj(w)q Fn(\()p Fj(s)p Fn(\)+)p Fj(\027)r Fn(\()p Fj(s)p Fn(\))p Fj(w)q Fn(\()p Fj(t)p Fn(\)\))730 1389 y Fo(\016)748 1396 y Fj(i\027)r Fn(\()p Fj(s)p Fn(\))p Fj(;j)848 1389 y Fo(\020)869 1371 y Fj(k)q(iq)916 1359 y Fh(0)927 1371 y 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y(assertion.)j(Since)g Fo(q)g Fp(and)f Fo(q)691 1791 y Fl(0)718 1806 y Fp(are)h(normalized,)d (w)o(e)j(get,)f(b)o(y)g(inserting)h Fo(s)e Fp(=)h Fo(t)f Fp(=)g(1)h(in)g(the)257 1856 y(previous)g(equation,)e(the)h(fact)g (that)g Fo(w)q Fp(\(1\))d(=)h(0.)h Fi(2)257 1990 y Fq(4.6)48 b Fp(In)18 b(this)g(paragraph,)f(w)o(e)h(con)o(tin)o(ue)g(our)g (analysis)f(of)h(the)g(case)h Fo(R)f Fp(=)g Fk(Z)1544 1996 y Fj(p)1578 1990 y Fp(for)g(an)257 2040 y(o)q(dd)13 b(prime)e Fo(p)p Fp(,)h(but)h(sp)q(ecialize)g(the)g(situation)f (further)i(to)e(the)h(case)h(where)f(also)f Fo(G)f Fp(=)h Fk(Z)1662 2046 y Fj(p)1678 2040 y Fp(.)257 2089 y(W)m(e)19 b(w)o(an)o(t)f(to)h(determine)g(the)g(isomorphism)d(classes)k(of)f(non) o(trivial)e(Y)m(etter-Drinfel'd)257 2139 y(Hopf)d(algebras)g(among)e (the)i(algebras)g Fo(A)919 2145 y Fj(p)939 2139 y Fp(\()p Fo(\013;)7 b(\014)r(;)g(q)q Fp(\).)13 b(W)m(e)g(shall)h(use)h(the)f (notation)f(of)h(the)257 2189 y(previous)h(paragraph.)257 2273 y(The)g(algebras)e Fo(A)535 2279 y Fj(p)555 2273 y Fp(\()p Fo(\013;)7 b(\014)r(;)g(q)q Fp(\))13 b(are)i(de\014ned)f (with)g(resp)q(ect)i(to)e(the)g(c)o(haracters)577 2366 y Fo(\037)d Fp(:)g Fk(Z)668 2372 y Fj(p)696 2366 y Fm(!)g Fo(K)q(;)c(i)k Fm(7!)g Fo(\020)903 2349 y Fj(i=)p Fn(2)1034 2366 y Fo(\021)h Fp(:)f Fk(Z)1121 2372 y Fj(p)1149 2366 y Fm(!)g Fo(K)q(;)c(i)k Fm(7!)g Fo(\020)1356 2349 y Fj(i)257 2453 y Fp(and)j(a)g(group)f(homomorphism)861 2503 y Fo(\027)h Fp(:)d Fk(Z)949 2509 y Fj(p)977 2503 y Fm(!)g Fk(Z)1061 2486 y Fl(\002)1061 2513 y Fj(p)953 2628 y Fp(58)p eop %%Page: 59 59 59 58 bop 257 262 a Fp(Ob)o(viously)m(,)14 b(the)h(only)g(group)f (homomorphism)d(of)j(this)h(t)o(yp)q(e)h(is)f(the)g(trivial)f(homomo)o (r-)257 311 y(phism)d(that)g(is)h(iden)o(tically)e(equal)i(to)f(one.)h (Therefore,)g(in)f(this)h(situation,)e Fk(Z)1480 317 y Fj(p)1508 311 y Fp(is)h(a)h(trivial)257 361 y Fk(Z)288 367 y Fj(p)304 361 y Fp(-mo)q(dule,)17 b(and)h(the)h(1-co)q(cycles)g Fo(\013)f Fp(and)g Fo(\014)j Fp(are)e(ordinary)f(group)g(homomorphism)o (s.)257 411 y(If)h(w)o(e)h(require)g(that)f Fo(A)642 417 y Fj(p)661 411 y Fp(\()p Fo(\013;)7 b(\014)r(;)g(q)q Fp(\))19 b(b)q(e)h(non)o(trivial,)d Fo(\013)i Fp(and)g Fo(\014)j Fp(ha)o(v)o(e)d(to)g(b)q(e)h(nonzero)g(b)o(y)257 461 y(Prop)q(osition)14 b(1.11.)257 543 y(F)m(or)g Fo(m)e Fm(2)f Fk(Z)450 528 y Fl(\002)450 553 y Fj(p)475 543 y Fp(,)i(w)o(e)h(denote)h(b)o(y)f Fo(\013)780 549 y Fj(m)825 543 y Fp(the)g(group)g(homomo)o(rphism)774 626 y Fo(\013)801 632 y Fj(m)844 626 y Fp(:)d Fk(Z)897 632 y Fj(p)925 626 y Fm(!)g Fk(Z)1009 632 y Fj(p)1025 626 y Fo(;)c(i)k Fm(7!)g Fo(mi)257 709 y Fp(Then,)19 b Fo(\013)409 715 y Fn(1)427 709 y Fo(;)7 b(:)g(:)g(:)12 b(;)7 b(\013)554 715 y Fj(p)p Fl(\000)p Fn(1)634 709 y Fp(are)19 b(all)e(nonzero)j(group)e (homomorphism)o(s)e(from)h Fk(Z)1501 715 y Fj(p)1536 709 y Fp(to)h(itself.)257 759 y(F)m(rom)f(Prop)q(osition)i(1.13.2,)d(w) o(e)j(kno)o(w)f(that)h(the)g(second)h(cohomology)c(group)j(of)f(the)257 809 y(trivial)11 b Fk(Z)409 815 y Fj(p)425 809 y Fp(-mo)q(dule)f Fo(H)621 794 y Fn(2)640 809 y Fp(\()p Fk(Z)687 815 y Fj(p)703 809 y Fo(;)d Fk(Z)752 815 y Fj(p)768 809 y Fp(\))12 b(is)f(isomorphic)f(to)i Fk(Z)1122 815 y Fj(p)1139 809 y Fp(.)f(W)m(e)g(c)o(ho)q(ose)h(a)g(complete)f(system)257 859 y(of)19 b(represen)o(tativ)o(es)i(for)d(these)j(cohomology)16 b(classes,)k(i.)e(e.,)g(co)q(cycles)i Fo(q)1444 865 y Fn(0)1463 859 y Fo(;)7 b(:)g(:)g(:)k(;)c(q)1581 865 y Fj(p)p Fl(\000)p Fn(1)1662 859 y Fm(2)257 909 y Fo(Z)288 894 y Fn(2)307 909 y Fp(\()p Fk(Z)354 915 y Fj(p)370 909 y Fo(;)g Fk(Z)419 915 y Fj(p)436 909 y Fp(\))14 b(suc)o(h)g(that)g Fo(H)687 894 y Fn(2)706 909 y Fp(\()p Fk(Z)752 915 y Fj(p)769 909 y Fo(;)7 b Fk(Z)818 915 y Fj(p)834 909 y Fp(\))12 b(=)f Fm(f)s Fp(\026)-24 b Fo(q)945 915 y Fn(0)963 909 y Fo(;)7 b(:)g(:)g(:)12 b(;)e Fp(\026)-24 b Fo(q)1082 915 y Fj(p)p Fl(\000)p Fn(1)1143 909 y Fm(g)p Fp(.)257 999 y Fq(Prop)q(osition)308 1048 y Fp(1.)20 b(Supp)q(ose)14 b(that)e 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(algebras)g Fo(A)1054 1284 y Fj(p)1073 1278 y Fp(\()p Fo(\013)1116 1284 y Fj(m)1147 1278 y Fo(;)7 b Fp(id)o Fo(;)g(q)1238 1284 y Fj(n)1260 1278 y Fp(\),)16 b(for)h Fo(m)g Fp(=)g(1)p Fo(;)7 b(:)g(:)g(:)12 b(;)7 b(p)j Fm(\000)i Fp(1)361 1327 y(and)i Fo(n)d Fp(=)h(0)p Fo(;)7 b(:)g(:)g(:)12 b(;)7 b(p)h Fm(\000)i Fp(1,)j(are)h(m)o(utually)d(nonisomorphic.)257 1422 y Fq(Pro)q(of.)36 b Fp(T)m(o)15 b(pro)o(v)o(e)h(the)g(\014rst)g (statemen)o(t,)f(de\014ne)i Fo(k)e Fp(:=)f Fo(\014)r Fp(\(1\).)i(Since)g(group)g(endomor-)257 1472 y(phisms)g(of)f Fk(Z)479 1478 y Fj(p)511 1472 y Fp(are)i(determined)g(b)o(y)f(their)g (v)n(alue)g(on)g(1,)g(w)o(e)g(then)h(ha)o(v)o(e)f Fo(\014)j Fp(=)c Fo(k)8 b Fp(id)16 b(and,)257 1522 y(since)f Fo(k)q(\013)409 1528 y Fj(m)440 1522 y Fp(\(1\))d(=)h Fo(k)q(m)f Fp(=)h Fo(\013)p Fp(\(1\),)g(w)o(e)h(also)g(ha)o(v)o(e)g Fo(\013)e Fp(=)g Fo(k)q(\013)1145 1528 y Fj(m)1176 1522 y Fp(.)i(Since)h Fo(q)g Fp(is)f(cohomologous)e(to)257 1572 y Fo(k)q(q)299 1578 y Fj(n)321 1572 y Fp(,)i(there)h(is)f(a)f(1-co)q(c)o(hain)h Fo(w)e Fp(:)f Fk(Z)810 1578 y Fj(p)838 1572 y Fm(!)g Fk(Z)921 1578 y Fj(p)951 1572 y Fp(suc)o(h)k(that)555 1655 y Fo(q)q Fp(\()p Fo(i;)7 b(j)r Fp(\))12 b(=)g Fo(k)q(q)757 1661 y Fj(n)779 1655 y Fp(\()p Fo(i;)7 b(j)r Fp(\))j(+)f Fo(k)q(w)q Fp(\()p Fo(j)r Fp(\))h Fm(\000)f Fo(k)q(w)q Fp(\()p Fo(i)h Fp(+)f Fo(j)r Fp(\))h(+)g Fo(k)q(w)q Fp(\()p Fo(i)p Fp(\))257 1738 y(It)k(no)o(w)g(follo)o(ws)e(from)g(Prop)q (osition)i(4.3)f(that)h(the)g(map)448 1822 y Fo(f)468 1828 y Fj(A)507 1822 y Fp(:)d Fo(A)561 1828 y Fj(p)580 1822 y Fp(\()p Fo(\013;)c(\014)r(;)g(q)q Fp(\))k Fm(!)g Fo(A)817 1828 y Fj(p)837 1822 y Fp(\()p Fo(\013)880 1828 y Fj(m)911 1822 y Fo(;)c Fp(id)o Fo(;)g(q)1002 1828 y Fj(n)1023 1822 y Fp(\))p Fo(;)g(e)1077 1828 y Fj(i)1100 1822 y Fm(\012)j Fo(c)1160 1828 y Fj(j)1189 1822 y Fm(7!)h Fo(\020)1263 1804 y Fj(k)q(iw)q Fn(\()p Fj(j)r Fn(\))1361 1822 y Fo(e)1380 1828 y Fj(k)q(i)1422 1822 y Fm(\012)f Fo(c)1482 1828 y Fj(j)257 1905 y Fp(is)k(an)g(isomorphism.)257 1987 y(T)m(o)f(pro)o(v)o(e)h(the)h(second)g(statemen)o(t,)e(supp)q(ose) i(that)636 2070 y Fo(f)656 2076 y Fj(A)695 2070 y Fp(:)c Fo(A)749 2076 y Fj(p)768 2070 y Fp(\()p Fo(\013)811 2076 y Fj(m)842 2070 y Fo(;)c Fp(id)o Fo(;)g(q)933 2076 y Fj(n)955 2070 y Fp(\))k Fm(!)g Fo(A)1066 2076 y Fj(p)1086 2070 y Fp(\()p Fo(\013)1129 2076 y Fj(m)1158 2068 y Fh(0)1171 2070 y Fo(;)c Fp(id)o Fo(;)g(q)1262 2076 y Fj(n)1283 2068 y Fh(0)1295 2070 y Fp(\))257 2154 y(is)20 b(an)f(isomorphism.)d(F) m(rom)i(Prop)q(osition)h(4.5,)f(w)o(e)h(kno)o(w)g(that)h(there)h (exists)f(an)f(ele-)257 2203 y(men)o(t)14 b Fo(k)f Fm(2)f Fk(Z)468 2188 y Fl(\002)468 2214 y Fj(p)493 2203 y Fp(,)i(a)g(group)g (isomorphism)e Fo(f)17 b Fp(:)11 b Fk(Z)1007 2209 y Fj(p)1035 2203 y Fm(!)h Fk(Z)1120 2209 y Fj(p)1136 2203 y Fp(,)i(and)g(a)g(1-co)q (c)o(hain)g Fo(w)f Fp(:)f Fk(Z)1560 2209 y Fj(p)1589 2203 y Fm(!)g Fk(Z)1673 2209 y Fj(p)257 2253 y Fp(satisfying)h Fo(w)q Fp(\(0\))f(=)g(0)h(suc)o(h)i(that)f 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Fl(0)1063 583 y Fp(.)i Fi(2)257 698 y Fp(Therefore,)h(the)g (algebras)f Fo(A)719 704 y Fj(p)738 698 y Fp(\()p Fo(\013;)7 b(\014)r(;)g(q)q Fp(\))12 b(fall)g(in)o(to)g Fo(p)p Fp(\()p Fo(p)7 b Fm(\000)h Fp(1\))13 b(isomorphism)d(classes,)k(whic)o(h)257 748 y(are)j(represen)o(ted)j(b)o(y)c(the)h(algebras)g Fo(A)883 754 y Fj(p)902 748 y Fp(\()p Fo(\013)945 754 y Fj(m)976 748 y Fo(;)7 b Fp(id)o Fo(;)g(q)1067 754 y Fj(n)1089 748 y Fp(\),)16 b(for)h Fo(m)f Fp(=)g(1)p Fo(;)7 b(:)g(:)g(:)12 b(;)7 b(p)j Fm(\000)i Fp(1)k(and)g Fo(n)g Fp(=)257 797 y(0)p Fo(;)7 b(:)g(:)g(:)12 b(;)7 b(p)i Fm(\000)i Fp(1.)j(Note)i(that,)f(b)o(y)g(Prop)q(osition)g(1.13.2,)d (these)17 b(algebras)e(are)h(also)e(comm)o(u-)257 847 y(tativ)o(e.)257 980 y Fq(4.7)48 b Fp(The)18 b(framew)o(ork)f (considered)i(in)e(P)o(aragraph)h(4.3)e(can)i(also)g(b)q(e)g(applied)f (to)h(the)257 1030 y(case)g Fo(p)f Fp(=)g(2.)g(Supp)q(ose)h(that)f Fo(\023)g Fp(is)g(a)f(primitiv)o(e)f(fourth)i(ro)q(ot)g(of)g(unit)o(y.) f(W)m(e)h(denote)h(b)o(y)257 1080 y(^)-21 b Fo(\023)12 b Fp(:)f Fk(Z)338 1086 y Fn(2)365 1080 y Fm(!)g Fk(Z)449 1086 y Fn(4)479 1080 y Fp(the)j(unique)g(injectiv)o(e)g(group)g (homomorphism)c(and)j(b)o(y)j(^)-23 b Fo(\031)13 b Fp(:)e Fk(Z)1491 1086 y Fn(4)1518 1080 y Fm(!)h Fk(Z)1602 1086 y Fn(2)1632 1080 y Fp(the)257 1130 y(unique)i(surjectiv)o(e)h(group)f (homomorphi)o(sm)o(.)257 1213 y(Supp)q(ose)f(that)f Fo(G)539 1198 y Fl(0)563 1213 y Fp(is)g(a)f(\014nite)i(group.)e(W)m(e)h(regard)g Fk(Z)1095 1219 y Fn(2)1123 1213 y Fp(as)g(a)g(trivial)e Fo(G)1359 1198 y Fl(0)1371 1213 y Fp(-mo)q(dule.)g(Supp)q(ose)257 1263 y(that)15 b Fo(\013)375 1248 y Fl(0)400 1263 y Fp(:)e Fo(G)458 1248 y Fl(0)482 1263 y Fm(!)g Fk(Z)567 1269 y Fn(2)598 1263 y Fp(and)h Fo(\014)704 1248 y Fl(0)730 1263 y Fp(:)f Fo(G)788 1248 y Fl(0)812 1263 y Fm(!)g Fk(Z)897 1269 y Fn(2)928 1263 y Fp(are)i(1-co)q(cycles;)h(since)f(the)h Fo(G)1413 1248 y Fl(0)1424 1263 y Fp(-mo)q(dule)d(struc-)257 1313 y(ture)j(is)f(trivial,)e(these)j(are)g(just)f(group)g(homom)o (orphism)o(s.)d(In)j(addition,)e(supp)q(ose)j(that)257 1362 y Fo(q)277 1347 y Fl(0)301 1362 y Fm(2)11 b Fo(Z)371 1347 y Fn(2)390 1362 y Fp(\()p Fo(G)439 1347 y Fl(0)450 1362 y Fo(;)c Fk(Z)499 1368 y Fn(4)515 1362 y Fp(\))14 b(is)g(a)f(normalized)f(2-co)q(cycle)j(of)e(the)i(trivial)d Fo(G)1286 1347 y Fl(0)1298 1362 y Fp(-mo)q(dule)g Fk(Z)1489 1368 y Fn(4)1518 1362 y Fp(satisfying)839 1448 y(^)-23 b Fo(\031)10 b Fm(\016)f Fo(q)921 1431 y Fl(0)944 1448 y Fp(=)j Fo(\014)1013 1431 y Fl(0)1035 1448 y Fm([)d Fo(\013)1099 1431 y Fl(0)257 1534 y Fp(F)m(or)k Fo(i)f Fm(2)f Fk(Z)427 1540 y Fn(2)443 1534 y Fp(,)i(w)o(e)g(denote)i(the)f (corresp)q(onding)g(primitiv)o(e)d(idemp)q(oten)o(tin)i Fo(K)1468 1519 y Fc(Z)1490 1523 y Fd(2)1519 1534 y Fp(b)o(y)g Fo(e)1595 1540 y Fj(i)1623 1534 y Fp(and)257 1584 y(the)20 b(corresp)q(onding)h(canonical)d(basis)i(v)o(ector)g(of)e Fo(K)s Fp([)p Fk(Z)1165 1590 y Fn(2)1180 1584 y Fp(])h(b)o(y)g Fo(c)1292 1590 y Fj(i)1306 1584 y Fp(;)g(the)h(canonical)e(basis)257 1634 y(v)o(ectors)e(of)d Fo(K)s Fp([)p Fo(G)529 1619 y Fl(0)540 1634 y Fp(])h(are)g(denoted)h(b)o(y)f Fo(x)874 1619 y Fl(0)874 1645 y Fj(s)890 1637 y Fh(0)903 1634 y Fp(,)f(for)h Fo(s)1011 1619 y Fl(0)1035 1634 y Fm(2)d Fo(G)1107 1619 y Fl(0)1119 1634 y Fp(.)i(W)m(e)h(de\014ne)h Fo(\033)1361 1619 y Fl(0)1360 1645 y Fj(i)1385 1634 y Fp(:)d Fo(G)1442 1619 y Fl(0)1462 1634 y Fm(\002)e Fo(G)1537 1619 y Fl(0)1560 1634 y Fm(!)i Fo(K)1652 1619 y Fc(Z)1674 1623 y Fd(2)257 1684 y Fp(b)o(y)638 1734 y Fo(\033)663 1717 y Fl(0)662 1744 y Fn(0)681 1734 y Fp(\()p Fo(s)716 1717 y Fl(0)728 1734 y Fo(;)7 b(t)762 1717 y Fl(0)773 1734 y Fp(\))12 b(:=)f(1)83 b Fo(\033)985 1717 y Fl(0)984 1744 y Fn(1)1002 1734 y Fp(\()p Fo(s)1037 1717 y Fl(0)1050 1734 y Fo(;)7 b(t)1084 1717 y Fl(0)1095 1734 y Fp(\))k(:=)h Fo(\023)1193 1717 y Fj(q)1209 1704 y Fh(0)1220 1717 y Fn(\()p Fj(s)1249 1704 y Fh(0)1260 1717 y Fj(;t)1283 1704 y Fh(0)1293 1717 y Fn(\))257 1805 y Fp(and)i(set:)653 1855 y Fo(\033)678 1838 y Fl(0)690 1855 y Fp(\()p Fo(s)725 1838 y Fl(0)737 1855 y Fo(;)7 b(t)771 1838 y Fl(0)782 1855 y Fp(\))12 b(:=)f Fo(\033)890 1838 y Fl(0)889 1865 y Fn(0)908 1855 y Fp(\()p Fo(s)943 1838 y Fl(0)955 1855 y Fo(;)c(t)989 1838 y Fl(0)1000 1855 y Fp(\))p Fo(e)1035 1861 y Fn(0)1063 1855 y Fp(+)j Fo(\033)1130 1838 y Fl(0)1129 1865 y Fn(1)1147 1855 y Fp(\()p Fo(s)1182 1838 y Fl(0)1194 1855 y Fo(;)d(t)1228 1838 y Fl(0)1240 1855 y Fp(\))p Fo(e)1275 1861 y Fn(1)257 1926 y Fp(As)17 b(explained)f(in)g(P)o (aragraph)g(3.5,)f(these)j(data)e(can)h(b)q(e)g(used)g(to)f(construct)i (a)f(Y)m(etter-)257 1976 y(Drinfel'd)c(Hopf)g(algebra)h(o)o(v)o(er)g Fo(H)g Fp(:=)e Fo(K)s Fp([)p Fk(Z)952 1982 y Fn(2)968 1976 y Fp(],)h(whic)o(h)g(w)o(e)h(denote)h(b)o(y)f Fo(A)1407 1961 y Fl(0)1419 1976 y Fp(.)257 2060 y(No)o(w)j(supp)q(ose)h(that)f Fo(G)g Fp(is)g(another)g(\014nite)h(group)e(and)h(that)g Fo(f)22 b Fp(:)16 b Fo(G)h Fm(!)f Fo(G)1475 2044 y Fl(0)1503 2060 y 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b(t)p Fp(\))k Fm(7!)g Fo(q)q Fp(\()p Fo(s;)c(t)p Fp(\))12 b(:=)g Fo(q)964 748 y Fl(0)975 765 y Fp(\()p Fo(f)t Fp(\()p Fo(s)p Fp(\))p Fo(;)7 b(f)t Fp(\()p Fo(t)p Fp(\)\))12 b(+)d(^)-21 b Fo(\023)p Fp(\()p Fo(w)q Fp(\()p Fo(t)p Fp(\))9 b Fm(\000)h Fo(w)q Fp(\()p Fo(st)p Fp(\))g(+)f Fo(w)q Fp(\()p Fo(s)p Fp(\)\))257 841 y(F)m(rom)f Fo(\013)p Fp(,)h Fo(\014)r Fp(,)h(and)g Fo(q)h Fp(w)o(e)f(can)g(construct)i (another)e(Y)m(etter-Drinfel'd)f(Hopf)h(algebra)f(o)o(v)o(er)h Fo(H)s Fp(,)257 890 y(whic)o(h)k(w)o(e)g(denote)h(b)o(y)f Fo(A)p Fp(.)257 981 y Fq(Prop)q(osition)33 b Fp(The)15 b(map)591 1056 y Fo(f)611 1062 y Fj(A)650 1056 y Fp(:)c Fo(A)g Fm(!)h Fo(A)800 1039 y Fl(0)811 1056 y Fo(;)7 b(e)849 1062 y Fj(i)872 1056 y Fm(\012)j Fo(x)938 1062 y Fj(s)967 1056 y Fm(7!)h Fp(\()p Fm(\000)p Fp(1\))1105 1039 y Fj(iw)q Fn(\()p Fj(s)p Fn(\))1185 1056 y Fo(e)1204 1062 y Fj(i)1228 1056 y Fm(\012)e Fo(x)1293 1039 y Fl(0)1293 1067 y Fj(f)s Fn(\()p Fj(s)p Fn(\))257 1131 y Fp(is)14 b(a)g(morphism)d(of)i(Y)m(etter-Drinfel'd)h(Hopf)f(algebras.)257 1221 y Fq(Pro)q(of.)36 b Fp(By)14 b(de\014nition,)f Fo(A)715 1206 y Fl(0)740 1221 y Fp(is)h(the)g(Y)m(etter-Drinfel'd)g(Hopf)f (algebra)g(arising)g(from)f(the)257 1271 y(construction)23 b(describ)q(ed)g(in)e(P)o(aragraph)g(3.2,)f(using)i(the)g(structure)h (elemen)o(ts)f Fo(z)1626 1256 y Fl(0)1662 1271 y Fm(2)257 1321 y Fo(Z)288 1306 y Fn(1)307 1321 y Fp(\()p Fo(G)356 1306 y Fl(0)368 1321 y Fo(;)7 b Fp(Hom)n(\()p Fk(Z)519 1327 y Fn(2)535 1321 y Fo(;)g Fk(Z)584 1327 y Fn(2)600 1321 y Fp(\)\))14 b(and)g Fo(\015)750 1306 y Fl(0)774 1321 y Fm(2)d Fo(Z)844 1306 y Fn(1)863 1321 y Fp(\()p Fo(G)912 1306 y Fl(0)923 1321 y Fo(;)c Fp(Hom)n(\()p Fk(Z)1075 1327 y Fn(2)1091 1321 y Fo(;)1112 1310 y Fp(^)1110 1321 y Fk(Z)1140 1327 y Fn(2)1155 1321 y Fp(\)\))14 b(de\014ned)h(as:) 569 1402 y Fo(z)590 1385 y Fl(0)588 1412 y Fj(s)604 1404 y Fh(0)618 1402 y Fp(\()p Fo(i)p Fp(\))d(:=)f Fo(i\014)770 1385 y Fl(0)783 1402 y Fp(\()p Fo(s)818 1385 y Fl(0)830 1402 y Fp(\))83 b(\()p Fo(\015)968 1385 y Fl(0)966 1412 y Fj(s)982 1404 y Fh(0)996 1402 y Fp(\()p Fo(i)p Fp(\)\)\()p Fo(j)r Fp(\))13 b(:=)e(\()p Fm(\000)p Fp(1\))1262 1385 y Fj(ij)r(\013)1312 1372 y Fh(0)1322 1385 y Fn(\()p Fj(s)1351 1372 y Fh(0)1362 1385 y Fn(\))257 1477 y Fp(and)f Fo(\033)359 1462 y Fl(0)383 1477 y Fm(2)h Fo(Z)453 1462 y Fn(2)472 1477 y Fp(\()p Fo(G)521 1462 y Fl(0)532 1477 y Fo(;)c(U)e Fp(\()p Fo(K)638 1462 y Fc(Z)660 1466 y Fd(2)676 1477 y Fp(\)\))10 b(as)h(de\014ned)g(ab)q(o)o(v)o(e.)f(De\014ne,)g(for)g Fo(i)i Fm(2)f Fk(Z)1323 1483 y Fn(2)1349 1477 y Fp(and)f Fo(s)i Fm(2)f Fo(G)p Fp(,)e Fo(\034)1568 1483 y Fj(i)1582 1477 y Fp(\()p Fo(s)p Fp(\))k(:=)257 1527 y(\()p Fm(\000)p Fp(1\))342 1512 y Fj(iw)q Fn(\()p Fj(s)p Fn(\))423 1527 y Fp(.)e(Using)h(the)g(maps)e Fo(f)756 1533 y Fc(Z)779 1537 y Fd(2)806 1527 y Fp(:=)h(id)g(and)h Fo(f)1006 1533 y Fj(G)1046 1527 y Fp(:=)f Fo(f)t Fp(,)h(w)o(e)g(get)g(new)g(structure) i(elemen)o(ts)257 1581 y Fo(z)j Fm(2)c 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Fn(0)403 1933 y Fp(\()p Fo(s;)7 b(t)p Fp(\))12 b(=)g(1)83 b Fo(\033)672 1939 y Fn(1)690 1933 y Fp(\()p Fo(s;)7 b(t)p Fp(\))12 b(=)g Fo(\023)846 1918 y Fj(q)q Fn(\()p Fj(s;t)p Fn(\))257 2036 y Fp(where)j Fo(\033)q Fp(\()p Fo(s;)7 b(t)p Fp(\))12 b(=)543 2005 y Fg(P)587 2016 y Fn(1)587 2049 y Fj(i)p Fn(=0)650 2036 y Fo(\033)674 2042 y Fj(i)687 2036 y Fp(\()p Fo(s;)7 b(t)p Fp(\))p Fo(e)791 2042 y Fj(i)806 2036 y Fp(.)257 2115 y(Here,)15 b(the)g(last)e(equation)h(follo)o(ws)e(from)g (the)j(de\014nition)e(of)g Fo(\033)1248 2121 y Fj(i)1262 2115 y Fp(:)607 2190 y Fo(\033)631 2196 y Fj(i)645 2190 y Fp(\()p Fo(s;)7 b(t)p Fp(\))k(:=)h Fo(\033)822 2173 y Fl(0)821 2201 y Fj(i)834 2190 y Fp(\()p Fo(f)t Fp(\()p Fo(s)p Fp(\))p Fo(;)7 b(f)t Fp(\()p Fo(t)p Fp(\)\))p Fo(\034)1049 2196 y Fj(i)1066 2190 y Fp(\()p Fo(t)p Fp(\))p Fo(\034)1131 2196 y Fj(i)1145 2190 y Fp(\()p Fo(st)p Fp(\))1211 2173 y Fl(\000)p Fn(1)1256 2190 y Fo(\034)1274 2196 y Fj(i)1288 2190 y Fp(\()p Fo(s)p Fp(\))257 2266 y(This)14 b(ob)o(viously)f(yields)g Fo(\033)676 2272 y Fn(0)695 2266 y Fp(\()p Fo(s;)7 b(t)p Fp(\))k(=)h(1,)h(whereas)j(for) d Fo(i)f Fp(=)g(1)h(it)h(reads:)544 2347 y Fo(\033)568 2353 y Fn(1)587 2347 y Fp(\()p Fo(s;)7 b(t)p Fp(\))k(=)h Fo(\023)742 2330 y Fj(q)758 2317 y Fh(0)769 2330 y Fn(\()p Fj(f)s Fn(\()p Fj(s)p Fn(\))p Fj(;f)s Fn(\()p Fj(t)p Fn(\)\))926 2347 y Fp(\()p Fm(\000)p Fp(1\))1011 2330 y Fj(w)q Fn(\()p Fj(t)p Fn(\))p Fl(\000)p Fj(w)q Fn(\()p Fj(st)p Fn(\)+)p Fj(w)q Fn(\()p Fj(s)p Fn(\))683 2420 y Fp(=)g Fo(\023)742 2403 y Fj(q)758 2390 y Fh(0)769 2403 y Fn(\()p Fj(f)s Fn(\()p Fj(s)p Fn(\))p Fj(;f)s Fn(\()p Fj(t)p Fn(\)\)+^)-17 b Fj(\023)q Fn(\()p Fj(w)q Fn(\()p Fj(t)p Fn(\))p Fl(\000)p Fj(w)q Fn(\()p Fj(st)p Fn(\)+)p Fj(w)q Fn(\()p Fj(s)p Fn(\)\))1262 2420 y Fp(=)11 b Fo(\023)1320 2403 y Fj(q)q Fn(\()p Fj(s;t)p Fn(\))257 2495 y Fp(No)o(w,)i(the)i(assertion)f(follo)o(ws)f(from)f(Prop)q (osition)h(4.2.)g Fi(2)953 2628 y Fp(61)p eop %%Page: 62 62 62 61 bop 257 262 a Fq(4.8)48 b Fp(In)13 b(the)g(preceding)h (paragraph,)e(w)o(e)h(ha)o(v)o(e)f(constructed)j(homomo)o(rphism)o(s)10 b(of)i(Y)m(et-)257 311 y(ter-Drinfel'd)k(Hopf)g(algebras)g(in)g(the)h (case)h Fo(p)d Fp(=)h(2,)g(whic)o(h,)g(in)g(sp)q(ecial)g(cases,)h(ma)o (y)e(b)q(e)257 361 y(isomorphisms.)10 b(W)m(e)i(no)o(w)g(consider)h (the)g(question)g(whether)h(all)d(isomorphisms)f(are)j(nec-)257 411 y(essarily)d(of)g(this)g(form.)d(W)m(e)j(con)o(tin)o(ue)g(to)f(use) i(the)g(notation)e(of)g(the)h(preceding)h(paragraph.)257 494 y(Supp)q(ose)16 b(that)f Fo(G)f Fp(and)h Fo(G)674 479 y Fl(0)700 494 y Fp(are)g(\014nite)g(groups.)g(W)m(e)f(regard)h Fk(Z)1257 500 y Fn(2)1287 494 y Fp(as)g(a)g(trivial)e Fo(G)p Fp(-mo)q(dule,)257 544 y(resp.)18 b(as)g(a)f(trivial)f Fo(G)613 529 y Fl(0)624 544 y Fp(-mo)q(dule.)g(Supp)q(ose)i(that)f Fo(\013)h Fp(:)f Fo(G)g Fm(!)f Fk(Z)1272 550 y Fn(2)1305 544 y Fp(and)h Fo(\014)j Fp(:)d Fo(G)g Fm(!)g Fk(Z)1601 550 y Fn(2)1634 544 y Fp(are)257 594 y(1-co)q(cycles,)g(i.)f(e.,)g (group)g(homomo)o(rphism)o(s.)e(Similarl)o(y)m(,)f(supp)q(ose)k(that)g Fo(\013)1472 579 y Fl(0)1499 594 y Fp(:)e Fo(G)1559 579 y Fl(0)1586 594 y Fm(!)g Fk(Z)1674 600 y Fn(2)257 643 y Fp(and)h Fo(\014)365 628 y Fl(0)391 643 y Fp(:)e Fo(G)450 628 y Fl(0)476 643 y Fm(!)g Fk(Z)562 649 y Fn(2)593 643 y Fp(are)i(1-co)q(cycles.)g(In)g(addition,)e(supp)q(ose)j(that)e Fo(q)h Fm(2)e Fo(Z)1459 628 y Fn(2)1477 643 y Fp(\()p Fo(G;)7 b Fk(Z)1575 649 y Fn(4)1591 643 y Fp(\))16 b(and)257 693 y Fo(q)277 678 y Fl(0)301 693 y Fm(2)11 b Fo(Z)371 678 y Fn(2)390 693 y Fp(\()p Fo(G)439 678 y Fl(0)450 693 y Fo(;)c Fk(Z)499 699 y Fn(4)515 693 y Fp(\))14 b(are)g(normalized) e(2-co)q(cycles)j(satisfying:)678 779 y(^)-23 b Fo(\031)11 b Fm(\016)e Fo(q)j Fp(=)g Fo(\014)g Fm([)d Fo(\013)85 b Fp(^)-24 b Fo(\031)11 b Fm(\016)e Fo(q)1082 761 y Fl(0)1105 779 y Fp(=)j Fo(\014)1174 761 y Fl(0)1195 779 y Fm([)d Fo(\013)1259 761 y Fl(0)257 864 y Fp(W)m(e)14 b(de\014ne)h Fo(\033)473 870 y Fj(i)498 864 y Fp(:)c Fo(G)e Fm(\002)g Fo(G)i Fm(!)g Fo(K)739 849 y Fc(Z)762 853 y Fd(2)791 864 y Fp(b)o(y)679 949 y Fo(\033)703 955 y Fn(0)721 949 y Fp(\()p Fo(s;)c(t)p Fp(\))12 b(:=)f(1)83 b Fo(\033)1001 955 y Fn(1)1019 949 y Fp(\()p Fo(s;)7 b(t)p Fp(\))12 b(:=)f Fo(\023)1186 932 y Fj(q)q Fn(\()p Fj(s;t)p Fn(\))257 1035 y Fp(and)j(set:)694 1084 y Fo(\033)q Fp(\()p Fo(s;)7 b(t)p Fp(\))12 b(:=)f Fo(\033)895 1090 y Fn(0)914 1084 y Fp(\()p Fo(s;)c(t)p Fp(\))p Fo(e)1018 1090 y Fn(0)1046 1084 y Fp(+)i Fo(\033)1111 1090 y Fn(1)1130 1084 y Fp(\()p Fo(s;)e(t)p Fp(\))p Fo(e)1234 1090 y Fn(1)257 1156 y Fp(In)15 b(a)f(similar)f(w)o(a)o(y)m(,)g(w)o(e)i(also)f(de\014ne)h Fo(\033)867 1140 y Fl(0)866 1166 y Fn(0)885 1156 y Fp(,)f Fo(\033)936 1140 y Fl(0)935 1166 y Fn(1)953 1156 y Fp(,)g(and)h Fo(\033)1086 1140 y Fl(0)1098 1156 y Fp(.)f(As)h(explained)f(in)h(P)o (aragraph)f(3.5,)257 1205 y(these)21 b(data)d(can)h(b)q(e)g(used)h(to)e (construct)j(t)o(w)o(o)d(Y)m(etter-Drinfel'd)g(Hopf)h(algebras)f(o)o(v) o(er)257 1255 y Fo(H)d Fp(:=)c Fo(K)s Fp([)p Fk(Z)443 1261 y Fn(2)458 1255 y Fp(],)i(whic)o(h)h(w)o(e)g(denote)h(b)o(y)e Fo(A)h Fp(and)g Fo(A)1023 1240 y Fl(0)1035 1255 y Fp(.)257 1338 y(As)h(in)e(P)o(aragraph)h(1.10,)e(w)o(e)i(in)o(tro)q(duce)h(the)f (notation)321 1423 y Fo(\036)p Fp(\()p Fo(e)381 1429 y Fj(i)404 1423 y Fm(\012)c Fo(x)470 1429 y Fj(s)487 1423 y Fp(\))i(:=)f Fo(c)588 1429 y Fn(1)618 1423 y Fm(!)h Fp(\()p Fo(e)707 1429 y Fj(i)730 1423 y Fm(\012)e Fo(x)796 1429 y Fj(s)813 1423 y Fp(\))83 b Fo( )q Fp(\()p Fo(e)975 1429 y Fj(i)999 1423 y Fm(\012)10 b Fo(x)1065 1429 y Fj(s)1082 1423 y Fp(\))i(:=)f Fo(\015)r Fp(\(\()p Fo(e)1239 1429 y Fj(i)1264 1423 y Fm(\012)e Fo(x)1329 1429 y Fj(s)1347 1423 y Fp(\))1363 1406 y Fn(\(1\))1407 1423 y Fp(\)\()p Fo(e)1458 1429 y Fj(i)1482 1423 y Fm(\012)h Fo(x)1548 1429 y Fj(s)1565 1423 y Fp(\))1581 1406 y Fn(\(2\))257 1509 y Fp(where)i Fo(\015)j Fm(2)c Fo(G)p Fp(\()p Fo(H)536 1494 y Fl(\003)555 1509 y Fp(\))f(is)h(the)g(unique)g(non)o(trivial)e (c)o(haracter)j(of)f Fo(H)s Fp(;)f(it)g(satis\014es)i Fo(\015)r Fp(\()p Fo(c)1534 1515 y Fn(1)1553 1509 y Fp(\))g(=)g Fm(\000)p Fp(1.)257 1559 y(Similarly)m(,)e(w)o(e)k(in)o(tro)q(duce)h (endomorphisms)c Fo(\036)1008 1544 y Fl(0)1034 1559 y Fp(and)i Fo( )1142 1544 y Fl(0)1168 1559 y Fp(of)h Fo(A)1247 1544 y Fl(0)1258 1559 y Fp(.)257 1642 y(No)o(w)g(supp)q(ose)h(that)f Fo(f)618 1648 y Fj(A)657 1642 y Fp(:)d Fo(A)h Fm(!)f Fo(A)807 1627 y Fl(0)832 1642 y Fp(is)j(an)g(isomorphism.)257 1738 y Fq(Prop)q(osition)33 b Fp(Supp)q(ose)19 b(that)f Fo(\013)f Fm(6)p Fp(=)h(0)g(or)f Fo(\014)k Fm(6)p Fp(=)d(0.)f(Then)h (there)h(is)e(a)h(group)f(isomor-)257 1787 y(phism)h Fo(f)25 b Fp(:)20 b Fo(G)f Fm(!)h Fo(G)609 1772 y Fl(0)639 1787 y Fp(and)f(a)g(1-co)q(c)o(hain)f Fo(w)j Fp(:)f Fo(G)g Fm(!)26 b Fk(Z)1189 1793 y Fn(2)1223 1787 y Fp(satisfying)19 b Fo(w)q Fp(\(1\))h(=)g(0)f(suc)o(h)257 1837 y(that)308 1949 y(1.)h Fm(8)p Fo(s)12 b Fm(2)f Fo(G)h Fp(:)f Fo(\013)p Fp(\()p Fo(s)p Fp(\))h(=)f Fo(\013)682 1934 y Fl(0)694 1949 y Fp(\()p Fo(f)t Fp(\()p Fo(s)p Fp(\)\))308 2029 y(2.)20 b Fm(8)p Fo(s)12 b Fm(2)f Fo(G)h Fp(:)f Fo(\014)r Fp(\()p Fo(s)p Fp(\))i(=)e Fo(\014)679 2014 y Fl(0)692 2029 y Fp(\()p Fo(f)t Fp(\()p Fo(s)p Fp(\)\))308 2110 y(3.)20 b Fm(8)p Fo(s;)7 b(t)12 b Fm(2)f Fo(G)g Fp(:)g Fo(q)q Fp(\()p Fo(s;)c(t)p Fp(\))i Fm(\000)h Fo(q)731 2095 y Fl(0)743 2110 y Fp(\()p Fo(f)t Fp(\()p Fo(s)p Fp(\))p Fo(;)d(f)t Fp(\()p Fo(t)p Fp(\)\))14 b(=)d(^)-21 b Fo(\023)p Fp(\()p Fo(w)q Fp(\()p Fo(t)p Fp(\))10 b Fm(\000)f Fo(w)q Fp(\()p Fo(st)p Fp(\))h(+)f Fo(w)q Fp(\()p Fo(s)p Fp(\)\))308 2191 y(4.)20 b Fm(8)p Fo(i)12 b Fm(2)f Fk(Z)480 2197 y Fn(2)502 2191 y Fm(8)p Fo(s)i Fm(2)e Fo(G)g Fp(:)g Fo(f)683 2197 y Fj(A)710 2191 y Fp(\()p Fo(e)745 2197 y Fj(i)769 2191 y Fm(\012)e Fo(x)834 2197 y Fj(s)852 2191 y Fp(\))i(=)h(\()p Fm(\000)p Fp(1\))1008 2176 y Fj(iw)q Fn(\()p Fj(s)p Fn(\))1089 2191 y Fo(e)1108 2197 y Fj(i)1131 2191 y Fm(\012)e Fo(x)1197 2176 y Fl(0)1197 2204 y Fj(f)s Fn(\()p Fj(s)p Fn(\))257 2293 y Fq(Pro)q(of.)36 b Fp(\(1\))21 b(W)m(e)11 b(follo)o(w)f(the)i(line)f(of)g(reasoning)h (in)f(P)o(aragraph)h(4.5.)e(In)i(the)g(algebras)f Fo(A)257 2343 y Fp(resp.)k Fo(A)388 2328 y Fl(0)400 2343 y Fp(,)e(w)o(e)h (consider)h(the)f(elemen)o(ts)543 2465 y Fo(u)d Fp(:=)655 2413 y Fn(1)633 2426 y Fg(X)635 2514 y Fj(j)r Fn(=0)693 2465 y Fp(\()p Fm(\000)p Fp(1\))778 2448 y Fj(j)796 2465 y Fo(e)815 2471 y Fj(j)842 2465 y Fm(\012)f Fo(x)908 2471 y Fn(1)1009 2465 y Fo(u)1033 2448 y Fl(0)1056 2465 y Fp(:=)1133 2413 y Fn(1)1111 2426 y Fg(X)1113 2514 y Fj(j)r Fn(=0)1171 2465 y Fp(\()p Fm(\000)p Fp(1\))1256 2448 y Fj(j)1274 2465 y Fo(e)1293 2471 y Fj(j)1320 2465 y Fm(\012)g Fo(x)1386 2448 y Fl(0)1386 2475 y Fn(1)953 2628 y Fp(62)p eop %%Page: 63 63 63 62 bop 257 262 a Fp(W)m(e)14 b(then)g(ha)o(v)o(e)463 375 y Fo(u)p Fp(\()p Fo(e)522 381 y Fj(i)545 375 y Fm(\012)c Fo(x)611 381 y Fj(s)628 375 y Fp(\))i(=)722 323 y Fn(1)700 335 y Fg(X)701 424 y Fj(j)r Fn(=0)760 375 y Fp(\()p Fm(\000)p Fp(1\))845 358 y Fj(j)863 375 y Fp(\()p Fo(e)898 381 y Fj(j)925 375 y Fm(\012)d Fo(x)990 381 y Fn(1)1009 375 y Fp(\)\()p Fo(e)1060 381 y Fj(i)1083 375 y Fm(\012)h Fo(x)1149 381 y Fj(s)1166 375 y Fp(\))656 525 y(=)722 473 y Fn(1)700 485 y Fg(X)701 574 y Fj(j)r Fn(=0)760 525 y Fp(\()p Fm(\000)p Fp(1\))845 508 y Fj(j)863 525 y Fo(\016)881 531 y Fj(j)r(i)910 525 y Fo(\033)934 531 y Fj(j)952 525 y Fp(\(1)p Fo(;)d(s)p Fp(\)\()p Fo(e)1078 531 y Fj(j)1105 525 y Fm(\012)i Fo(x)1170 531 y Fj(s)1188 525 y Fp(\))i(=)h(\()p Fm(\000)p Fp(1\))1344 508 y Fj(i)1358 525 y Fo(e)1377 531 y Fj(i)1401 525 y Fm(\012)d Fo(x)1466 531 y Fj(s)257 659 y Fp(and)16 b(similarly)c Fo(u)535 644 y Fl(0)547 659 y Fp(\()p Fo(e)582 665 y Fj(i)606 659 y Fm(\012)f Fo(x)673 644 y Fl(0)673 670 y Fj(s)689 662 y Fh(0)702 659 y Fp(\))j(=)h(\()p Fm(\000)p Fp(1\))864 644 y Fj(i)878 659 y Fo(e)897 665 y Fj(i)921 659 y Fm(\012)c Fo(x)988 644 y Fl(0)988 670 y Fj(s)1004 662 y Fh(0)1017 659 y Fp(.)k(Since)h Fo(\036)p Fp(\()p Fo(e)1214 665 y Fj(i)1238 659 y Fm(\012)11 b Fo(x)1305 665 y Fj(s)1322 659 y Fp(\))k(=)g(\()p Fm(\000)p Fp(1\))1485 644 y Fj(i\013)p Fn(\()p Fj(s)p Fn(\))1562 659 y Fo(e)1581 665 y Fj(i)1605 659 y Fm(\012)c Fo(x)1672 665 y Fj(s)257 709 y Fp(and)j Fo( )q Fp(\()p Fo(e)401 715 y Fj(i)425 709 y Fm(\012)c Fo(x)491 715 y Fj(s)508 709 y Fp(\))i(=)g(\()p Fm(\000)p Fp(1\))665 694 y Fj(i\014)q Fn(\()p Fj(s)p Fn(\))741 709 y Fo(e)760 715 y Fj(i)783 709 y Fm(\012)e Fo(x)849 715 y Fj(s)866 709 y Fp(,)j(w)o(e)i(see)g(that:)479 793 y Fo(\036)p Fp(\()p Fo(e)539 799 y Fj(i)562 793 y Fm(\012)9 b Fo(x)627 799 y Fj(s)645 793 y Fp(\))i(=)h Fo(u)740 776 y Fj(\013)p Fn(\()p Fj(s)p Fn(\))805 793 y Fo(e)824 799 y Fj(i)848 793 y Fm(\012)d Fo(x)913 799 y Fj(s)1014 793 y Fo( )q Fp(\()p Fo(e)1077 799 y Fj(i)1101 793 y Fm(\012)g Fo(x)1166 799 y Fj(s)1184 793 y Fp(\))i(=)h Fo(u)1279 776 y Fj(\014)q Fn(\()p Fj(s)p Fn(\))1343 793 y Fo(e)1362 799 y Fj(i)1385 793 y Fm(\012)e Fo(x)1451 799 y Fj(s)257 877 y Fp(Similarly)m(,)g(w)o(e)k(ha)o(v)o(e:)410 961 y Fo(\036)435 944 y Fl(0)447 961 y Fp(\()p Fo(e)482 967 y Fj(i)505 961 y Fm(\012)c Fo(x)571 944 y Fl(0)571 971 y Fj(s)587 963 y Fh(0)599 961 y Fp(\))i(=)g Fo(u)695 944 y Fl(0)q Fj(\013)728 931 y Fh(0)739 944 y Fn(\()p Fj(s)768 931 y Fh(0)779 944 y Fn(\))794 961 y Fo(e)813 967 y Fj(i)836 961 y Fm(\012)e Fo(x)902 944 y Fl(0)902 971 y Fj(s)918 963 y Fh(0)1014 961 y Fo( )1042 944 y Fl(0)1054 961 y Fp(\()p Fo(e)1089 967 y Fj(i)1112 961 y Fm(\012)g Fo(x)1178 944 y Fl(0)1178 971 y Fj(s)1194 963 y Fh(0)1207 961 y Fp(\))h(=)h Fo(u)1302 944 y Fl(0)q Fj(\014)1333 931 y Fh(0)1345 944 y Fn(\()p Fj(s)1374 931 y Fh(0)1385 944 y Fn(\))1400 961 y Fo(e)1419 967 y Fj(i)1442 961 y Fm(\012)e Fo(x)1508 944 y Fl(0)1508 971 y Fj(s)1524 963 y Fh(0)257 1060 y Fp(\(2\))21 b(The)c(grouplik)o(e) e(elemen)o(ts)h(of)g(the)h(dual)e(group)i(ring)e Fo(K)1237 1045 y Fc(Z)1260 1049 y Fd(2)1291 1060 y Fp(are)i(the)g(F)m(ourier)f (trans-)257 1114 y(formed)10 b(elemen)o(ts)561 1083 y Fg(P)604 1093 y Fn(1)604 1127 y Fj(j)r Fn(=0)664 1114 y Fp(\()p Fm(\000)p Fp(1\))749 1099 y Fj(ij)778 1114 y Fo(e)797 1120 y Fj(j)826 1114 y Fp(of)g(the)i(idemp)q(oten)o(ts)e Fo(e)1190 1120 y Fj(j)1208 1114 y Fp(.)g(Since)h(the)h(coalgebra)e (struc-)257 1164 y(ture)j(of)e Fo(A)h Fp(is)f(the)h(ordinary)f(tensor)i (pro)q(duct)g(coalgebra)e(structure,)i(the)f(elemen)o(ts)g(of)f(the)257 1214 y(form)353 1183 y Fg(P)397 1193 y Fn(1)397 1226 y Fj(j)r Fn(=0)456 1214 y Fp(\()p Fm(\000)p Fp(1\))541 1199 y Fj(ij)571 1214 y Fo(e)590 1220 y Fj(j)612 1214 y Fm(\012)t Fo(x)672 1220 y Fj(s)701 1214 y Fp(constitute)h(a)f(basis)h (of)e Fo(A)i Fp(consisting)f(of)g(grouplik)o(e)f(elemen)o(ts.)257 1264 y(These)16 b(elemen)o(ts)d(can)h(b)q(e)h(written)f(in)g(the)g (form:)724 1325 y Fn(1)703 1337 y Fg(X)704 1426 y Fj(j)r Fn(=0)763 1377 y Fp(\()p Fm(\000)p Fp(1\))848 1360 y Fj(ij)877 1377 y Fo(e)896 1383 y Fj(j)923 1377 y Fm(\012)c Fo(x)989 1383 y Fj(s)1018 1377 y Fp(=)h Fo(u)1085 1360 y Fj(i)1099 1377 y Fp(\(1)e Fm(\012)h Fo(x)1211 1383 y Fj(s)1228 1377 y Fp(\))257 1505 y Fo(A)288 1490 y Fl(0)317 1505 y Fp(has)17 b(a)f(similar)f(basis)h(consisting)h(of)f(grouplik)o (e)g(elemen)o(ts.)g(Since)i Fo(f)1419 1511 y Fj(A)1463 1505 y Fp(tak)o(es)f(group-)257 1555 y(lik)o(e)f(elemen)o(ts)h(to)g (grouplik)o(e)f(elemen)o(ts,)g(there)j(exists,)e(for)f(ev)o(ery)i Fo(s)f Fm(2)f Fo(G)p Fp(,)g(an)h(elemen)o(t)257 1605 y Fo(w)q Fp(\()p Fo(s)p Fp(\))12 b Fm(2)g Fk(Z)421 1611 y Fn(2)451 1605 y Fp(and)i(an)f(elemen)o(t)h Fo(f)t Fp(\()p Fo(s)p Fp(\))f Fm(2)e Fo(G)903 1590 y Fl(0)928 1605 y Fp(suc)o(h)k(that)702 1689 y Fo(f)722 1695 y Fj(A)750 1689 y Fp(\(1)9 b Fm(\012)g Fo(x)861 1695 y Fj(s)879 1689 y Fp(\))i(=)h Fo(u)974 1672 y Fl(0)r Fj(w)q Fn(\()p Fj(s)p Fn(\))1054 1689 y Fp(\(1)d Fm(\012)h Fo(x)1166 1672 y Fl(0)1166 1700 y Fj(f)s Fn(\()p Fj(s)p Fn(\))1229 1689 y Fp(\))257 1773 y(No)o(w)k(the)g(linearit)o(y)f(of)h Fo(f)655 1779 y Fj(A)696 1773 y Fp(o)o(v)o(er)g Fo(H)i Fp(implies:)330 1857 y Fo(f)350 1863 y Fj(A)377 1857 y Fp(\()p Fo(u)417 1840 y Fj(\013)p Fn(\()p Fj(s)p Fn(\))482 1857 y Fp(\))p Fo(u)522 1840 y Fl(0)r Fj(w)q Fn(\()p Fj(s)p Fn(\))603 1857 y Fp(\(1)9 b Fm(\012)g Fo(x)714 1840 y Fl(0)714 1868 y Fj(f)s Fn(\()p Fj(s)p Fn(\))777 1857 y Fp(\))j(=)f Fo(f)868 1863 y Fj(A)896 1857 y Fp(\()p Fo(u)936 1840 y Fj(\013)p Fn(\()p Fj(s)p Fn(\))1001 1857 y Fp(\(1)e Fm(\012)h Fo(x)1113 1863 y Fj(s)1130 1857 y Fp(\)\))i(=)g Fo(f)1238 1863 y Fj(A)1265 1857 y Fp(\()p Fo(\036)p Fp(\(1)d Fm(\012)h Fo(x)1418 1863 y Fj(s)1435 1857 y Fp(\)\))805 1933 y(=)h Fo(\036)873 1916 y Fl(0)885 1933 y Fp(\()p Fo(f)921 1939 y Fj(A)948 1933 y Fp(\(1)e Fm(\012)h Fo(x)1060 1939 y Fj(s)1077 1933 y Fp(\)\))i(=)g Fo(u)1189 1916 y Fl(0)r Fj(\013)1223 1904 y Fh(0)1233 1916 y Fn(\()p Fj(f)s Fn(\()p Fj(s)p Fn(\)\))1322 1933 y Fo(u)1346 1916 y Fl(0)r Fj(w)q Fn(\()p Fj(s)p Fn(\))1426 1933 y Fp(\(1)d Fm(\012)h Fo(x)1538 1916 y Fl(0)1538 1945 y Fj(f)s Fn(\()p Fj(s)p Fn(\))1601 1933 y Fp(\))257 2017 y(Since)15 b(grouplik)o(e)e(elemen)o(ts)g(are)i(in)o(v)o(ertible,) e(this)h(implies:)783 2101 y Fo(f)803 2107 y Fj(A)830 2101 y Fp(\()p Fo(u)870 2084 y Fj(\013)p Fn(\()p Fj(s)p Fn(\))935 2101 y Fp(\))e(=)g Fo(u)1031 2084 y Fl(0)q Fj(\013)1064 2072 y Fh(0)1075 2084 y Fn(\()p Fj(f)s Fn(\()p Fj(s)p Fn(\)\))257 2186 y Fp(Similarly)m(,)d(the)14 b(colinearit)o(y)e (of)h Fo(f)781 2192 y Fj(A)821 2186 y Fp(implies)e Fo(f)981 2192 y Fj(A)1008 2186 y Fp(\()p Fo(u)1048 2170 y Fj(\014)q Fn(\()p Fj(s)p Fn(\))1112 2186 y Fp(\))h(=)g Fo(u)1208 2170 y Fl(0)q Fj(\014)1239 2158 y Fh(0)1251 2170 y Fn(\()p Fj(f)s Fn(\()p Fj(s)p Fn(\)\))1340 2186 y Fp(.)g(No)o(w)h(supp)q(ose)h (that)257 2235 y Fo(\013)h Fm(6)p Fp(=)g(0.)g(Then)h(there)h(exists)g (an)e(elemen)o(t)h Fo(s)f Fm(2)f Fo(G)i Fp(suc)o(h)g(that)g Fo(\013)p Fp(\()p Fo(s)p Fp(\))f(=)g(1,)g(and,)h(since)g Fo(f)1662 2241 y Fj(A)257 2285 y Fp(is)i(injectiv)o(e,)f(w)o(e)g(also)g (ha)o(v)o(e)h Fo(\013)761 2270 y Fl(0)772 2285 y Fp(\()p Fo(f)t Fp(\()p Fo(s)p Fp(\)\))h(=)f(1.)f(Therefore,)h(w)o(e)g(ha)o(v)o (e)f Fo(f)1385 2291 y Fj(A)1412 2285 y Fp(\()p Fo(u)p Fp(\))h(=)g Fo(u)1560 2270 y Fl(0)1571 2285 y Fp(.)f(By)h(a)257 2335 y(similar)12 b(reasoning,)h(this)h(also)f(holds)h(if)f Fo(\014)h Fm(6)p Fp(=)e(0.)h(The)h(ab)q(o)o(v)o(e)g(equations)g(no)o(w) f(yield:)652 2419 y Fo(\013)p Fp(\()p Fo(s)p Fp(\))f(=)g Fo(\013)813 2402 y Fl(0)824 2419 y Fp(\()p Fo(f)t Fp(\()p Fo(s)p Fp(\)\))85 b Fo(\014)r Fp(\()p Fo(s)p Fp(\))13 b(=)f Fo(\014)1174 2402 y Fl(0)1186 2419 y Fp(\()p Fo(f)t Fp(\()p Fo(s)p Fp(\)\))257 2503 y(This)i(pro)o(v)o(es)g(the)h(\014rst)g (and)e(the)i(second)g(assertion.)953 2628 y(63)p eop %%Page: 64 64 64 63 bop 257 262 a Fp(\(3\))21 b(By)14 b(in)o(v)o(erting)g(the)g (discrete)i(F)m(ourier)d(transform)g(ab)q(o)o(v)o(e,)g(w)o(e)h(get:)684 392 y Fo(e)703 398 y Fj(i)726 392 y Fm(\012)c Fo(x)792 398 y Fj(s)821 392 y Fp(=)869 364 y(1)p 869 383 21 2 v 869 421 a(2)924 340 y Fn(1)902 353 y Fg(X)903 441 y Fj(j)r Fn(=0)962 392 y Fp(\()p Fm(\000)p Fp(1\))1047 375 y Fj(ij)1077 392 y Fo(u)1101 375 y Fj(j)1118 392 y Fp(\(1)f Fm(\012)h Fo(x)1230 398 y Fj(s)1247 392 y Fp(\))257 526 y(Therefore,)15 b(w)o(e)f(ha)o(v)o(e:)297 654 y Fo(f)317 660 y Fj(A)344 654 y Fp(\()p Fo(e)379 660 y Fj(i)403 654 y Fm(\012)9 b Fo(x)468 660 y Fj(s)485 654 y Fp(\))j(=)562 626 y(1)p 562 645 V 562 683 a(2)616 602 y Fn(1)595 615 y Fg(X)596 703 y Fj(j)r Fn(=0)655 654 y Fp(\()p Fm(\000)p Fp(1\))740 637 y Fj(ij)769 654 y Fo(f)789 660 y Fj(A)817 654 y Fp(\()p Fo(u)857 637 y Fj(j)874 654 y Fp(\(1)d Fm(\012)h Fo(x)986 660 y Fj(s)1003 654 y Fp(\)\))i(=)1096 626 y(1)p 1096 645 V 1096 683 a(2)1150 602 y Fn(1)1128 615 y Fg(X)1130 703 y Fj(j)r Fn(=0)1188 654 y Fp(\()p Fm(\000)p Fp(1\))1273 637 y Fj(ij)1303 654 y Fo(u)1327 637 y Fl(0)q Fj(j)1356 654 y Fo(u)1380 637 y Fl(0)q Fj(w)q Fn(\()p Fj(s)p Fn(\))1460 654 y Fp(\(1)d Fm(\012)g Fo(x)1571 637 y Fl(0)1571 665 y Fj(f)s Fn(\()p Fj(s)p Fn(\))1634 654 y Fp(\))513 804 y(=)j Fo(u)581 787 y Fl(0)q Fj(w)q Fn(\()p Fj(s)p Fn(\))666 776 y Fp(1)p 666 795 V 666 833 a(2)720 752 y Fn(1)699 765 y Fg(X)700 853 y Fj(j)r Fn(=0)758 804 y Fp(\()p Fm(\000)p Fp(1\))843 787 y Fj(ij)873 804 y Fo(u)897 787 y Fl(0)q Fj(j)926 804 y Fp(\(1)d Fm(\012)h Fo(x)1038 787 y Fl(0)1038 816 y Fj(f)s Fn(\()p Fj(s)p Fn(\))1100 804 y Fp(\))i(=)g Fo(u)1196 787 y Fl(0)q Fj(w)q Fn(\()p Fj(s)p Fn(\))1276 804 y Fo(e)1295 810 y Fj(i)1318 804 y Fm(\012)e Fo(x)1384 787 y Fl(0)1384 816 y Fj(f)s Fn(\()p Fj(s)p Fn(\))513 919 y Fp(=)i(\()p Fm(\000)p Fp(1\))642 901 y Fj(iw)q Fn(\()p Fj(s)p Fn(\))723 919 y Fo(e)742 925 y Fj(i)765 919 y Fm(\012)d Fo(x)830 901 y Fl(0)830 930 y Fj(f)s Fn(\()p Fj(s)p Fn(\))257 1010 y Fp(This)14 b(pro)o(v)o(es)g(that)g Fo(f)590 1016 y Fj(A)632 1010 y Fp(has)g(the)g(form)e(giv)o(en)i(in)f (the)i(fourth)e(assertion.)257 1093 y(\(4\))21 b(W)m(e)14 b(ha)o(v)o(e:)384 1184 y Fo(f)404 1190 y Fj(A)431 1184 y Fp(\()p Fo(e)466 1190 y Fn(1)495 1184 y Fm(\012)9 b Fo(x)560 1190 y Fj(s)577 1184 y Fp(\))p Fo(f)613 1190 y Fj(A)641 1184 y Fp(\()p Fo(e)676 1190 y Fn(1)704 1184 y Fm(\012)h Fo(x)770 1190 y Fj(t)784 1184 y Fp(\))i(=)g(\()p Fm(\000)p Fp(1\))941 1167 y Fj(w)q Fn(\()p Fj(s)p Fn(\)+)p Fj(w)q Fn(\()p Fj(t)p Fn(\))1098 1184 y Fp(\()p Fo(e)1133 1190 y Fn(1)1162 1184 y Fm(\012)d Fo(x)1227 1167 y Fl(0)1227 1196 y Fj(f)s Fn(\()p Fj(s)p Fn(\))1290 1184 y Fp(\)\()p Fo(e)1341 1190 y Fn(1)1369 1184 y Fm(\012)h Fo(x)1435 1167 y Fl(0)1435 1196 y Fj(f)s Fn(\()p Fj(t)p Fn(\))1495 1184 y Fp(\))812 1261 y(=)i(\()p Fm(\000)p Fp(1\))941 1243 y Fj(w)q Fn(\()p Fj(s)p Fn(\)+)p Fj(w)q Fn(\()p Fj(t)p Fn(\))1098 1261 y Fo(\023)1113 1243 y Fj(q)1129 1231 y Fh(0)1140 1243 y Fn(\()p Fj(f)s Fn(\()p Fj(s)p Fn(\))p Fj(;f)s Fn(\()p Fj(t)p Fn(\)\))1297 1261 y Fp(\()p Fo(e)1332 1267 y Fn(1)1361 1261 y Fm(\012)d Fo(x)1426 1243 y Fl(0)1426 1272 y Fj(f)s Fn(\()p Fj(s)p Fn(\))p Fj(f)s Fn(\()p Fj(t)p Fn(\))1547 1261 y Fp(\))257 1352 y(On)15 b(the)f(other)h(hand,)e(w)o(e)h(ha)o(v)o(e:)503 1443 y Fo(f)523 1449 y Fj(A)551 1443 y Fp(\(\()p Fo(e)602 1449 y Fn(1)630 1443 y Fm(\012)c Fo(x)696 1449 y Fj(s)713 1443 y Fp(\)\()p Fo(e)764 1449 y Fn(1)792 1443 y Fm(\012)g Fo(x)858 1449 y Fj(t)872 1443 y Fp(\)\))i(=)g(\()p Fm(\000)p Fp(1\))1045 1426 y Fj(w)q Fn(\()p Fj(st)p Fn(\))1126 1443 y Fo(\023)1141 1426 y Fj(q)q Fn(\()p Fj(s;t)p Fn(\))1223 1443 y Fp(\()p Fo(e)1258 1449 y Fn(1)1287 1443 y Fm(\012)d Fo(x)1352 1426 y Fl(0)1352 1455 y Fj(f)s Fn(\()p Fj(st)p Fn(\))1428 1443 y Fp(\))257 1535 y(Since)15 b Fo(f)386 1541 y Fj(A)427 1535 y Fp(is)e(an)h(algebra)f(homomo)o(rphism)o(,)d(b)q (oth)k(expressions)i(m)o(ust)c(b)q(e)j(equal.)d(There-)257 1584 y(fore,)k(w)o(e)h(see)h(that)e Fo(f)t Fp(\()p Fo(st)p Fp(\))i(=)e Fo(f)t Fp(\()p Fo(s)p Fp(\))p Fo(f)t Fp(\()p Fo(t)p Fp(\),)i(i.)e(e.,)g Fo(f)21 b Fp(is)16 b(a)g(group)h(homom)o (orphism)o(,)c(and)j(w)o(e)257 1634 y(ha)o(v)o(e)f Fo(q)q Fp(\()p Fo(s;)7 b(t)p Fp(\))j Fm(\000)g Fo(q)531 1619 y Fl(0)543 1634 y Fp(\()p Fo(f)t Fp(\()p Fo(s)p Fp(\))p Fo(;)d(f)t Fp(\()p Fo(t)p Fp(\)\))15 b(=)e(^)-20 b Fo(\023)o Fp(\()p Fo(w)q Fp(\()p Fo(t)p Fp(\))11 b Fm(\000)f Fo(w)q Fp(\()p Fo(st)p Fp(\))g(+)g Fo(w)q Fp(\()p Fo(s)p Fp(\)\).)15 b(This)g(pro)o(v)o(es)g(the)h(third)f(as-)257 1684 y(sertion.)j(Since)f Fo(q)i Fp(and)e Fo(q)662 1669 y Fl(0)691 1684 y Fp(are)g(normalized,)f (w)o(e)h(get,)g(b)o(y)g(inserting)g Fo(s)h Fp(=)g Fo(t)f Fp(=)g(1)g(in)g(the)257 1734 y(previous)e(equation,)e(the)h(fact)g (that)g Fo(w)q Fp(\(1\))d(=)h(0.)h Fi(2)257 1869 y Fq(4.9)48 b Fp(As)13 b(an)e(application)g(of)g(the)i(preceding)g(considerations,) f(w)o(e)g(consider)h(the)g(Y)m(etter-)257 1919 y(Drinfel'd)f(Hopf)h (algebras)g Fo(A)723 1925 y Fn(+)764 1919 y Fp(and)g Fo(A)875 1925 y Fl(\000)916 1919 y Fp(constructed)j(in)d(P)o(aragraph)f (3.6.)g(W)m(e)h(k)o(eep)h(the)257 1969 y(notation)f(of)h(P)o(aragraph)f (4.7)g(and)h(P)o(aragraph)f(4.8.)257 2068 y Fq(Prop)q(osition)33 b Fo(A)567 2074 y Fn(+)609 2068 y Fp(and)13 b Fo(A)720 2074 y Fl(\000)762 2068 y Fp(are)i(not)f(isomorphic.)257 2168 y Fq(Pro)q(of.)36 b Fp(As)20 b(in)g(P)o(aragraph)f(3.6,)f(w)o(e)i (denote)h(the)f(co)q(cycles)i(used)e(in)f(the)i(de\014nition)257 2218 y(of)13 b Fo(A)335 2224 y Fn(+)376 2218 y Fp(resp.)h Fo(A)506 2224 y Fl(\000)547 2218 y Fp(b)o(y)f Fo(q)623 2224 y Fn(+)663 2218 y Fp(resp.)h Fo(q)781 2224 y Fl(\000)809 2218 y Fp(.)f(W)m(e)f(ha)o(v)o(e)i Fo(q)1019 2224 y Fl(\000)1054 2218 y Fm(\000)8 b Fo(q)1113 2224 y Fn(+)1152 2218 y Fp(=)j(^)-20 b Fo(\023)1211 2224 y Fn(2)1229 2218 y Fp(\()p Fo(q)q Fp(\),)13 b(where)h Fo(q)f Fm(2)e Fo(Z)1527 2203 y Fn(2)1546 2218 y Fp(\()p Fk(Z)1593 2224 y Fn(2)1609 2218 y Fo(;)c Fk(Z)1658 2224 y Fn(2)1673 2218 y Fp(\))257 2268 y(is)14 b(the)h(2-co)q(cycle)f(of)f(the)i(trivial)d Fk(Z)821 2274 y Fn(2)837 2268 y Fp(-mo)q(dule)g Fk(Z)1028 2274 y Fn(2)1057 2268 y Fp(de\014ned)j(as:)678 2394 y Fo(q)q Fp(\()p Fo(i;)7 b(j)r Fp(\))12 b(:=)850 2323 y Fg(\()883 2366 y Fp(0)41 b(if)13 b Fo(i)f Fp(=)g(0)i(or)f Fo(j)h Fp(=)e(0)883 2426 y(1)41 b(if)13 b Fo(i)f Fp(=)g(1)i(and)f Fo(j)h Fp(=)e(1)953 2628 y(64)p eop %%Page: 65 65 65 64 bop 257 262 a Fp(If)16 b Fo(f)321 268 y Fj(A)362 262 y Fp(:)e Fo(A)419 268 y Fl(\000)461 262 y Fm(!)g Fo(A)548 268 y Fn(+)591 262 y Fp(w)o(ere)j(an)e(isomorphism,)d(w)o(e)k (kno)o(w)f(from)f(Prop)q(osition)h(4.8)f(that)i(it)257 311 y(w)o(ould)d(ha)o(v)o(e)h(the)h(form)704 403 y Fo(f)724 409 y Fj(A)751 403 y Fp(\()p Fo(e)786 409 y Fj(i)810 403 y Fm(\012)9 b Fo(c)869 409 y Fj(j)887 403 y Fp(\))i(=)h(\()p Fm(\000)p Fp(1\))1043 386 y Fj(iw)q Fn(\()p Fj(j)r Fn(\))1124 403 y Fo(e)1143 409 y Fj(i)1166 403 y Fm(\012)d Fo(c)1225 409 y Fj(j)257 494 y Fp(for)14 b(some)f(1-co)q(c)o(hain)g Fo(w)g Fp(:)e Fk(Z)706 500 y Fn(2)733 494 y Fm(!)18 b Fk(Z)823 500 y Fn(2)853 494 y Fp(satisfying)13 b Fo(w)q Fp(\(0\))e(=)h(0)i(and)689 585 y Fo(q)q Fp(\()p Fo(i;)7 b(j)r Fp(\))12 b(=)g Fo(w)q Fp(\()p Fo(j)r Fp(\))e Fm(\000)f Fo(w)q Fp(\()p Fo(i)h Fp(+)f Fo(j)r Fp(\))h(+)g Fo(w)q Fp(\()p Fo(i)p Fp(\))257 677 y(for)18 b(all)e Fo(i;)7 b(j)21 b Fm(2)c Fk(Z)533 683 y Fn(2)549 677 y Fp(.)g(Therefore,)h Fo(q)h Fp(w)o(ere)g(the)f(cob)q(oundary)g(arising)f(from)f Fo(w)q Fp(.)h(Ho)o(w)o(ev)o(er,)257 726 y(this)j(is)f(not)g(the)h (case,)g(since)g(from)d Fo(w)q Fp(\(0\))j(=)h(0)e(w)o(e)h(get,)f(as)g (in)g(P)o(aragraph)g(3.6,)e(that)257 776 y(1)12 b(=)f Fo(q)q Fp(\(1)p Fo(;)c Fp(1\))k(=)h Fo(w)q Fp(\(1\))d(+)h Fo(w)q Fp(\(1\))h(=)h(0,)h(whic)o(h)h(is)g(a)f(con)o(tradiction.)g Fi(2)953 2628 y Fp(65)p eop %%Page: 66 66 66 65 bop 257 262 a Fr(5)67 b(Constructions)257 575 y Fq(5.1)48 b Fp(Y)m(etter-Drinfel'd)19 b(Hopf)f(algebras)h(can)g(b)q(e)g (used)h(to)e(construct)j(ordinary)d(Hopf)257 625 y(algebras.)h(W)m(e)f (ha)o(v)o(e)h(already)g(seen)h(one)f(of)f(these)j(constructions)f(in)e (P)o(aragraph)h(1.6,)257 675 y(namely)12 b(the)i(Radford)f(bipro)q (duct)g(construction.)h(The)g(second)h(construction)f(from)e([78)o(]) 257 724 y(is)h(another)g(example)f(of)g(suc)o(h)i(a)f(construction;)g (it)f(yields)h(a)g(Hopf)f(algebra)h(in)f(whic)o(h)h(the)257 774 y(Radford)g(bipro)q(duct)i(app)q(ears)f(as)g(a)g(kind)f(of)h(Borel) g(subalgebra.)257 858 y(In)k(this)g(section,)g(w)o(e)g(apply)f(these)i (constructions)g(to)f(the)g(Y)m(etter-Drinfel'd)g(Hopf)f(al-)257 907 y(gebra)e Fo(A)402 913 y Fj(G)430 907 y Fp(\()p Fo(\013;)7 b(\014)r(;)g(q)q Fp(\))15 b(considered)h(in)f(P)o(aragraph)f(3.3.)f(W)m (e)i(describ)q(e)h(the)g(resulting)f(Hopf)257 957 y(algebras)g(b)o(y)g (exhibiting)g(a)f(basis)i(for)f(whic)o(h)g(the)g(structure)j(elemen)o (ts)d(can)g(b)q(e)h(written)257 1007 y(do)o(wn)c(explicitly)m(.)f (Afterw)o(ards,)i(w)o(e)f(determine)h(when)f(the)h(resulting)g(Hopf)f (algebras)g(are)257 1057 y(semisimple.)257 1140 y(In)j(the)h(whole)e (section,)h(w)o(e)g(assume)f(that)h Fo(G)g Fp(is)f(a)h(\014nite)g (group)f(and)h(that)g Fo(R)f Fp(is)h(a)f(\014nite)257 1190 y(ring.)e(As)h(in)e(P)o(aragraph)h(3.3,)f(w)o(e)i(assume)f(that)g (w)o(e)g(are)h(giv)o(en)f(a)g(group)g(homomorphism)847 1276 y Fo(\027)i Fp(:)d Fo(G)g Fm(!)g Fo(U)5 b Fp(\()p Fo(R)p Fp(\))257 1362 y(from)13 b Fo(G)g Fp(to)h(the)h(m)o (ultiplicativ)o(e)c(group)j Fo(U)5 b Fp(\()p Fo(R)p Fp(\))14 b(of)g(units)g(of)g Fo(R)p Fp(,)f(and)h(use)h(it)f(to)g(in)o(tro)q (duce)257 1412 y(the)f(left)e Fo(G)p Fp(-mo)q(dule)f(structures)780 1418 y Fj(G)808 1412 y Fo(R)i Fp(and)f Fo(R)962 1418 y Fj(G)1002 1412 y Fp(on)g Fo(R)h Fp(describ)q(ed)h(there.)g(Also,)e(w) o(e)h(assume)257 1462 y(that)i(the)h(follo)o(wing)c(additional)h (structure)k(elemen)o(ts)e(are)g(giv)o(en:)308 1574 y(1.)20 b(Tw)o(o)14 b(1-co)q(cycles)g Fo(\013;)7 b(\014)14 b Fm(2)d Fo(Z)801 1559 y Fn(1)820 1574 y Fp(\()p Fo(G;)894 1580 y Fj(G)922 1574 y Fo(R)p Fp(\).)308 1655 y(2.)20 b(A)14 b(normalized)e(2-co)q(cycle)j Fo(q)e Fm(2)e Fo(Z)898 1640 y Fn(2)917 1655 y Fp(\()p Fo(G;)991 1661 y Fj(G)1019 1655 y Fo(R)p Fp(\).)308 1736 y(3.)20 b(Tw)o(o)15 b(c)o(haracters)i Fo(\037;)7 b(\021)15 b Fm(2)787 1726 y Fp(^)778 1736 y Fo(R)g Fp(of)g(the)h(additiv)o(e)e(group)i(of)f Fo(R)p Fp(,)f(where)j Fo(\037)e Fp(is)h(required)361 1786 y(to)e(satisfy)f Fo(\037)p Fp(\()p Fo(uv)q(w)q Fp(\))f(=)g Fo(\037)p Fp(\()p Fo(v)q(uw)q Fp(\))j(for)e(all)g Fo(u;)7 b(v)q(;)g(w)k Fm(2)h Fo(R)p Fp(.)257 1899 y(F)m(rom)f(P)o(aragraph)g(5.5)g(on,)g (where)i(w)o(e)g(b)q(egin)f(to)f(consider)i(the)g(second)g (construction,)f(w)o(e)257 1948 y(require)j(that)f Fo(\013)g Fp(and)f Fo(\014)k Fp(are)d(compatible)e(in)i(the)g(sense)i(that)e(w)o (e)g(ha)o(v)o(e)712 2035 y Fo(\037)p Fp(\()p Fo(u\013)p Fp(\()p Fo(s)p Fp(\))p Fo(\014)r Fp(\()p Fo(t)p Fp(\)\))f(=)f Fo(\037)p Fp(\()p Fo(u\014)r Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(t)p Fp(\)\))257 2121 y(for)i(all)f Fo(s;)7 b(t)k Fm(2)g Fo(G)j Fp(and)f(all)g Fo(u)e Fm(2)h Fo(R)p Fp(.)257 2204 y(W)m(e)j(use)i(the)f(notation)e Fo(H)j Fp(:=)d Fo(K)s Fp([)p Fo(R)p Fp(])h(for)g(the)h(group)f(ring)g(of)g (the)h(additiv)o(e)f(group)g(of)g Fo(R)257 2254 y Fp(and)i Fo(A)e Fp(:=)h Fo(A)478 2260 y Fj(G)506 2254 y Fp(\()p Fo(\013;)7 b(\014)r(;)g(q)q Fp(\))15 b(=)h Fo(K)749 2239 y Fj(R)788 2254 y Fm(\012)11 b Fo(K)s Fp([)p Fo(G)p Fp(])k(for)h(the)h (tensor)h(pro)q(duct)f(of)f(the)h(dual)f(group)257 2304 y(ring)d(of)f(the)h(additiv)o(e)f(group)h(of)f Fo(R)g Fp(and)h(the)g(group)g(ring)f(of)g Fo(G)p Fp(,)g(considered)i(as)f(a)g (Y)m(etter-)257 2354 y(Drinfel'd)e(Hopf)g(algebra)g(with)h(the)g (structure)i(elemen)o(ts)d(describ)q(ed)j(in)d(Prop)q(osition)g(3.3.) 257 2403 y(The)18 b(canonical)e(basis)h(elemen)o(ts)g(of)g Fo(K)s Fp([)p Fo(R)p Fp(])f(resp.)i Fo(K)s Fp([)p Fo(G)p Fp(])e(are)h(denoted)h(b)o(y)f Fo(c)1495 2409 y Fj(u)1534 2403 y Fp(resp.)g Fo(x)1660 2409 y Fj(s)1678 2403 y Fp(,)257 2453 y(where)h Fo(u)f Fm(2)f Fo(R)h Fp(and)f Fo(s)h Fm(2)g Fo(G)p Fp(,)f(and)g(the)i(primitiv)o(e)c(idemp)q(oten)o(ts)j(of)f Fo(K)1404 2438 y Fj(R)1449 2453 y Fp(resp.)h Fo(K)1589 2438 y Fj(G)1634 2453 y Fp(are)257 2503 y(denoted)e(b)o(y)f Fo(e)491 2509 y Fj(u)527 2503 y Fp(resp.)g Fo(d)648 2509 y Fj(s)666 2503 y Fp(.)953 2628 y(66)p eop %%Page: 67 67 67 66 bop 257 262 a Fq(5.2)48 b Fp(In)16 b(this)h(paragraph,)e(w)o(e)h (consider)h(the)g(Radford)f(bipro)q(duct)g Fo(B)i Fp(:=)d Fo(A)c Fm(\012)g Fo(H)s Fp(.)16 b(W)m(e)257 311 y(in)o(tro)q(duce)f (the)f(basis)775 361 y Fo(b)793 367 y Fj(uv)832 361 y Fp(\()p Fo(s)p Fp(\))e(:=)f Fo(e)969 367 y Fj(u)1001 361 y Fm(\012)e Fo(x)1066 367 y Fj(s)1093 361 y Fm(\012)g Fo(c)1152 367 y Fj(v)257 426 y Fp(of)17 b(the)h(Radford)f(bipro)q (duct,)g(where)i Fo(u;)7 b(v)18 b Fm(2)f Fo(R)g Fp(are)h(elemen)o(ts)f (of)g(the)h(\014nite)g(ring)f(and)257 475 y Fo(s)h Fm(2)f Fo(G)f Fp(is)h(an)g(elemen)o(t)g(of)g(the)g(\014nite)h(group)f(under)h (consideration.)f(With)f(resp)q(ect)k(to)257 525 y(this)14 b(basis,)g(the)g(structure)i(elemen)o(ts)e(of)f Fo(B)k Fp(tak)o(e)d(the)g(follo)o(wing)d(form:)257 606 y Fq(Prop)q(osition)308 656 y Fp(1.)20 b(Multiplication:)12 b Fo(b)662 662 y Fj(uv)701 656 y Fp(\()p Fo(s)p Fp(\))p Fo(b)770 662 y Fj(u)790 654 y Fh(0)801 662 y Fj(v)819 654 y Fh(0)832 656 y Fp(\()p Fo(s)867 641 y Fl(0)879 656 y Fp(\))g(=)361 719 y Fo(\016)379 726 y Fj(u\027)r Fn(\()p Fj(s)p Fn(\))p Fj(;u)490 718 y Fh(0)502 719 y Fo(\021)q Fp(\()p Fo(uq)q Fp(\()p Fo(s;)7 b(s)657 704 y Fl(0)669 719 y Fp(\)\))p Fo(\037)p Fp(\(2)p Fo(v)q(u)809 704 y Fl(0)822 719 y Fo(\013)p Fp(\()p Fo(s)884 704 y Fl(0)896 719 y Fp(\))i(+)g Fo(u)986 704 y Fn(2)1005 719 y Fo(\027)s Fp(\()p Fo(s)p Fp(\))p Fo(\014)r Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)1218 704 y Fl(0)1231 719 y Fp(\)\))p Fo(b)1281 725 y Fj(u;v)q Fn(+)p Fj(v)1372 717 y Fh(0)1385 719 y Fp(\()p Fo(ss)1439 704 y Fl(0)1451 719 y Fp(\))308 795 y(2.)20 b(Unit:)13 b(1)489 801 y Fj(B)529 795 y Fp(=)573 764 y Fg(P)617 807 y Fj(u)p Fl(2)p Fj(R)693 795 y Fo(b)711 801 y Fj(u)p Fn(0)749 795 y Fp(\(1\))308 871 y(3.)20 b(Com)o(ultiplicatio)o(n:)11 b(\001)725 877 y Fj(B)753 871 y Fp(\()p Fo(b)787 877 y Fj(uv)826 871 y Fp(\()p Fo(s)p Fp(\)\))i(=)949 840 y Fg(P)993 883 y Fj(w)q Fl(2)p Fj(R)1075 871 y 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Fp(\))257 1176 y Fq(Pro)q(of.)36 b Fp(This)14 b(follo)o(ws)e(b)o(y)i(direct)h(computation:)257 1273 y(\(1\))21 b(Multiplication:)294 1348 y Fo(b)312 1354 y Fj(uv)351 1348 y Fp(\()p Fo(s)q Fp(\))p Fo(b)421 1354 y Fj(u)441 1346 y Fh(0)452 1354 y Fj(v)470 1346 y Fh(0)483 1348 y Fp(\()p Fo(s)518 1331 y Fl(0)530 1348 y Fp(\))12 b(=)f(\()p Fo(e)636 1354 y Fj(u)668 1348 y Fm(\012)e Fo(x)733 1354 y Fj(s)751 1348 y Fp(\)\()p Fo(c)801 1354 y Fj(v)832 1348 y Fm(!)i Fp(\()p Fo(e)920 1354 y Fj(u)940 1346 y Fh(0)963 1348 y Fm(\012)e Fo(x)1028 1354 y Fj(s)1044 1346 y Fh(0)1057 1348 y Fp(\)\))h Fm(\012)f Fo(c)1158 1354 y Fj(v)1178 1348 y Fo(c)1196 1354 y Fj(v)1214 1346 y Fh(0)398 1415 y Fp(=)j Fo(\037)p Fp(\()p Fo(v)q(u)529 1398 y Fl(0)541 1415 y Fo(\013)p Fp(\()p Fo(s)603 1398 y Fl(0)615 1415 y Fp(\)\))647 1398 y Fn(2)666 1415 y Fo(\016)684 1422 y Fj(u\027)r Fn(\()p Fj(s)p Fn(\))p Fj(;u)795 1414 y Fh(0)807 1415 y Fo(\021)q Fp(\()p Fo(uq)q Fp(\()p Fo(s;)7 b(s)962 1398 y Fl(0)974 1415 y Fp(\)\))p 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Fj(B)361 1652 y Fp(\()p Fo(e)396 1658 y Fj(u)427 1652 y Fm(\012)10 b Fo(x)493 1658 y Fj(s)519 1652 y Fm(\012)g Fo(c)579 1658 y Fj(v)599 1652 y Fp(\))h(=)676 1613 y Fg(X)670 1702 y Fj(w)q Fl(2)p Fj(R)743 1652 y Fp(\()p Fo(e)778 1658 y Fj(u)p Fl(\000)p Fj(w)860 1652 y Fm(\012)e Fo(x)925 1658 y Fj(s)952 1652 y Fm(\012)g Fp(\()p Fo(e)1028 1658 y Fj(w)1065 1652 y Fm(\012)g Fo(x)1130 1658 y Fj(s)1148 1652 y Fp(\))1164 1635 y Fn(\(1\))1209 1652 y Fo(c)1227 1658 y Fj(v)1246 1652 y Fp(\))h Fm(\012)f Fp(\(\()p Fo(e)1364 1658 y Fj(w)1401 1652 y Fm(\012)g Fo(x)1466 1658 y Fj(s)1484 1652 y Fp(\))1500 1635 y Fn(\(2\))1554 1652 y Fm(\012)g Fo(c)1613 1658 y Fj(v)1633 1652 y Fp(\))626 1769 y(=)676 1729 y Fg(X)670 1818 y Fj(w)q Fl(2)p Fj(R)743 1769 y Fp(\()p Fo(e)778 1775 y Fj(u)p Fl(\000)p Fj(w)860 1769 y Fm(\012)g Fo(x)925 1775 y Fj(s)952 1769 y Fm(\012)g Fo(c)1011 1776 y Fj(w)q(\014)q Fn(\()p Fj(s)p Fn(\)+)p Fj(v)1143 1769 y Fp(\))h Fm(\012)f Fp(\()p Fo(e)1245 1775 y Fj(w)1282 1769 y Fm(\012)g Fo(x)1347 1775 y Fj(s)1374 1769 y Fm(\012)g Fo(c)1433 1775 y Fj(v)1453 1769 y Fp(\))257 1897 y(\(3\))21 b(An)o(tip)q(o)q(de:)293 1972 y Fo(S)318 1978 y Fj(B)347 1972 y Fp(\()p Fo(e)382 1978 y Fj(u)413 1972 y Fm(\012)10 b Fo(x)479 1978 y Fj(s)505 1972 y Fm(\012)g Fo(c)565 1978 y Fj(v)585 1972 y Fp(\))h(=)h(\(1)693 1978 y Fj(A)729 1972 y Fm(\012)e Fo(S)796 1978 y Fj(H)828 1972 y Fp(\(\()p Fo(e)879 1978 y Fj(u)910 1972 y Fm(\012)g Fo(x)976 1978 y Fj(s)993 1972 y Fp(\))1009 1955 y Fn(\(1\))1054 1972 y Fo(c)1072 1978 y Fj(v)1092 1972 y Fp(\)\)\()p Fo(S)1165 1978 y Fj(A)1192 1972 y Fp(\(\()p Fo(e)1243 1978 y Fj(u)1275 1972 y Fm(\012)g Fo(x)1341 1978 y Fj(s)1358 1972 y Fp(\))1374 1955 y Fn(\(2\))1419 1972 y Fp(\))f Fm(\012)h Fp(1)1507 1978 y Fj(H)1538 1972 y Fp(\))304 2039 y(=)i Fo(\021)q Fp(\()p Fo(uq)q Fp(\()p Fo(s;)7 b(s)503 2022 y Fl(\000)p Fn(1)548 2039 y Fp(\)\))p Fo(\037)p Fp(\()p Fo(u)646 2022 y Fn(2)665 2039 y Fo(\014)r Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)p Fp(\)\)\(1)872 2045 y Fj(A)910 2039 y Fm(\012)j Fo(c)970 2046 y Fl(\000)p Fj(u\014)q Fn(\()p Fj(s)p Fn(\))p Fl(\000)p Fj(v)1123 2039 y Fp(\)\()p Fo(e)1174 2046 y Fl(\000)p Fj(u\027)r Fn(\()p Fj(s)p Fn(\))1291 2039 y Fm(\012)g Fo(x)1357 2047 y Fj(s)1373 2039 y Fh(\000)p Fd(1)j Fm(\012)d Fp(1)1485 2045 y Fj(H)1516 2039 y Fp(\))304 2107 y(=)i Fo(\021)q Fp(\()p Fo(uq)q Fp(\()p Fo(s;)7 b(s)503 2089 y Fl(\000)p Fn(1)548 2107 y Fp(\)\))p Fo(\037)p Fp(\()p Fo(u)646 2089 y Fn(2)665 2107 y Fo(\014)r Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)p Fp(\)\))p Fo(\037)p Fp(\()p Fo(u\027)s Fp(\()p Fo(s)p Fp(\)\()p Fo(u\014)r Fp(\()p Fo(s)p Fp(\))12 b(+)e Fo(v)q Fp(\))p Fo(\013)p Fp(\()p Fo(s)1245 2089 y Fl(\000)p Fn(1)1290 2107 y Fp(\)\))1322 2089 y Fn(2)1052 2169 y Fp(\()p Fo(e)1087 2176 y Fl(\000)p Fj(u\027)r Fn(\()p Fj(s)p Fn(\))1204 2169 y Fm(\012)g Fo(x)1270 2177 y Fj(s)1286 2168 y Fh(\000)p Fd(1)1335 2169 y Fm(\012)g Fo(c)1395 2176 y Fl(\000)p Fj(u\014)q Fn(\()p Fj(s)p Fn(\))p Fl(\000)p Fj(v)1548 2169 y Fp(\))304 2236 y(=)i Fo(\021)q Fp(\()p Fo(uq)q Fp(\()p Fo(s;)7 b(s)503 2219 y Fl(\000)p Fn(1)548 2236 y Fp(\)\))p Fo(\037)p Fp(\()p Fo(u)646 2219 y Fn(2)665 2236 y Fo(\014)r Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)p Fp(\))k Fm(\000)f Fp(2)p Fo(u)p Fp(\()p Fo(u\014)r Fp(\()p Fo(s)p Fp(\))g(+)f Fo(v)q Fp(\))p Fo(\013)p Fp(\()p Fo(s)p Fp(\)\))1052 2298 y(\()p Fo(e)1087 2305 y Fl(\000)p Fj(u\027)r Fn(\()p Fj(s)p Fn(\))1204 2298 y Fm(\012)h Fo(x)1270 2306 y Fj(s)1286 2298 y Fh(\000)p Fd(1)j Fm(\012)d Fo(c)1395 2305 y Fl(\000)p Fj(u\014)q Fn(\()p Fj(s)p Fn(\))p Fl(\000)p Fj(v)1548 2298 y Fp(\))304 2366 y(=)i Fo(\021)q Fp(\()p Fo(uq)q Fp(\()p Fo(s;)7 b(s)503 2349 y Fl(\000)p Fn(1)548 2366 y Fp(\)\))p Fo(\037)p Fp(\()p Fm(\000)p Fo(u)678 2349 y Fn(2)697 2366 y Fo(\014)r Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)p Fp(\))k Fm(\000)f Fp(2)p Fo(uv)q(\014)r Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)p Fp(\)\)\()p Fo(e)1175 2373 y Fl(\000)p Fj(u\027)r Fn(\()p Fj(s)p Fn(\))1294 2366 y Fm(\012)g Fo(x)1360 2374 y Fj(s)1376 2365 y Fh(\000)p Fd(1)1425 2366 y Fm(\012)g Fo(c)1485 2373 y Fl(\000)p Fj(u\014)q Fn(\()p Fj(s)p Fn(\))p Fl(\000)p Fj(v)1638 2366 y Fp(\))257 2440 y(where)15 b(w)o(e)g(ha)o(v)o(e)e(used) i(the)f(equation)g Fo(\013)p Fp(\()p Fo(s)p Fp(\))e(=)g Fm(\000)p Fo(\027)s Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)1173 2425 y Fl(\000)p Fn(1)1218 2440 y Fp(\))i(from)e (Lemma)f(1.13.)257 2503 y(W)m(e)j(lea)o(v)o(e)f(the)i(v)o (eri\014cation)f(of)f(the)h(form)o(ulas)e(for)i(unit)f(and)h(counit)g (to)f(the)i(reader.)g Fi(2)953 2628 y Fp(67)p eop %%Page: 68 68 68 67 bop 257 262 a Fp(The)14 b(reader)h(should)e(compare)g(the)h(ab)q (o)o(v)o(e)f(description)h(with)f(the)h(description)g(in)f(P)o(ara-)257 311 y(graph)h(8.5,)f(where)i(a)e(sligh)o(tly)g(di\013eren)o(t)i(basis)e (for)h(the)g(Radford)f(bipro)q(duct)i(is)f(used.)257 447 y Fq(5.3)48 b Fp(It)14 b(is)f(kno)o(wn)h(that)f(the)h(crossed)i (pro)q(duct)e(of)f(a)h(semisimple)d(algebra)i(and)g(a)h(semi-)257 497 y(simple)19 b(group)h(ring)f(is)h(semisimple)d(\(cf.)j([64)o(],)f (Thm.)f(4.4,)g(p.)i(31,)f([57)o(],)g(Thm.)f(7.4.2.)257 546 y(p.)c(116\).)f(Therefore,)h(the)h(algebra)e Fo(A)h Fp(constructed)i(in)e(P)o(aragraph)f(3.3)g(is)h(semisimple)d(if)257 596 y(the)k(c)o(haracteristic)h(of)e Fo(K)j Fp(do)q(es)f(not)e(divide)g (the)h(cardinalit)o(y)e(of)h Fo(G)p Fp(.)f(Here)j(w)o(e)e(approac)o(h) 257 646 y(this)g(issue)h(in)e(a)h(di\013eren)o(t)h(w)o(a)o(y)e(that)h (pro)o(vides)g(sligh)o(tly)f(more)f(information:)257 737 y Fq(Prop)q(osition)g(1)308 787 y Fp(1.)20 b(\003)390 793 y Fj(A)440 787 y Fp(:=)507 755 y Fg(P)551 799 y Fj(s)p Fl(2)p Fj(G)624 787 y Fo(e)643 793 y Fn(0)675 787 y Fm(\012)14 b Fo(x)745 793 y Fj(s)784 787 y Fp(is)20 b(a)h(t)o(w)o(o-sided)f(in)o (tegral)g(of)h Fo(A)p Fp(.)f(It)h(is)f(in)o(v)n(arian)o(t)g(and)361 836 y(coin)o(v)n(arian)o(t.)308 919 y(2.)g Fo(A)15 b Fp(is)f(semisimple)e(if)i(and)g(only)g(if)g(the)h(c)o(haracteristic)h (of)d Fo(K)18 b Fp(do)q(es)d(not)g(divide)f(the)361 969 y(cardinalit)o(y)f(of)g Fo(G)p Fp(.)308 1052 y(3.)20 b Fo(B)e Fp(is)d(semisimple)d(if)j(and)g(only)f(if)g(the)i(c)o (haracteristic)h(of)d Fo(K)19 b Fp(neither)d(divides)f(the)361 1102 y(cardinalit)o(y)e(of)g Fo(G)h Fp(nor)f(the)i(cardinalit)o(y)e(of) g Fo(R)p Fp(.)257 1202 y Fq(Pro)q(of.)36 b Fp(\003)452 1208 y Fj(A)493 1202 y Fp(is)14 b(a)f(left)h(in)o(tegral)f(since)i(w)o (e)f(ha)o(v)o(e:)321 1299 y(\()p Fo(e)356 1305 y Fj(u)387 1299 y Fm(\012)c Fo(x)453 1305 y Fj(s)470 1299 y Fp(\)\003)515 1305 y Fj(A)554 1299 y Fp(=)598 1260 y Fg(X)597 1349 y Fj(t)p Fl(2)p Fj(G)658 1299 y Fp(\()p Fo(e)693 1305 y Fj(u)725 1299 y Fm(\012)f Fo(x)790 1305 y Fj(s)808 1299 y Fp(\)\()p Fo(e)859 1305 y Fn(0)887 1299 y Fm(\012)h Fo(x)953 1305 y Fj(t)967 1299 y Fp(\))i(=)1039 1260 y Fg(X)1038 1349 y Fj(t)p Fl(2)p Fj(G)1106 1299 y Fo(\016)1124 1305 y Fj(u)p Fn(0)1163 1299 y Fo(e)1182 1305 y Fn(0)1210 1299 y Fm(\012)d Fo(x)1275 1305 y Fj(st)1317 1299 y Fp(=)j Fo(\017)1378 1305 y Fj(A)1405 1299 y Fp(\()p Fo(e)1440 1305 y Fj(u)1471 1299 y Fm(\012)e Fo(x)1537 1305 y Fj(s)1554 1299 y Fp(\)\003)1599 1305 y Fj(A)257 1430 y Fp(The)21 b(fact)f(that)g(\003)562 1436 y Fj(A)609 1430 y Fp(is)g(also)g(a)f (righ)o(t)h(in)o(tegral)g(follo)o(ws)e(from)g(a)i(similar)e (calculation.)257 1479 y(It)h(is)g(ob)o(vious)e(that)i(\003)632 1485 y Fj(A)678 1479 y Fp(is)f(in)o(v)n(arian)o(t)f(and)h(coin)o(v)n (arian)o(t.)f(Since)i Fo(\017)1348 1485 y Fj(A)1375 1479 y Fp(\(\003)1420 1485 y Fj(A)1447 1479 y Fp(\))h(=)g(card\()p Fo(G)p Fp(\),)257 1529 y(the)c(second)g(assertion)f(follo)o(ws)f(from)f (Masc)o(hk)o(e's)i(theorem)g(for)f(Y)m(etter-Drinfel'd)h(Hopf)257 1579 y(algebras)f(\(cf.)f([20)o(],)g(Cor.)g(5.8,)f(p.)i(4885,)e([81)o (],)h(Prop.)g(2.14,)f(p.)h(22\).)g(The)h(third)g(assertion)257 1629 y(on)g(the)h(Radford)d(bipro)q(duct)j Fo(B)h Fp(follo)o(ws)d(from) f([65)o(],)h(Prop.)g(3,)g(p.)h(333.)f Fi(2)257 1748 y Fp(Since)21 b(the)f(coalgebra)g(structure)i(of)d Fo(A)h Fp(is)f(the)i(ordinary)e(tensor)i(pro)q(duct)g(coalgebra)257 1797 y(structure,)16 b(it)d(is)h(easy)h(to)e(describ)q(e)j(when)e Fo(A)g Fp(is)g(cosemisimple:)257 1888 y Fq(Prop)q(osition)e(2)308 1938 y Fp(1.)20 b(The)14 b(linear)g(form)e Fo(\025)684 1944 y Fj(A)723 1938 y Fp(:)f Fo(A)h Fm(!)f Fo(K)17 b Fp(determined)d(b)o(y)863 2029 y Fo(\025)887 2035 y Fj(A)914 2029 y Fp(\()p Fo(e)949 2035 y Fj(u)981 2029 y Fm(\012)9 b Fo(x)1046 2035 y Fj(s)1064 2029 y Fp(\))i(=)h Fo(\016)1153 2035 y Fj(s)p Fn(1)361 2121 y Fp(is)i(a)f(t)o(w)o(o-sided)h(in)o (tegral)f(of)h Fo(A)852 2106 y Fl(\003)871 2121 y Fp(.)f(It)h(is)g Fo(H)s Fp(-linear)f(and)h(colinear.)308 2204 y(2.)20 b Fo(A)d Fp(is)g(cosemisimple)e(if)h(and)h(only)f(if)g(the)i(c)o (haracteristic)h(of)d Fo(K)k Fp(do)q(es)e(not)f(divide)361 2254 y(the)e(cardinalit)o(y)d(of)i Fo(R)p Fp(.)308 2337 y(3.)20 b Fo(B)g Fp(is)c(cosemisimple)f(if)h(and)g(only)h(if)f(the)h(c) o(haracteristic)h(of)f Fo(K)j Fp(do)q(es)d(not)g(divide)361 2386 y(the)e(cardinalit)o(y)d(of)i Fo(R)p Fp(.)953 2628 y(68)p eop %%Page: 69 69 69 68 bop 257 262 a Fq(Pro)q(of.)36 b Fp(The)12 b(linear)g(form)e Fo(\025)740 268 y Fj(A)778 262 y Fp(is)i(the)g(tensor)h(pro)q(duct)g (of)e(the)h(in)o(tegral)f Fo(\025)1451 268 y Fj(R)1490 262 y Fp(on)h Fo(K)1584 246 y Fj(R)1623 262 y Fp(and)257 311 y(the)j(in)o(tegral)e Fo(\025)504 317 y Fj(G)546 311 y Fp(on)h Fo(K)s Fp([)p Fo(G)p Fp(])e(that)i(are)h(determined)f(b)o (y)f(the)i(conditions:)715 403 y Fo(\025)739 409 y Fj(R)766 403 y Fp(\()p Fo(e)801 409 y Fj(u)823 403 y Fp(\))d(=)g(1)82 b Fo(\025)1022 409 y Fj(G)1051 403 y Fp(\()p Fo(x)1091 409 y Fj(s)1108 403 y Fp(\))12 b(=)g Fo(\016)1198 409 y Fj(s)p Fn(1)257 494 y Fp(Therefore,)18 b(it)f(is)g(a)g(t)o(w)o (o-sided)g(in)o(tegral)f(itself;)g(the)i(fact)f(that)g(it)g(is)g (linear)f(and)h(colin-)257 544 y(ear)g(follo)o(ws)e(from)g(the)i(fact)g (that)f Fo(\013)p Fp(\(1\))g(=)g Fo(\014)r Fp(\(1\))h(=)f(0,)g(whic)o (h)g(w)o(e)h(ha)o(v)o(e)g(established)g(in)257 594 y(Lemma)f(1.13.)g (Since)i Fo(\025)649 600 y Fj(A)677 594 y Fp(\(1)714 600 y Fj(A)741 594 y Fp(\))g(=)h(card\()p Fo(R)p Fp(\),)f(the)g(second) i(assertion)e(follo)o(ws)f(from)f(the)257 643 y(dual)c(of)h(Masc)o(hk)o (e's)g(theorem)f(for)h(Y)m(etter-Drinfel'd)f(Hopf)h(algebras)f(\(cf.)h ([20)o(],)f(Cor.)g(5.8,)257 693 y(p.)h(4885,)e([81)o(],)h(Cor.)g(2.14,) g(p.)g(23\).)g(The)h(third)g(assertion)h(on)e(the)i(Radford)e(bipro)q (duct)h Fo(B)257 743 y Fp(follo)o(ws)g(from)f([65)o(],)h(Prop.)g(4,)h (p.)f(335.)g Fi(2)257 878 y Fq(5.4)48 b Fp(The)12 b(v)o(ector)h(space)g (underlying)e(the)h(Radford)f(bipro)q(duct)h(is)g Fo(K)1379 863 y Fj(R)1411 878 y Fm(\012)5 b Fo(K)s Fp([)p Fo(G)p Fp(])g Fm(\012)g Fo(K)s Fp([)p Fo(R)p Fp(],)257 928 y(where)17 b(the)f(\014rst)h(t)o(w)o(o)e(tensor)i(factors)e(constitute)i(the)f(Y)m (etter-Drinfel'd)g(Hopf)f(algebra)257 978 y(and)f(the)g(last)g(tensor)g (factor)g(represen)o(ts)j(its)c(base)i(Hopf)e(algebra.)g(A)g(sligh)o(t) g(shift)h(of)f(the)257 1028 y(viewp)q(oin)o(t)k(yields)g(a)f(rather)i (di\013eren)o(t)g(picture:)g(Dividing)d(the)j(triple)f(tensor)h(pro)q (duct)257 1078 y(in)o(to)13 b(t)o(w)o(o)h(parts)g(consisting)f(of)h (the)g(\014rst)g(tensor)h(factor)f(on)f(the)h(one)g(hand)g(and)f(the)i (last)257 1127 y(t)o(w)o(o)k(tensor)g(factors)g(on)g(the)g(other)g (hand,)f(w)o(e)h(get)g(a)g(Hopf)f(algebra)g(extension:)h(The)257 1177 y(\014rst)d(tensor)g(factor)f Fo(K)628 1162 y Fj(R)670 1177 y Fp(is)g(a)g(Hopf)f(subalgebra)h(of)g(the)g(Radford)f(bipro)q (duct,)h(whereas)257 1227 y(the)i(last)f(t)o(w)o(o)f(tensor)i(factors)f (app)q(ear)g(as)g(a)g(Hopf)f(algebra)h(quotien)o(t.)f(W)m(e)h(mak)o(e)e (these)257 1277 y(assertions)h(precise)h(in)d(the)i(follo)o(wing:)257 1377 y Fq(Prop)q(osition)33 b Fp(De\014ne)15 b(the)f(linear)g(mappings) 612 1468 y Fo(\023)d Fp(:)g Fo(K)699 1451 y Fj(R)738 1468 y Fm(!)g Fo(B)r(;)c(e)862 1474 y Fj(u)896 1468 y Fm(7!)k Fo(b)967 1474 y Fj(u)p Fn(0)1005 1468 y Fp(\(1\))h(=)f Fo(e)1132 1474 y Fj(u)1164 1468 y Fm(\012)e Fo(x)1229 1474 y Fn(1)1257 1468 y Fm(\012)g Fo(c)1316 1474 y Fn(0)257 1559 y Fp(and)448 1609 y Fo(\031)k Fp(:)e Fo(B)j Fm(!)d Fo(K)s Fp([)p Fo(G)p Fp(])d Fm(\012)h Fo(K)s Fp([)p Fo(R)p Fp(])p Fo(;)e(b)881 1615 y Fj(uv)919 1609 y Fp(\()p Fo(s)p Fp(\))12 b(=)g Fo(e)1045 1615 y Fj(u)1076 1609 y Fm(\012)e Fo(x)1142 1615 y Fj(s)1169 1609 y Fm(\012)f Fo(c)1228 1615 y Fj(v)1259 1609 y Fm(7!)i Fo(\016)1330 1615 y Fj(u)p Fn(0)1369 1609 y Fo(x)1393 1615 y Fj(s)1420 1609 y Fm(\012)e Fo(c)1479 1615 y Fj(v)257 1684 y Fp(Then)15 b Fo(\023)e Fp(and)h Fo(\031)h Fp(are)f(Hopf)g(algebra)f(homomorphism)o(s)e(and)738 1784 y Fo(K)776 1767 y Fj(R)832 1754 y(\023)815 1784 y Fe(\032)h Fo(B)928 1754 y Fj(\031)918 1784 y Fe(\020)f Fo(K)s Fp([)p Fo(G)p Fp(])d Fm(\012)i Fo(K)s Fp([)p Fo(R)p Fp(])257 1875 y(is)k(a)g(short)g(exact)h(sequence)h(of)d(Hopf)g (algebras.)257 1975 y Fq(Pro)q(of.)36 b Fp(Using)17 b(the)g(fact)f (that)h Fo(\013)p Fp(\(1\))e(=)h Fo(\014)r Fp(\(1\))h(=)f(0)g(from)f (Lemma)f(1.13,)g(it)i(is)h(easy)g(to)257 2025 y(see)e(that)f Fo(\023)g Fp(is)g(an)f(injectiv)o(e)h(Hopf)g(algebra)f(homomorphism)o (.)d(W)m(e)j(ha)o(v)o(e:)364 2116 y Fo(\031)q Fp(\()p Fo(b)423 2122 y Fj(uv)462 2116 y Fp(\()p Fo(s)p Fp(\))p Fo(b)531 2122 y Fj(u)551 2114 y Fh(0)563 2122 y Fj(v)581 2114 y Fh(0)594 2116 y Fp(\()p Fo(s)629 2099 y Fl(0)641 2116 y Fp(\)\))375 2184 y(=)f Fo(\016)437 2191 y Fj(u\027)r Fn(\()p Fj(s)p Fn(\))p Fj(;u)548 2183 y Fh(0)560 2184 y Fo(\021)q Fp(\()p Fo(uq)q Fp(\()p Fo(s;)7 b(s)715 2166 y Fl(0)727 2184 y Fp(\)\))p Fo(\037)p Fp(\(2)p Fo(v)q(u)867 2166 y Fl(0)880 2184 y Fo(\013)p Fp(\()p Fo(s)942 2166 y Fl(0)954 2184 y Fp(\))i(+)g Fo(u)1044 2166 y Fn(2)1063 2184 y Fo(\027)s Fp(\()p Fo(s)p Fp(\))p Fo(\014)r Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)1276 2166 y Fl(0)1289 2184 y Fp(\)\))p Fo(\031)q Fp(\()p Fo(b)1380 2190 y Fj(u;v)q Fn(+)p Fj(v)1471 2182 y Fh(0)1484 2184 y Fp(\()p Fo(ss)1538 2166 y Fl(0)1551 2184 y Fp(\)\))375 2246 y(=)j Fo(\016)437 2252 y Fj(u)p Fn(0)476 2246 y Fo(\016)494 2252 y Fj(u)514 2244 y Fh(0)525 2252 y Fn(0)544 2246 y Fp(\()p Fo(x)584 2252 y Fj(s)611 2246 y Fm(\012)d Fo(c)670 2252 y Fj(v)690 2246 y Fp(\)\()p Fo(x)746 2252 y Fj(s)762 2244 y Fh(0)784 2246 y Fm(\012)h Fo(c)844 2252 y Fj(v)862 2244 y Fh(0)874 2246 y Fp(\))i(=)g Fo(\031)q Fp(\()p Fo(b)1005 2252 y Fj(uv)1044 2246 y Fp(\()p Fo(s)p Fp(\)\))p Fo(\031)q Fp(\()p Fo(b)1170 2252 y Fj(u)1190 2244 y Fh(0)1202 2252 y Fj(v)1220 2244 y Fh(0)1233 2246 y Fp(\()p Fo(s)1268 2229 y Fl(0)1280 2246 y Fp(\)\))257 2337 y(and)k(therefore)i Fo(\031)f Fp(is)f(an)g(algebra)g(homom)o(orphism)o(,)d(since)k(it)e (also)h(preserv)o(es)i(the)f(unit.)257 2387 y(The)k(fact)f(that)g Fo(\031)h Fp(is)e(a)h(coalgebra)g(homom)o(orphism)c(follo)o(ws)i(from)h (a)g(similar)f(direct)257 2437 y(computation.)953 2628 y(69)p eop %%Page: 70 70 70 69 bop 257 262 a Fp(T)m(o)13 b(pro)o(v)o(e)g(the)h(exactness)h(of)d (the)i(ab)q(o)o(v)o(e)f(sequence,)h(it)f(su\016ces)i(to)d(sho)o(w)i 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2392 y Fp(.)953 2628 y(71)p eop %%Page: 72 72 72 71 bop 257 262 a Fp(With)20 b(resp)q(ect)j(to)e(the)g(basis)g Fo(c)783 268 y Fj(u)818 262 y Fm(\012)14 b Fo(d)886 268 y Fj(s)924 262 y Fp(of)20 b Fo(A)1009 246 y Fl(\003)1029 262 y Fp(,)g(these)i(structure)h(elemen)o(ts)d(tak)o(e)h(the)257 311 y(follo)o(wing)12 b(form:)257 394 y Fq(Prop)q(osition)308 444 y Fp(1.)20 b(Multiplication:)12 b(\()p Fo(c)678 450 y Fj(u)709 444 y Fm(\012)d Fo(d)772 450 y Fj(s)790 444 y Fp(\)\()p Fo(c)840 450 y Fj(v)869 444 y Fm(\012)g Fo(d)932 450 y Fj(t)946 444 y Fp(\))j(=)g Fo(\016)1036 450 y Fj(st)1067 444 y Fo(c)1085 450 y Fj(u)p Fn(+)p Fj(v)1159 444 y Fm(\012)d Fo(d)1222 450 y Fj(t)308 522 y Fp(2.)20 b(Unit:)13 b(1)489 528 y Fj(A)514 520 y Fh(\003)545 522 y Fp(=)589 491 y Fg(P)633 535 y Fj(s)p Fl(2)p Fj(G)706 522 y Fo(c)724 528 y Fn(0)751 522 y Fm(\012)d Fo(d)815 528 y Fj(s)308 600 y Fp(3.)20 b(Com)o(ultiplicatio)o(n:)11 b(\001)725 606 y Fj(A)750 598 y Fh(\003)769 600 y Fp(\()p Fo(c)803 606 y Fj(u)834 600 y Fm(\012)e Fo(d)897 606 y Fj(s)915 600 y Fp(\))i(=)370 633 y Fg(P)361 701 y Fj(t)p Fl(2)p Fj(G)429 664 y Fo(\021)q Fp(\()p Fo(uq)q Fp(\()p Fo(t;)c(t)576 649 y Fl(\000)p Fn(1)620 664 y Fo(s)p Fp(\)\))p Fo(\037)p Fp(\()p Fm(\000)p Fo(u)769 649 y Fn(2)789 664 y Fo(\027)s Fp(\()p Fo(t)p Fp(\))p Fo(\014)r Fp(\()p Fo(t)p Fp(\))p Fo(\013)p Fp(\()p Fo(t)990 649 y Fl(\000)p Fn(1)1034 664 y Fo(s)p Fp(\)\)\()p Fo(c)1119 671 y Fj(u\027)r Fn(\()p Fj(t)p Fn(\))1208 664 y Fm(\012)j Fo(d)1272 672 y Fj(t)1285 663 y Fh(\000)p Fd(1)1322 672 y Fj(s)1340 664 y Fp(\))g Fm(\012)f Fp(\()p Fo(c)1441 670 y Fj(u)1472 664 y Fm(\012)h Fo(d)1536 670 y Fj(t)1550 664 y Fp(\))308 771 y(4.)20 b(Counit:)13 b Fo(\017)528 777 y Fj(A)553 769 y Fh(\003)572 771 y Fp(\()p Fo(c)606 777 y Fj(u)637 771 y Fm(\012)d Fo(d)701 777 y Fj(s)718 771 y Fp(\))i(=)g Fo(\016)808 777 y Fj(s)p Fn(1)308 849 y Fp(5.)20 b(An)o(tip)q(o)q(de:)14 b Fo(S)580 855 y Fj(A)605 846 y Fh(\003)625 849 y Fp(\()p Fo(c)659 855 y Fj(u)690 849 y Fm(\012)9 b 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1011 y Fl(\000)p Fj(u\014)q Fn(\()p Fj(s)p Fn(\))257 1096 y Fq(Pro)q(of.)36 b Fp(This)13 b(rests)i(on)d(straigh)o(tforw)o(ard)h(v)o(eri\014cation.)f(W)m(e)g (therefore)j(only)d(pro)o(v)o(e)h(the)257 1146 y(more)e(complicated)e (parts)j(concerning)g(the)g(com)o(ultiplication)c(and)j(the)h(an)o(tip) q(o)q(de.)e(F)m(rom)257 1196 y(the)15 b(bilinear)f(form)f Fm(h\001)p Fo(;)7 b Fm(\001i)654 1202 y Fj(A)679 1196 y Fp(,)14 b(w)o(e)h(get)g(as)f(in)g([79)o(],)f(Subsec.)j(2.5)d(a)h (nondegenerate)i(bilinear)257 1246 y(pairing)10 b(b)q(et)o(w)o(een)j Fo(A)t Fm(\012)t Fo(A)e Fp(and)h Fo(A)776 1231 y Fl(\003)799 1246 y Fm(\012)t Fo(A)866 1231 y Fl(\003)885 1246 y Fp(,)f(whic)o(h)g (w)o(e)g(also)g(denote)h(b)o(y)f Fm(h\001)p Fo(;)c Fm(\001i)1424 1252 y Fj(A)1450 1246 y Fp(.)j(If)h(w)o(e)h(denote)257 1295 y(the)j(m)o(ultiplicatio)o(n)c(mapping)h(of)h Fo(A)h Fp(b)o(y)g Fo(\026)941 1301 y Fj(A)968 1295 y Fp(,)f(w)o(e)h(ha)o(v)o (e)g(b)o(y)g([79)o(],)f(Subsec.)h(4.4:)263 1374 y Fm(h)p Fp(\()p Fo(e)314 1380 y Fj(v)343 1374 y Fm(\012)c Fo(x)409 1380 y Fj(p)428 1374 y Fp(\))f Fm(\012)h Fp(\()p Fo(e)530 1380 y Fj(w)566 1374 y Fm(\012)g Fo(x)632 1380 y Fj(r)650 1374 y Fp(\))p Fo(;)d Fp(\001)720 1380 y Fj(A)745 1372 y Fh(\003)764 1374 y Fp(\()p Fo(c)798 1380 y Fj(u)829 1374 y Fm(\012)i Fo(d)892 1380 y Fj(s)909 1374 y Fp(\))p Fm(i)941 1380 y Fj(A)274 1442 y Fp(=)j Fm(h)p Fo(\026)359 1448 y Fj(A)396 1442 y Fm(\016)d Fo(\033)451 1424 y Fl(\000)p Fn(1)450 1454 y Fj(A;A)511 1442 y Fp(\(\()p Fo(e)562 1448 y Fj(v)592 1442 y Fm(\012)g Fo(x)657 1448 y Fj(p)676 1442 y Fp(\))h Fm(\012)f Fp(\()p Fo(e)778 1448 y Fj(w)815 1442 y Fm(\012)g Fo(x)880 1448 y Fj(r)898 1442 y Fp(\)\))p Fo(;)e(c)967 1448 y Fj(u)998 1442 y Fm(\012)i Fo(d)1061 1448 y Fj(s)1079 1442 y Fm(i)1095 1448 y Fj(A)274 1512 y Fp(=)j Fo(\037)p Fp(\()p Fm(\000)p Fo(w)q(v)q(\014)r Fp(\()p Fo(r)q Fp(\))p Fo(\013)p Fp(\()p Fo(p)p Fp(\)\))617 1495 y Fn(2)638 1512 y Fm(h)p Fp(\()p Fo(e)689 1518 y Fj(w)725 1512 y Fm(\012)e Fo(x)791 1518 y 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Fp(\))p Fo(\014)r Fp(\()p Fo(t)p Fp(\))p Fo(\013)p Fp(\()p Fo(t)704 1638 y Fl(\000)p Fn(1)749 1655 y Fo(s)p Fp(\)\))800 1638 y Fn(2)819 1655 y Fo(\021)q Fp(\()p Fo(uq)q Fp(\()p Fo(t;)d(t)966 1638 y Fl(\000)p Fn(1)1010 1655 y Fo(s)p Fp(\)\))p Fo(\037)p Fp(\()p Fo(u)1127 1638 y Fn(2)1146 1655 y Fo(\027)s Fp(\()p Fo(t)p Fp(\))p Fo(\014)r Fp(\()p Fo(t)p Fp(\))p Fo(\013)p Fp(\()p Fo(t)1347 1638 y Fl(\000)p Fn(1)1392 1655 y Fo(s)p Fp(\)\))720 1762 y Fm(h)p Fp(\()p Fo(e)771 1768 y Fj(v)800 1762 y Fm(\012)j Fo(x)866 1768 y Fj(p)885 1762 y Fp(\))f Fm(\012)g Fp(\()p Fo(e)986 1768 y Fj(w)1023 1762 y Fm(\012)g Fo(x)1088 1768 y Fj(r)1107 1762 y Fp(\))p Fo(;)e Fp(\()p Fo(c)1176 1769 y Fj(u\027)r Fn(\()p Fj(t)p Fn(\))1263 1762 y Fm(\012)j Fo(d)1327 1770 y Fj(t)1340 1762 y Fh(\000)p Fd(1)1378 1770 y Fj(s)1395 1762 y Fp(\))g Fm(\012)f Fp(\()p Fo(c)1496 1768 y Fj(u)1527 1762 y Fm(\012)h Fo(d)1591 1768 y Fj(t)1605 1762 y Fp(\))p Fm(i)1637 1768 y Fj(A)274 1824 y Fp(=)i Fm(h)p Fp(\()p Fo(e)369 1830 y Fj(v)399 1824 y Fm(\012)d Fo(x)464 1830 y Fj(p)483 1824 y Fp(\))h Fm(\012)f Fp(\()p Fo(e)585 1830 y Fj(w)622 1824 y Fm(\012)g Fo(x)687 1830 y Fj(r)705 1824 y Fp(\))p Fo(;)436 1860 y Fg(X)436 1949 y Fj(t)p Fl(2)p Fj(G)504 1900 y Fo(\021)q Fp(\()p Fo(uq)q Fp(\()p Fo(t;)e(t)651 1882 y Fl(\000)p Fn(1)695 1900 y Fo(s)p Fp(\)\))p Fo(\037)p Fp(\()p Fm(\000)p Fo(u)844 1882 y Fn(2)863 1900 y Fo(\027)s Fp(\()p Fo(t)p Fp(\))p Fo(\014)r Fp(\()p Fo(t)p Fp(\))p Fo(\013)p Fp(\()p Fo(t)1064 1882 y Fl(\000)p Fn(1)1109 1900 y Fo(s)p Fp(\)\)\()p Fo(c)1194 1907 y Fj(u\027)r Fn(\()p Fj(t)p Fn(\))1283 1900 y Fm(\012)i Fo(d)1346 1907 y Fj(t)1359 1899 y Fh(\000)p Fd(1)1397 1907 y Fj(s)1415 1900 y Fp(\))g Fm(\012)h Fp(\()p Fo(c)1516 1906 y Fj(u)1547 1900 y Fm(\012)f Fo(d)1610 1906 y Fj(t)1625 1900 y Fp(\))p Fm(i)1657 1906 y Fj(A)257 2017 y Fp(Since)14 b(the)f(pairing)e(b)q(et)o(w)o(een)j Fo(A)7 b Fm(\012)g Fo(A)13 b Fp(and)g Fo(A)965 2002 y Fl(\003)990 2017 y Fm(\012)7 b Fo(A)1060 2002 y Fl(\003)1092 2017 y Fp(is)13 b(nondegenerate,)h(this)e(establishes)257 2067 y(the)j(form)d(of)h(the)i(com)o(ultiplicatio)o(n.)257 2147 y(T)m(o)g(establish)g(the)h(form)e(of)g(the)i(an)o(tip)q(o)q(de,)f (w)o(e)g(tak)o(e)g(the)h(form)o(ula)d(giv)o(en)i(ab)q(o)o(v)o(e)g(as)g (the)257 2197 y(de\014nition)d(of)f(a)g(linear)h(endomorphism)d Fo(S)933 2203 y Fj(A)958 2195 y Fh(\003)990 2197 y Fp(of)i Fo(A)1066 2182 y Fl(\003)1085 2197 y Fp(.)g(It)h(is)g(then)g(easy)h(to) e(v)o(erify)h(that)f(this)257 2247 y(endomorphism)h(satis\014es:)517 2325 y Fm(h)p Fo(S)558 2331 y Fj(A)585 2325 y Fp(\()p Fo(e)620 2331 y Fj(u)652 2325 y Fm(\012)d Fo(x)717 2331 y Fj(s)735 2325 y Fp(\))p Fo(;)e(S)795 2331 y Fj(A)820 2323 y Fh(\003)839 2325 y Fp(\()p Fo(c)873 2331 y Fj(v)902 2325 y Fm(\012)j Fo(d)966 2331 y Fj(t)980 2325 y Fp(\))p Fm(i)1012 2331 y Fj(A)1051 2325 y Fp(=)i Fm(h)p Fo(e)1130 2331 y Fj(u)1161 2325 y Fm(\012)d Fo(x)1226 2331 y Fj(s)1244 2325 y Fo(;)e(c)1281 2331 y Fj(v)1309 2325 y Fm(\012)j Fo(d)1373 2331 y Fj(t)1387 2325 y Fm(i)1403 2331 y Fj(A)257 2403 y Fp(This)18 b(implies)d(that)i(w)o(e)h(ha)o(v)o(e)f Fm(h)p Fo(S)800 2386 y Fl(\000)p Fn(1)798 2416 y Fj(A)845 2403 y Fp(\()p Fo(a)p Fp(\))p Fo(;)7 b(b)p Fm(i)952 2409 y Fj(A)996 2403 y Fp(=)18 b Fm(h)p Fo(a;)7 b(S)1128 2409 y Fj(A)1153 2401 y Fh(\003)1172 2403 y Fp(\()p Fo(b)p Fp(\))p Fm(i)1238 2409 y Fj(A)1282 2403 y Fp(for)17 b(all)f Fo(a)i Fm(2)e Fo(A)i Fp(and)f Fo(b)g Fm(2)257 2453 y Fo(A)288 2438 y Fl(\003)308 2453 y Fp(.)d(Therefore,)i(it)f(follo)o(ws) f(from)f([79)o(],)h(Subsec.)j(4.4)d(that)h Fo(S)1240 2459 y Fj(A)1265 2451 y Fh(\003)1300 2453 y Fp(really)g(is)g(the)h(an)o (tip)q(o)q(de)257 2503 y(of)e Fo(A)336 2488 y Fl(\003)355 2503 y Fp(.)f Fi(2)953 2628 y Fp(72)p eop %%Page: 73 73 73 72 bop 257 262 a Fq(5.7)48 b Fp(The)15 b(adjoin)o(t)d(action)i(of)g (a)f(Hopf)h(algebra)g(on)f(itself)h(has)g(v)n(arious)g(generalizations) 257 311 y(for)f(Y)m(etter-Drinfel'd)g(Hopf)f(algebras,)g(dep)q(ending)i (on)f(ho)o(w)f(the)i(necessary)g(in)o(terc)o(hang-)257 361 y(ing)d(of)h(the)g(tensor)h(factors)f(is)g(accomplished)e(and)i (whether)h(the)g(an)o(tip)q(o)q(de)e(or)h(its)g(in)o(v)o(erse)257 411 y(is)g(used)h(in)e(the)i(de\014nition.)e(As)h(in)g([79)o(],)e (Subsec.)j(4.6,)e(w)o(e)h(here)h(consider)g(the)f(left)g(adjoin)o(t)257 461 y(action)18 b(constructed)h(with)e(the)i(in)o(v)o(erse)f (quasisymmetry)d(and)j(the)g(in)o(v)o(erse)g(an)o(tip)q(o)q(de,)257 511 y(whic)o(h)c(w)o(e)g(denote)h(b)o(y)f Fo(+)p Fp(:)510 646 y Fo(a)d(+)g(a)618 629 y Fl(0)641 646 y Fp(:=)g Fo(\026)721 652 y Fj(A)749 646 y Fp(\()p Fo(\026)790 652 y Fj(A)826 646 y Fm(\012)e Fo(S)894 628 y Fl(\000)p Fn(1)892 658 y Fj(A)940 646 y Fp(\))p Fo(\033)981 628 y Fl(\000)p Fn(1)980 658 y Fj(A)p Fl(\012)p Fj(A;A)1093 646 y Fp(\(\001)1144 652 y Fj(A)1180 646 y Fm(\012)g Fp(id)1256 652 y Fj(A)1283 646 y Fp(\)\()p Fo(a)g Fm(\012)h Fo(a)1410 629 y Fl(0)1421 646 y Fp(\))257 756 y(Via)15 b(the)g(left)g(adjoin)o(t)f(action,)g Fo(A)h Fp(b)q(ecomes)h(a)e(left)h Fo(A)p Fp(-mo)q(dule.)e(Similarly)l (,)f Fo(A)1504 741 y Fl(\003)1538 756 y Fp(b)q(ecomes)257 806 y(a)i(righ)o(t)f Fo(A)423 791 y Fl(\003)443 806 y Fp(-mo)q(dule)f(via)h(the)h(righ)o(t)g(adjoin)o(t)f(action:)463 898 y Fo(b)481 880 y Fl(0)504 898 y Fo(\))e(b)g Fp(:=)h Fo(\026)667 904 y Fj(A)692 895 y Fh(\003)711 898 y Fp(\()p Fo(S)754 880 y Fl(\000)p Fn(1)752 910 y Fj(A)777 901 y Fh(\003)809 898 y Fm(\012)d Fo(\026)875 904 y Fj(A)900 895 y Fh(\003)920 898 y Fp(\))p Fo(\033)961 880 y Fl(\000)p Fn(1)960 910 y Fj(A)985 901 y Fh(\003)1002 910 y Fj(;A)1037 901 y Fh(\003)1054 910 y Fl(\012)p Fj(A)1105 901 y Fh(\003)1124 898 y Fp(\(id)1175 904 y Fj(A)1200 895 y Fh(\003)1226 898 y Fm(\012)p Fp(\001)1293 904 y Fj(A)1318 895 y Fh(\003)1338 898 y Fp(\)\()p Fo(b)1388 880 y Fl(0)1409 898 y Fm(\012)g Fo(b)p Fp(\))257 989 y(where)j(w)o(e)f(use)g(the)g(mo)q(di\014ed)f (structures)j(from)8 b(the)k(previous)e(paragraph.)g(With)g(resp)q(ect) 257 1039 y(to)17 b(the)g(bases)g Fo(e)515 1045 y Fj(u)548 1039 y Fm(\012)11 b Fo(x)615 1045 y Fj(s)649 1039 y Fp(resp.)17 b Fo(c)769 1045 y Fj(u)802 1039 y Fm(\012)11 b Fo(d)867 1045 y Fj(s)901 1039 y Fp(of)16 b Fo(A)g Fp(resp.)h Fo(A)1131 1024 y Fl(\003)1151 1039 y Fp(,)f(the)h(adjoin)o(t)e(actions)h(tak)o(e) h(the)257 1089 y(follo)o(wing)12 b(form:)257 1188 y Fq(Prop)q(osition) 33 b Fp(F)m(or)14 b Fo(u;)7 b(v)12 b Fm(2)f Fo(R)j Fp(and)g Fo(s;)7 b(t)k Fm(2)g Fo(G)p Fp(,)i(w)o(e)h(ha)o(v)o(e:)308 1307 y(1.)20 b(\()p Fo(e)396 1313 y Fj(u)428 1307 y Fm(\012)9 b Fo(x)493 1313 y Fj(s)511 1307 y Fp(\))i Fo(+)g Fp(\()p Fo(e)626 1313 y Fj(v)656 1307 y Fm(\012)e Fo(x)721 1313 y Fj(t)736 1307 y Fp(\))i(=)361 1357 y Fo(\016)379 1364 y Fj(u\027)r Fn(\()p Fj(s)p Fn(\))p Fj(;v)q Fl(\000)p Fj(v)q(\027)r Fn(\()p Fj(t)p Fn(\))590 1357 y Fo(\021)q Fp(\()p Fo(v)q(\027)s Fp(\()p Fo(s)708 1342 y Fl(\000)p Fn(1)753 1357 y Fp(\)\()p Fo(q)q Fp(\()p Fo(s;)c(t)p Fp(\))j(+)f Fo(q)q Fp(\()p Fo(st;)e(s)1049 1342 y Fl(\000)p Fn(1)1094 1357 y Fp(\))j Fm(\000)f Fo(\027)s Fp(\()p Fo(sts)1254 1342 y Fl(\000)p Fn(1)1299 1357 y Fp(\))p Fo(q)q Fp(\()p Fo(s;)e(s)1408 1342 y Fl(\000)p Fn(1)1453 1357 y Fp(\)\)\))361 1407 y Fo(\037)p Fp(\()p Fo(v)424 1391 y Fn(2)443 1407 y Fo(\027)s Fp(\()p Fo(s)502 1391 y Fl(\000)p Fn(2)547 1407 y Fp(\)\()p Fo(\027)s Fp(\()p Fo(s)p Fp(\))p Fo(\014)r Fp(\()p Fo(t)p Fp(\))j(+)g Fo(\027)s Fp(\()p Fo(t)p Fp(\))p Fo(\014)r Fp(\()p Fo(st)p Fp(\))g Fm(\000)f Fo(\027)s Fp(\()p Fo(t)1046 1391 y Fn(2)1064 1407 y Fp(\))p Fo(\014)r Fp(\()p Fo(s)p Fp(\)\))p Fo(\013)p Fp(\()p Fo(s)p Fp(\)\))p Fo(e)1285 1414 y Fj(v)q(\027)r Fn(\()p Fj(s)1351 1406 y Fh(\000)p Fd(1)1391 1414 y Fn(\))1416 1407 y Fm(\012)g Fo(x)1481 1414 y Fj(sts)1526 1406 y Fh(\000)p Fd(1)308 1490 y Fp(2.)20 b(\()p Fo(c)395 1496 y Fj(v)424 1490 y Fm(\012)10 b Fo(d)488 1496 y Fj(t)502 1490 y Fp(\))i Fo(\))f Fp(\()p Fo(c)617 1496 y Fj(u)648 1490 y Fm(\012)e Fo(d)711 1496 y Fj(s)729 1490 y Fp(\))i(=)h Fo(\016)818 1496 y Fj(s)p Fn(1)853 1490 y Fo(\037)p Fp(\()p Fm(\000)p Fo(v)q(u\027)s Fp(\()p Fo(t)p Fp(\))p Fo(\014)r Fp(\()p Fo(t)p Fp(\))p Fo(\013)p Fp(\()p Fo(t)p Fp(\)\))1205 1474 y Fn(2)1225 1490 y Fo(c)1243 1496 y Fj(v)1272 1490 y Fm(\012)d Fo(d)1335 1496 y Fj(t)257 1589 y Fq(Pro)q(of.)36 b Fp(Using)15 b(Heyneman-Sw)o(eedler)f(sigma)f(notation,)g(the)i(left)g (adjoin)o(t)e(action)h(can)257 1639 y(also)g(b)q(e)g(written)h(in)e (the)h(form:)534 1730 y Fo(a)d(+)g(a)642 1713 y Fl(0)666 1730 y Fp(=)g Fo(a)731 1737 y Fn(\(2\))776 1713 y(\(2\))820 1730 y Fo(a)842 1713 y Fl(0)r Fn(\(2\))899 1730 y Fo(S)926 1713 y Fl(\000)p Fn(1)924 1743 y Fj(A)971 1730 y Fp(\()p Fo(S)1012 1736 y Fj(H)1044 1730 y Fp(\()p Fo(a)1082 1737 y Fn(\(2\))1127 1713 y(\(1\))1171 1730 y Fo(a)1193 1713 y Fl(0)r Fn(\(1\))1250 1730 y Fp(\))g Fm(!)g Fo(a)1352 1737 y Fn(\(1\))1397 1730 y Fp(\))257 1822 y(With)20 b(Lemma)e(1.13,)h(w)o(e)i(see)h(that)f(the)g(in)o(v)o(erse)g(of)f(the)h (an)o(tip)q(o)q(de)g(is)f(giv)o(en)g(b)o(y)h(the)257 1871 y(equation:)437 1963 y Fo(S)464 1945 y Fl(\000)p Fn(1)462 1975 y Fj(A)510 1963 y Fp(\()p Fo(e)545 1969 y Fj(u)576 1963 y Fm(\012)9 b Fo(x)641 1969 y Fj(s)659 1963 y Fp(\))j(=)f Fo(\021)q Fp(\()p Fo(uq)q Fp(\()p Fo(s;)c(s)885 1946 y Fl(\000)p Fn(1)931 1963 y Fp(\)\))p Fo(\037)p Fp(\()p Fm(\000)p Fo(u)1061 1946 y Fn(2)1080 1963 y Fo(\014)r Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)p Fp(\)\))p Fo(e)1269 1970 y Fl(\000)p Fj(u\027)r Fn(\()p Fj(s)p Fn(\))1388 1963 y Fm(\012)j Fo(x)1454 1971 y Fj(s)1470 1962 y Fh(\000)p Fd(1)257 2054 y Fp(Using)20 b(this)f(lemma)d(again,)i(together)j(with)e(our)g(assumption)f(that)i Fo(\037)p Fp(\()p Fo(u\013)p Fp(\()p Fo(s)p Fp(\))p Fo(\014)r Fp(\()p Fo(t)p Fp(\)\))i(=)257 2104 y Fo(\037)p Fp(\()p Fo(u\014)r Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(t)p Fp(\)\),)15 b(w)o(e)f(get:)300 2195 y(\()p Fo(e)335 2201 y Fj(u)367 2195 y Fm(\012)9 b Fo(x)432 2201 y Fj(s)450 2195 y Fp(\))i Fo(+)g Fp(\()p Fo(e)565 2201 y Fj(v)595 2195 y Fm(\012)e Fo(x)660 2201 y Fj(t)675 2195 y Fp(\))312 2270 y(=)362 2231 y Fg(X)356 2320 y Fj(w)q Fl(2)p Fj(R)428 2270 y Fp(\()p Fo(e)463 2276 y Fj(w)500 2270 y Fm(\012)g Fo(x)565 2276 y Fj(s)583 2270 y Fp(\))599 2253 y Fn(\(2\))643 2270 y Fp(\()p Fo(e)678 2276 y Fj(v)708 2270 y Fm(\012)g Fo(x)773 2276 y Fj(t)788 2270 y Fp(\))804 2253 y Fn(\(2\))623 2382 y Fo(S)650 2364 y Fl(\000)p Fn(1)648 2394 y Fj(A)696 2382 y Fp(\()p Fo(S)737 2388 y Fj(H)769 2382 y Fp(\(\()p Fo(e)820 2388 y Fj(w)856 2382 y Fm(\012)h Fo(x)922 2388 y Fj(s)939 2382 y Fp(\))955 2365 y Fn(\(1\))1000 2382 y Fp(\()p Fo(e)1035 2388 y Fj(v)1065 2382 y Fm(\012)f Fo(x)1130 2388 y Fj(t)1144 2382 y Fp(\))1160 2365 y Fn(\(1\))1205 2382 y Fp(\))j Fm(!)f Fp(\()p Fo(e)1321 2388 y Fj(u)p Fl(\000)p Fj(w)1403 2382 y Fm(\012)e Fo(x)1468 2388 y Fj(s)1486 2382 y Fp(\)\))312 2457 y(=)362 2418 y Fg(X)356 2507 y Fj(w)q Fl(2)p Fj(R)428 2457 y Fp(\()p Fo(e)463 2463 y Fj(w)500 2457 y Fm(\012)g Fo(x)565 2463 y Fj(s)583 2457 y Fp(\)\()p Fo(e)634 2463 y Fj(v)663 2457 y Fm(\012)h Fo(x)729 2463 y Fj(t)743 2457 y Fp(\))p Fo(S)786 2439 y Fl(\000)p Fn(1)784 2469 y Fj(A)832 2457 y Fp(\()p Fo(c)866 2464 y Fl(\000)p Fj(w)q(\014)q Fn(\()p Fj(s)p Fn(\))p Fl(\000)p Fj(v)q(\014)q Fn(\()p Fj(t)p Fn(\))1094 2457 y Fm(!)h Fp(\()p Fo(e)1182 2463 y Fj(u)p Fl(\000)p Fj(w)1264 2457 y Fm(\012)f Fo(x)1330 2463 y Fj(s)1347 2457 y Fp(\)\))953 2628 y(73)p eop %%Page: 74 74 74 73 bop 312 264 a Fp(=)362 224 y Fg(X)356 313 y Fj(w)q Fl(2)p Fj(R)435 264 y Fo(\016)453 271 y Fj(w)q(\027)r Fn(\()p Fj(s)p Fn(\))p Fj(;v)568 264 y Fo(\021)q Fp(\()p Fo(w)q(q)q Fp(\()p Fo(s;)7 b(t)p Fp(\)\))p Fo(\037)p Fp(\()p Fo(w)831 246 y Fn(2)850 264 y Fo(\027)s Fp(\()p Fo(s)p Fp(\))p Fo(\014)r Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(t)p Fp(\)\))383 373 y Fo(\037)p Fp(\()p Fm(\000)p Fp(\()p Fo(w)q(\014)r Fp(\()p Fo(s)p Fp(\))k(+)f Fo(v)q(\014)r Fp(\()p Fo(t)p Fp(\)\)\()p Fo(u)h Fm(\000)e Fo(w)q Fp(\))p Fo(\013)p Fp(\()p Fo(s)p Fp(\)\))975 355 y Fn(2)994 373 y Fp(\()p Fo(e)1029 379 y Fj(w)1066 373 y Fm(\012)h Fo(x)1132 379 y Fj(st)1162 373 y Fp(\))p Fo(S)1205 355 y Fl(\000)p Fn(1)1203 385 y Fj(A)1250 373 y Fp(\()p Fo(e)1285 379 y Fj(u)p Fl(\000)p Fj(w)1367 373 y Fm(\012)g Fo(x)1433 379 y Fj(s)1450 373 y Fp(\))312 440 y(=)i Fo(\021)q Fp(\()p Fo(v)q(\027)s Fp(\()p Fo(s)474 423 y Fl(\000)p Fn(1)519 440 y Fp(\))p Fo(q)q Fp(\()p Fo(s;)7 b(t)p Fp(\)\))p Fo(\037)p Fp(\()p Fo(v)719 423 y Fn(2)739 440 y Fo(\027)s Fp(\()p Fo(s)798 423 y Fl(\000)p Fn(1)842 440 y Fp(\))p Fo(\014)r Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(t)p Fp(\)\))383 507 y Fo(\037)p Fp(\()p Fo(v)q Fp(\()p Fo(\027)s Fp(\()p Fo(s)521 490 y Fl(\000)p Fn(1)567 507 y Fp(\))p Fo(\014)r Fp(\()p Fo(s)p Fp(\))j(+)g Fo(\014)r Fp(\()p Fo(t)p Fp(\)\)\()p Fo(v)q(\027)s Fp(\()p Fo(s)895 490 y Fl(\000)p Fn(1)941 507 y Fp(\))f Fm(\000)h Fo(u)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)p Fp(\)\))1142 490 y Fn(2)383 575 y Fp(\()p Fo(e)418 583 y Fj(v)q(\027)r Fn(\()p Fj(s)484 575 y Fh(\000)p Fd(1)522 583 y Fn(\))547 575 y Fm(\012)f Fo(x)612 581 y Fj(st)642 575 y Fp(\))p Fo(S)685 558 y Fl(\000)p Fn(1)683 588 y Fj(A)731 575 y Fp(\()p Fo(e)766 583 y Fj(u)p Fl(\000)p Fj(v)q(\027)r Fn(\()p Fj(s)878 575 y Fh(\000)p Fd(1)915 583 y Fn(\))940 575 y Fm(\012)g Fo(x)1005 581 y Fj(s)1023 575 y Fp(\))312 643 y(=)j Fo(\021)q Fp(\()p Fo(v)q(\027)s Fp(\()p Fo(s)474 626 y Fl(\000)p Fn(1)519 643 y Fp(\))p Fo(q)q Fp(\()p Fo(s;)7 b(t)p Fp(\)\))p Fo(\037)p Fp(\()p Fo(v)719 626 y Fn(2)739 643 y Fo(\027)s Fp(\()p Fo(s)798 626 y Fl(\000)p Fn(1)842 643 y Fp(\))p Fo(\014)r Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(t)p Fp(\)\))383 710 y Fo(\037)p Fp(\()p Fo(v)q(\027)s Fp(\()p Fo(s)505 693 y Fl(\000)p Fn(1)550 710 y Fp(\)\()p Fo(\014)r Fp(\()p Fo(s)p Fp(\))k(+)f Fo(\027)s Fp(\()p Fo(s)p Fp(\))p Fo(\014)r Fp(\()p Fo(t)p Fp(\)\)\()p Fo(v)q(\027)s Fp(\()p Fo(s)970 693 y Fl(\000)p Fn(1)1016 710 y Fp(\))f Fm(\000)h Fo(u)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)p Fp(\)\))1217 693 y Fn(2)383 778 y Fo(\021)q Fp(\(\()p Fo(u)g Fm(\000)f Fo(v)q(\027)s Fp(\()p Fo(s)592 761 y Fl(\000)p Fn(1)637 778 y Fp(\)\))p Fo(q)q Fp(\()p Fo(s;)e(s)762 761 y Fl(\000)p Fn(1)808 778 y Fp(\)\))p Fo(\037)p Fp(\()p Fm(\000)p Fp(\()p Fo(u)i Fm(\000)h Fo(v)q(\027)s Fp(\()p Fo(s)1085 761 y Fl(\000)p Fn(1)1130 778 y Fp(\)\))1162 761 y Fn(2)1181 778 y Fo(\014)r Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)p Fp(\)\))383 840 y(\()p Fo(e)418 848 y Fj(v)q(\027)r Fn(\()p Fj(s)484 840 y Fh(\000)p Fd(1)522 848 y Fn(\))547 840 y Fm(\012)f Fo(x)612 846 y Fj(st)642 840 y Fp(\)\()p Fo(e)693 847 y Fj(v)q Fl(\000)p Fj(u\027)r Fn(\()p Fj(s)p Fn(\))829 840 y Fm(\012)g Fo(x)894 848 y Fj(s)910 840 y Fh(\000)p Fd(1)t Fp(\))312 908 y(=)j Fo(\021)q Fp(\()p Fo(v)q(\027)s Fp(\()p Fo(s)474 891 y Fl(\000)p Fn(1)519 908 y Fp(\))p Fo(q)q Fp(\()p Fo(s;)7 b(t)p Fp(\)\))p Fo(\037)p Fp(\()p Fo(v)719 891 y Fn(2)739 908 y Fo(\027)s Fp(\()p Fo(s)798 891 y Fl(\000)p Fn(1)842 908 y Fp(\))p Fo(\014)r Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(t)p Fp(\)\))383 975 y Fo(\037)p Fp(\()p Fo(v)q(\027)s Fp(\()p Fo(s)505 958 y Fl(\000)p Fn(1)550 975 y Fp(\))p Fo(\014)r Fp(\()p Fo(st)p Fp(\)\()p Fo(v)q(\027)s Fp(\()p Fo(s)753 958 y Fl(\000)p Fn(1)800 975 y Fp(\))i Fm(\000)h Fo(u)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)p Fp(\)\))1001 958 y Fn(2)1020 975 y Fo(\021)q Fp(\(\()p Fo(u)f Fm(\000)h Fo(v)q(\027)s Fp(\()p Fo(s)1229 958 y Fl(\000)p Fn(1)1274 975 y Fp(\)\))p Fo(q)q Fp(\()p Fo(s;)d(s)1399 958 y Fl(\000)p Fn(1)1444 975 y Fp(\)\))383 1043 y Fo(\037)p Fp(\()p Fm(\000)p Fp(\()p Fo(u)j Fm(\000)f Fo(v)q(\027)s Fp(\()p Fo(s)628 1025 y Fl(\000)p Fn(1)673 1043 y Fp(\)\))705 1025 y Fn(2)724 1043 y Fo(\014)r Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)p Fp(\)\))p Fo(\016)912 1050 y Fj(v)q(\027)r Fn(\()p Fj(t)p Fn(\))p Fj(;v)q Fl(\000)p Fj(u\027)r Fn(\()p Fj(s)p Fn(\))1125 1043 y Fo(\021)q Fp(\()p Fo(v)q(\027)s Fp(\()p Fo(s)1243 1025 y Fl(\000)p Fn(1)1288 1043 y Fp(\))p Fo(q)q Fp(\()p Fo(st;)e(s)1412 1025 y Fl(\000)p Fn(1)1457 1043 y Fp(\)\))383 1110 y Fo(\037)p Fp(\()p Fo(v)446 1093 y Fn(2)466 1110 y Fo(\027)s Fp(\()p Fo(s)525 1093 y Fl(\000)p Fn(2)569 1110 y Fp(\))p Fo(\027)s Fp(\()p Fo(st)p Fp(\))p Fo(\014)r Fp(\()p Fo(st)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)828 1093 y Fl(\000)p Fn(1)874 1110 y Fp(\)\))p Fo(e)925 1118 y Fj(v)q(\027)r Fn(\()p Fj(s)991 1110 y Fh(\000)p Fd(1)1030 1118 y Fn(\))1054 1110 y Fm(\012)i Fo(x)1119 1118 y Fj(sts)1164 1110 y Fh(\000)p Fd(1)312 1178 y Fp(=)j Fo(\016)374 1185 y Fj(u\027)r Fn(\()p Fj(s)p Fn(\))p Fj(;v)q Fl(\000)p Fj(v)q(\027)r Fn(\()p Fj(t)p Fn(\))584 1178 y Fo(\021)q Fp(\()p Fo(v)q(\027)s Fp(\()p Fo(s)702 1160 y Fl(\000)p Fn(1)747 1178 y Fp(\))p Fo(q)q Fp(\()p Fo(s;)7 b(t)p Fp(\)\))p Fo(\021)q Fp(\()p Fm(\000)p Fo(v)q(\027)s Fp(\()p Fo(ts)1049 1160 y Fl(\000)p Fn(1)1095 1178 y Fp(\))p Fo(q)q Fp(\()p Fo(s;)g(s)1204 1160 y Fl(\000)p Fn(1)1250 1178 y Fp(\)\))p Fo(\021)q Fp(\()p Fo(v)q(\027)s Fp(\()p Fo(s)1400 1160 y Fl(\000)p Fn(1)1445 1178 y Fp(\))p Fo(q)q Fp(\()p Fo(st;)g(s)1569 1160 y Fl(\000)p Fn(1)1614 1178 y Fp(\)\))383 1245 y Fo(\037)p Fp(\()p Fo(v)446 1228 y Fn(2)466 1245 y Fo(\027)s Fp(\()p Fo(s)525 1228 y Fl(\000)p Fn(1)569 1245 y Fp(\))p Fo(\014)r Fp(\()p Fo(t)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)p Fp(\)\))p Fo(\037)p Fp(\()p Fo(v)814 1228 y Fn(2)835 1245 y Fo(\027)s Fp(\()p Fo(s)894 1228 y Fl(\000)p Fn(1)938 1245 y Fp(\))p Fo(\014)r Fp(\()p Fo(st)p Fp(\))p Fo(\027)s Fp(\()p Fo(ts)1119 1228 y Fl(\000)p Fn(1)1165 1245 y Fp(\))p Fo(\013)p Fp(\()p Fo(s)p Fp(\)\))1275 1228 y Fn(2)383 1312 y Fo(\037)p Fp(\()p Fm(\000)p Fo(v)478 1295 y Fn(2)498 1312 y Fo(\027)s Fp(\()p Fo(ts)572 1295 y Fl(\000)p Fn(1)616 1312 y Fp(\))632 1295 y Fn(2)651 1312 y Fo(\014)r Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)p Fp(\)\))p Fo(\037)p Fp(\()p Fm(\000)p Fo(v)916 1295 y Fn(2)938 1312 y Fo(\027)s Fp(\()p Fo(s)997 1295 y Fl(\000)p Fn(2)1041 1312 y Fo(t)p Fp(\))p Fo(\014)r Fp(\()p Fo(st)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)p Fp(\)\))p Fo(e)1276 1320 y Fj(v)q(\027)r Fn(\()p Fj(s)1342 1312 y Fh(\000)p Fd(1)1382 1320 y Fn(\))1406 1312 y Fm(\012)j Fo(x)1472 1320 y Fj(sts)1517 1312 y Fh(\000)p Fd(1)312 1380 y Fp(=)i Fo(\016)374 1387 y Fj(u\027)r Fn(\()p Fj(s)p Fn(\))p Fj(;v)q Fl(\000)p Fj(v)q(\027)r Fn(\()p Fj(t)p Fn(\))584 1380 y Fo(\021)q Fp(\()p Fo(v)q(\027)s Fp(\()p Fo(s)702 1363 y Fl(\000)p Fn(1)747 1380 y Fp(\)\()p Fo(q)q Fp(\()p Fo(s;)7 b(t)p Fp(\))j(+)g Fo(q)q Fp(\()p Fo(st;)d(s)1044 1363 y Fl(\000)p Fn(1)1089 1380 y Fp(\))i Fm(\000)h Fo(\027)s Fp(\()p Fo(sts)1249 1363 y Fl(\000)p Fn(1)1293 1380 y Fp(\))p Fo(q)q Fp(\()p Fo(s;)d(s)1402 1363 y Fl(\000)p Fn(1)1448 1380 y Fp(\)\)\))383 1447 y Fo(\037)p Fp(\()p Fo(v)446 1430 y Fn(2)466 1447 y Fo(\027)s Fp(\()p Fo(s)525 1430 y Fl(\000)p Fn(2)569 1447 y Fp(\)\()p Fo(\027)s Fp(\()p Fo(s)p Fp(\))p Fo(\014)r Fp(\()p Fo(t)p Fp(\))j(+)g Fo(\027)s Fp(\()p Fo(t)p Fp(\))p Fo(\014)r Fp(\()p Fo(st)p Fp(\))g Fm(\000)f Fo(\027)s Fp(\()p Fo(t)1068 1430 y Fn(2)1086 1447 y Fp(\))p Fo(\014)r Fp(\()p Fo(s)p Fp(\)\))p Fo(\013)p Fp(\()p Fo(s)p Fp(\)\))p Fo(e)1307 1455 y Fj(v)q(\027)r Fn(\()p Fj(s)1373 1447 y Fh(\000)p Fd(1)1414 1455 y Fn(\))1438 1447 y Fm(\012)g Fo(x)1503 1455 y Fj(sts)1548 1447 y Fh(\000)p Fd(1)257 1614 y Fp(This)16 b(pro)o(v)o(es)f(the)h(\014rst)h (statemen)o(t;)d(w)o(e)i(no)o(w)f(turn)h(to)f(the)h(second.)g(Using)f (Heyneman-)257 1664 y(Sw)o(eedler)20 b(sigma)d(notation,)h(the)h(righ)o (t)g(adjoin)o(t)e(action)i(can)g(also)f(b)q(e)i(written)f(in)g(the)257 1713 y(form:)548 1763 y Fo(b)566 1746 y Fl(0)589 1763 y Fo(\))11 b(b)h Fp(=)g Fo(S)743 1746 y Fl(\000)p Fn(1)741 1776 y Fj(A)766 1767 y Fh(\003)788 1763 y Fp(\()p Fo(b)822 1770 y Fn(\(2\))878 1763 y Fm( )f Fo(S)956 1769 y Fj(H)988 1763 y Fp(\()p Fo(b)1022 1746 y Fl(0)r Fn(\(2\))1078 1763 y Fo(b)1096 1770 y Fn(\(1\))1141 1746 y(\(2\))1185 1763 y Fp(\)\))p Fo(b)1235 1746 y Fl(0)r Fn(\(1\))1291 1763 y Fo(b)1309 1770 y Fn(\(1\))1354 1746 y(\(1\))257 1834 y Fp(Again)j(with)f(Lemma)f(1.13,)g(w)o(e)j(see)g(that)f(the)h(in) o(v)o(erse)g(of)e(the)i(an)o(tip)q(o)q(de)f(of)g Fo(A)1520 1819 y Fl(\003)1553 1834 y Fp(is)g(giv)o(en)257 1883 y(b)o(y)g(the)h(equation:)441 1968 y Fo(S)468 1950 y Fl(\000)p Fn(1)466 1980 y Fj(A)491 1971 y Fh(\003)513 1968 y Fp(\()p Fo(c)547 1974 y Fj(u)578 1968 y Fm(\012)10 b Fo(d)642 1974 y Fj(s)659 1968 y Fp(\))i(=)f Fo(\021)q Fp(\()p Fm(\000)p Fo(uq)q Fp(\()p Fo(s;)c(s)917 1950 y Fl(\000)p Fn(1)963 1968 y Fp(\)\))p Fo(\037)p Fp(\()p Fo(u)1061 1950 y Fn(2)1080 1968 y Fo(\014)r Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)p Fp(\)\))p Fo(c)1268 1975 y Fl(\000)p Fj(u\027)r Fn(\()p Fj(s)p Fn(\))1387 1968 y Fm(\012)i Fo(d)1450 1975 y Fj(s)1466 1967 y Fh(\000)p Fd(1)257 2052 y Fp(W)m(e)14 b(therefore)h(ha)o(v)o(e:)258 2136 y(\()p Fo(c)292 2142 y Fj(v)321 2136 y Fm(\012)9 b Fo(d)384 2142 y Fj(t)399 2136 y Fp(\))i Fo(\))g Fp(\()p Fo(c)513 2142 y Fj(u)544 2136 y Fm(\012)f Fo(d)608 2142 y Fj(s)625 2136 y Fp(\))286 2211 y(=)332 2171 y Fg(X)329 2260 y Fj(r)q Fl(2)p Fj(G)401 2211 y Fo(\021)q Fp(\()p Fo(uq)q Fp(\()p Fo(r)o(;)d(r)556 2194 y Fl(\000)p Fn(1)599 2211 y Fo(s)p Fp(\)\))p Fo(\037)p Fp(\()p Fm(\000)p Fo(u)748 2194 y Fn(2)768 2211 y Fo(\027)s Fp(\()p Fo(r)q Fp(\))p Fo(\014)r Fp(\()p Fo(r)q Fp(\))p Fo(\013)p Fp(\()p Fo(r)984 2194 y Fl(\000)p Fn(1)1028 2211 y Fo(s)p Fp(\)\))357 2322 y Fo(S)384 2305 y Fl(\000)p Fn(1)382 2334 y Fj(A)407 2326 y Fh(\003)429 2322 y Fp(\(\()p Fo(c)479 2328 y Fj(u)511 2322 y Fm(\012)i Fo(d)574 2328 y Fj(r)592 2322 y Fp(\))j Fm( )f Fo(S)698 2328 y Fj(H)730 2322 y Fp(\(\()p Fo(c)780 2328 y Fj(v)809 2322 y Fm(\012)e Fo(d)872 2328 y Fj(t)887 2322 y Fp(\))903 2305 y Fn(\(2\))947 2322 y Fp(\()p Fo(c)981 2329 y Fj(u\027)r Fn(\()p Fj(r)q Fn(\))1073 2322 y Fm(\012)h Fo(d)1137 2330 y Fj(r)1153 2322 y Fh(\000)p Fd(1)1191 2330 y Fj(s)1209 2322 y Fp(\))1225 2305 y Fn(\(2\))1270 2322 y Fp(\)\))1021 2393 y(\()p Fo(c)1055 2399 y Fj(v)1084 2393 y Fm(\012)g Fo(d)1148 2399 y Fj(t)1162 2393 y Fp(\))1178 2376 y Fn(\(1\))1223 2393 y Fp(\()p Fo(c)1257 2400 y Fj(u\027)r Fn(\()p Fj(r)q Fn(\))1349 2393 y Fm(\012)f Fo(d)1412 2401 y Fj(r)1428 2392 y Fh(\000)p Fd(1)1467 2401 y Fj(s)1484 2393 y Fp(\))1500 2376 y Fn(\(1\))286 2468 y Fp(=)332 2428 y Fg(X)329 2518 y Fj(r)q Fl(2)p Fj(G)401 2468 y Fo(\021)q Fp(\()p Fo(uq)q Fp(\()p Fo(r)o(;)e(r)556 2451 y Fl(\000)p Fn(1)599 2468 y Fo(s)p Fp(\)\))p Fo(\037)p Fp(\()p Fm(\000)p Fo(u)748 2451 y Fn(2)768 2468 y Fo(\027)s Fp(\()p Fo(r)q Fp(\))p Fo(\014)r Fp(\()p Fo(r)q Fp(\))p Fo(\013)p Fp(\()p Fo(r)984 2451 y Fl(\000)p Fn(1)1028 2468 y Fo(s)p Fp(\)\))953 2628 y(74)p eop %%Page: 75 75 75 74 bop 357 262 a Fo(S)384 244 y Fl(\000)p Fn(1)382 274 y Fj(A)407 265 y Fh(\003)429 262 y Fp(\(\()p Fo(c)479 268 y Fj(u)511 262 y Fm(\012)9 b Fo(d)574 268 y Fj(r)592 262 y Fp(\))j Fm( )f Fo(c)691 269 y Fj(v)q(\014)q Fn(\()p Fj(t)p Fn(\)+)p Fj(u\027)r Fn(\()p Fj(r)q Fn(\))p Fj(\014)q Fn(\()p Fj(r)923 261 y Fh(\000)p Fd(1)961 269 y Fj(s)p Fn(\))992 262 y Fp(\)\()p Fo(c)1042 268 y Fj(v)1071 262 y Fm(\012)f Fo(d)1135 268 y Fj(t)1149 262 y Fp(\)\()p Fo(c)1199 269 y Fj(u\027)r Fn(\()p Fj(r)q Fn(\))1291 262 y Fm(\012)g Fo(d)1355 269 y Fj(r)1371 261 y Fh(\000)p Fd(1)1409 269 y Fj(s)1427 262 y Fp(\))286 337 y(=)332 297 y Fg(X)329 386 y Fj(r)q Fl(2)p Fj(G)401 337 y Fo(\021)q Fp(\()p Fo(uq)q Fp(\()p Fo(r)o(;)d(r)556 320 y Fl(\000)p Fn(1)599 337 y Fo(s)p Fp(\)\))p Fo(\037)p Fp(\()p Fm(\000)p Fo(u)748 320 y Fn(2)768 337 y Fo(\027)s Fp(\()p Fo(r)q Fp(\))p Fo(\014)r Fp(\()p Fo(r)q Fp(\))p Fo(\013)p Fp(\()p Fo(r)984 320 y Fl(\000)p Fn(1)1028 337 y Fo(s)p Fp(\)\))357 446 y Fo(\037)p Fp(\(\()p Fo(v)q(\014)r Fp(\()p Fo(t)p Fp(\))k(+)f Fo(u\027)s Fp(\()p Fo(r)q Fp(\))p Fo(\014)r Fp(\()p Fo(r)722 429 y Fl(\000)p Fn(1)766 446 y Fo(s)p Fp(\)\))p Fo(u\013)p Fp(\()p Fo(r)q Fp(\)\))936 429 y Fn(2)955 446 y Fo(\016)973 454 y Fj(t;r)1012 445 y Fh(\000)p Fd(1)1051 454 y Fj(s)1069 446 y Fo(S)1096 428 y Fl(\000)p Fn(1)1094 458 y Fj(A)1119 450 y Fh(\003)1141 446 y Fp(\()p Fo(c)1175 452 y Fj(u)1206 446 y Fm(\012)g Fo(d)1270 452 y Fj(r)1288 446 y Fp(\)\()p Fo(c)1338 453 y Fj(v)q Fn(+)p Fj(u\027)r Fn(\()p Fj(r)q Fn(\))1473 446 y Fm(\012)f Fo(d)1536 452 y Fj(t)1550 446 y Fp(\))286 513 y(=)i Fo(\021)q Fp(\()p Fo(uq)q Fp(\()p Fo(st)461 496 y Fl(\000)p Fn(1)507 513 y Fo(;)c(t)p Fp(\)\))p Fo(\037)p Fp(\()p Fm(\000)p Fo(u)671 496 y Fn(2)689 513 y Fo(\027)s Fp(\()p Fo(st)763 496 y Fl(\000)p Fn(1)807 513 y Fp(\))p Fo(\014)r Fp(\()p Fo(st)898 496 y Fl(\000)p Fn(1)944 513 y Fp(\))p Fo(\013)p Fp(\()p Fo(t)p Fp(\)\))357 581 y Fo(\037)p Fp(\(\()p Fo(v)q(\014)r Fp(\()p Fo(t)p Fp(\))k(+)f Fo(u\027)s Fp(\()p Fo(st)659 564 y Fl(\000)p Fn(1)703 581 y Fp(\))p Fo(\014)r Fp(\()p Fo(t)p Fp(\)\))p Fo(u\013)p Fp(\()p Fo(st)908 564 y Fl(\000)p Fn(1)953 581 y Fp(\)\))985 564 y Fn(2)1004 581 y Fo(\021)q Fp(\()p Fm(\000)p Fo(uq)q Fp(\()p Fo(st)1168 564 y Fl(\000)p Fn(1)1214 581 y Fo(;)d(ts)1267 564 y Fl(\000)p Fn(1)1311 581 y Fp(\)\))357 648 y Fo(\037)p Fp(\()p Fo(u)423 631 y Fn(2)442 648 y Fo(\014)r Fp(\()p Fo(st)517 631 y Fl(\000)p Fn(1)562 648 y Fp(\))p Fo(\013)p Fp(\()p Fo(st)655 631 y Fl(\000)p Fn(1)700 648 y Fp(\)\)\()p Fo(c)766 656 y Fl(\000)p Fj(u\027)r Fn(\()p Fj(st)873 648 y Fh(\000)p Fd(1)911 656 y Fn(\))935 648 y Fm(\012)j Fo(d)999 656 y Fj(ts)1028 648 y Fh(\000)p Fd(1)s Fp(\)\()p Fo(c)1117 656 y Fj(v)q Fn(+)p Fj(u\027)r Fn(\()p Fj(st)1241 648 y Fh(\000)p Fd(1)1279 656 y Fn(\))1303 648 y Fm(\012)f Fo(d)1366 654 y Fj(t)1381 648 y Fp(\))286 716 y(=)i Fo(\016)347 722 y Fj(s)p Fn(1)382 716 y Fo(\021)q Fp(\()p Fo(uq)q Fp(\()p Fo(t)495 699 y Fl(\000)p Fn(1)540 716 y Fo(;)c(t)p Fp(\)\))p Fo(\037)p Fp(\()p Fm(\000)p Fo(u)704 699 y Fn(2)722 716 y Fo(\027)s Fp(\()p Fo(t)777 699 y Fl(\000)p Fn(1)821 716 y Fp(\))p Fo(\014)r Fp(\()p Fo(t)893 699 y Fl(\000)p Fn(1)939 716 y Fp(\))p Fo(\013)p Fp(\()p Fo(t)p Fp(\)\))357 783 y Fo(\037)p Fp(\(\()p Fo(v)q(\014)r Fp(\()p Fo(t)p Fp(\))k(+)f Fo(u\027)s Fp(\()p Fo(t)640 766 y Fl(\000)p Fn(1)683 783 y Fp(\))p Fo(\014)r Fp(\()p Fo(t)p Fp(\)\))p Fo(u\013)p Fp(\()p Fo(t)869 766 y Fl(\000)p Fn(1)915 783 y Fp(\)\))947 766 y Fn(2)965 783 y Fo(\021)q Fp(\()p Fm(\000)p Fo(uq)q Fp(\()p Fo(t)1110 766 y Fl(\000)p Fn(1)1155 783 y Fo(;)d(t)p Fp(\)\))p Fo(\037)p Fp(\()p Fo(u)1287 766 y Fn(2)1306 783 y Fo(\014)r Fp(\()p Fo(t)1362 766 y Fl(\000)p Fn(1)1407 783 y Fp(\))p Fo(\013)p Fp(\()p Fo(t)1481 766 y Fl(\000)p Fn(1)1526 783 y Fp(\)\))p Fo(c)1576 789 y Fj(v)1605 783 y Fm(\012)i Fo(d)1668 789 y Fj(t)286 851 y Fp(=)i Fo(\016)347 857 y Fj(s)p Fn(1)382 851 y Fo(\037)p Fp(\()p Fo(u)448 833 y Fn(2)467 851 y Fo(\014)r Fp(\()p Fo(t)523 833 y Fl(\000)p Fn(1)568 851 y Fp(\))p Fo(\013)p Fp(\()p Fo(t)642 833 y Fl(\000)p Fn(1)686 851 y Fp(\)\))718 833 y Fn(2)737 851 y Fo(\037)p Fp(\(\()p Fo(v)q(\014)r Fp(\()p Fo(t)p Fp(\))g Fm(\000)f Fo(u\014)r Fp(\()p Fo(t)1021 833 y Fl(\000)p Fn(1)1066 851 y Fp(\)\))p Fo(u\013)p Fp(\()p Fo(t)1180 833 y Fl(\000)p Fn(1)1225 851 y Fp(\)\))1257 833 y Fn(2)1275 851 y Fo(c)1293 857 y Fj(v)1322 851 y Fm(\012)g Fo(d)1386 857 y Fj(t)286 918 y Fp(=)h Fo(\016)347 924 y Fj(s)p Fn(1)382 918 y Fo(\037)p Fp(\()p Fo(v)q(u\014)r Fp(\()p Fo(t)p Fp(\))p Fo(\013)p Fp(\()p Fo(t)599 901 y Fl(\000)p Fn(1)645 918 y Fp(\)\))677 901 y Fn(2)696 918 y Fo(c)714 924 y Fj(v)743 918 y Fm(\012)e Fo(d)806 924 y Fj(t)832 918 y Fp(=)j Fo(\016)894 924 y Fj(s)p Fn(1)929 918 y Fo(\037)p Fp(\()p Fm(\000)p Fo(v)q(u\027)s Fp(\()p Fo(t)p Fp(\))p Fo(\014)r Fp(\()p Fo(t)p Fp(\))p Fo(\013)p Fp(\()p Fo(t)p Fp(\)\))1281 901 y Fn(2)1300 918 y Fo(c)1318 924 y Fj(v)1347 918 y Fm(\012)e Fo(d)1411 924 y Fj(t)1658 918 y Fi(2)257 1053 y Fq(5.8)48 b Fp(W)m(e)14 b(no)o(w)h(dualize)f(the)i(adjoin)o(t)d (actions)i(to)g(get)g(the)g(coadjoin)o(t)f(actions)h(\(cf.)f([79)o(],) 257 1103 y(Subsec.)19 b(4.7\).)d(Of)h(course,)h(this)f(dualization)f (dep)q(ends)j(on)e(the)h(bilinear)f(form)e(that)j(is)257 1153 y(used.)g(T)m(o)e(de\014ne)i(the)g(righ)o(t)e(coadjoin)o(t)g (action)h Fo(\()g Fp(of)f Fo(A)h Fp(on)g Fo(A)1292 1138 y Fl(\003)1311 1153 y Fp(,)g(w)o(e)g(use)h(the)f(bilinear)257 1203 y(form)11 b Fm(h\001)p Fo(;)c Fm(\001i)429 1209 y Fj(A)467 1203 y Fp(from)j(P)o(aragraph)i(5.6;)e(i.)i(e.,)f(w)o(e)i (de\014ne)g(the)f(righ)o(t)g(coadjoin)o(t)f(action)h(b)o(y)g(the)257 1253 y(condition:)730 1302 y Fm(h)p Fo(a)768 1285 y Fl(0)780 1302 y Fo(;)7 b(b)k(\()g(a)p Fm(i)919 1308 y Fj(A)957 1302 y Fp(=)h Fm(h)p Fo(a)g(+)f(a)1126 1285 y Fl(0)1137 1302 y Fo(;)c(b)p Fm(i)1190 1308 y Fj(A)257 1413 y Fp(F)m(or)13 b(the)h(dualization)e(of)g(the)i(righ)o(t)f(adjoin)o(t)e(action)i(of)g Fo(A)1174 1398 y Fl(\003)1206 1413 y Fp(on)g(itself,)f(w)o(e)i(use)g(a) e(di\013eren)o(t)257 1463 y(bilinear)h(form)g Fm(h\001)p Fo(;)7 b Fm(\001i)581 1469 y Fj(A)606 1461 y Fh(\003)638 1463 y Fp(that)13 b(is)h(de\014ned)h(as:)583 1554 y Fm(h\001)p Fo(;)7 b Fm(\001i)658 1560 y Fj(A)683 1552 y Fh(\003)713 1554 y Fp(:)k Fo(A)e Fm(\012)h Fo(A)849 1537 y Fl(\003)879 1554 y Fm(!)h Fo(K)q(;)c(a)i Fm(\012)g Fo(b)j Fm(7!)f(h)p Fo(S)1185 1536 y Fl(\000)p Fn(1)1183 1566 y Fj(A)1230 1554 y Fp(\()p Fo(a)p Fp(\))p Fo(;)c(b)p Fm(i)1337 1560 y Fj(A)257 1645 y Fp(This)13 b(bilinear)f(form)f(is)h(the)i(con)o(v)o (olution)d(in)o(v)o(erse)i(of)f(the)i(bilinear)d(form)g Fm(h\001)p Fo(;)c Fm(\001i)1502 1651 y Fj(A)1528 1645 y Fp(;)12 b(i.)g(e.,)g(w)o(e)257 1695 y(ha)o(v)o(e:)383 1786 y Fm(h)p Fo(a)421 1793 y Fn(\(1\))465 1786 y Fo(;)7 b(b)502 1793 y Fn(\(1\))546 1786 y Fm(i)562 1792 y Fj(A)589 1786 y Fm(h)p Fo(a)627 1793 y Fn(\(2\))672 1786 y Fo(;)g(b)709 1793 y Fn(\(2\))753 1786 y Fm(i)769 1792 y Fj(A)794 1784 y Fh(\003)825 1786 y Fp(=)12 b Fo(\017)886 1792 y Fj(A)913 1786 y Fp(\()p Fo(a)p Fp(\))p Fo(\017)984 1792 y Fj(A)1009 1784 y Fh(\003)1028 1786 y Fp(\()p Fo(b)p Fp(\))g(=)g Fm(h)p Fo(a)1172 1793 y Fn(\(1\))1216 1786 y Fo(;)7 b(b)1253 1793 y Fn(\(1\))1297 1786 y Fm(i)1313 1792 y Fj(A)1338 1784 y Fh(\003)1358 1786 y Fm(h)p Fo(a)1396 1793 y Fn(\(2\))1440 1786 y Fo(;)g(b)1477 1793 y Fn(\(2\))1521 1786 y Fm(i)1537 1792 y Fj(A)257 1878 y Fp(for)14 b(all)f Fo(a)e Fm(2)g Fo(A)j Fp(and)g Fo(b)d Fm(2)g Fo(A)676 1863 y Fl(\003)696 1878 y Fp(.)257 1963 y(W)m(e)j(then)g(de\014ne)h(the)g(left)e(coadjoin) o(t)g(action)h Fo(*)f Fp(of)h Fo(A)1126 1948 y Fl(\003)1159 1963 y Fp(on)f Fo(A)h Fp(b)o(y)g(the)g(condition:)717 2055 y Fm(h)p Fo(b)d(*)g(a;)c(b)874 2037 y Fl(0)885 2055 y Fm(i)901 2061 y Fj(A)926 2053 y Fh(\003)957 2055 y Fp(=)12 b Fm(h)p Fo(a;)7 b(b)1076 2037 y Fl(0)1099 2055 y Fo(\))k(b)p Fm(i)1186 2061 y Fj(A)1211 2053 y Fh(\003)257 2182 y Fp(Finally)m(,)h(w)o(e)i(in)o(tro)q(duce)g(the)h(mapping)315 2273 y Fo(])d Fp(:)f Fo(A)397 2256 y Fl(\003)425 2273 y Fm(\012)f Fo(A)h Fm(!)g Fo(H)q(;)c(b)h Fm(\012)i Fo(a)h Fm(7!)g Fo(b]a)h Fp(:=)f Fm(h)p Fo(a)932 2280 y Fn(\(1\))977 2273 y Fo(;)c(b)1014 2280 y Fn(\(1\))1058 2256 y(\(1\))1102 2273 y Fm(i)1118 2279 y Fj(A)1143 2271 y Fh(\003)1163 2273 y Fo(b)1181 2280 y Fn(\(1\))1225 2256 y(\(2\))1270 2273 y Fo(a)1292 2280 y Fn(\(2\))1336 2256 y(\(1\))1381 2273 y Fm(h)p Fo(a)1419 2280 y Fn(\(2\))1463 2256 y(\(2\))1508 2273 y Fo(;)g(b)1545 2280 y Fn(\(2\))1589 2273 y Fm(i)1605 2279 y Fj(A)953 2628 y Fp(75)p eop %%Page: 76 76 76 75 bop 257 262 a Fp(With)18 b(resp)q(ect)i(to)e(the)g(bases)h Fo(e)777 268 y Fj(u)811 262 y Fm(\012)13 b Fo(x)880 268 y Fj(s)915 262 y Fp(of)k Fo(A)p Fp(,)h(resp.)h Fo(c)1149 268 y Fj(u)1182 262 y Fm(\012)13 b Fo(d)1249 268 y Fj(s)1284 262 y Fp(of)k Fo(A)1366 246 y Fl(\003)1385 262 y Fp(,)h(these)h (structure)257 311 y(elemen)o(ts)14 b(tak)o(e)g(the)h(follo)o(wing)c (form:)257 411 y Fq(Prop)q(osition)33 b Fp(F)m(or)14 b Fo(u;)7 b(v)12 b Fm(2)f Fo(R)j Fp(and)g Fo(s;)7 b(t)k Fm(2)g Fo(G)p Fp(,)i(w)o(e)h(ha)o(v)o(e:)308 530 y(1.)20 b(\()p Fo(c)395 536 y Fj(u)426 530 y Fm(\012)10 b Fo(d)490 536 y Fj(s)507 530 y Fp(\))i Fo(*)f Fp(\()p Fo(e)623 536 y Fj(v)652 530 y Fm(\012)f Fo(x)718 536 y Fj(t)732 530 y Fp(\))i(=)g Fo(\016)822 536 y Fj(s)p Fn(1)856 530 y Fo(\037)p Fp(\()p Fo(v)q(u\027)s Fp(\()p Fo(t)998 515 y Fl(\000)p Fn(2)1043 530 y Fp(\))p Fo(\014)r Fp(\()p Fo(t)p Fp(\))p Fo(\013)p Fp(\()p Fo(t)p Fp(\)\))1221 515 y Fn(2)1240 530 y Fo(e)1259 536 y Fj(v)1289 530 y Fm(\012)d Fo(x)1354 536 y Fj(t)308 613 y Fp(2.)20 b(\()p Fo(c)395 619 y Fj(v)415 613 y Fm(\012)p Fo(d)469 619 y Fj(t)483 613 y Fp(\))12 b Fo(\()f Fp(\()p Fo(e)599 619 y Fj(u)621 613 y Fm(\012)p Fo(x)677 619 y Fj(s)695 613 y Fp(\))h(=)g Fo(\016)785 620 y Fj(u;v)q Fl(\000)p Fj(v)q(\027)r Fn(\()p Fj(t)p Fn(\))935 613 y Fo(\021)q Fp(\()p Fo(v)q Fp(\()p Fo(q)q Fp(\()p Fo(s;)7 b(s)1103 598 y Fl(\000)p Fn(1)1149 613 y Fo(ts)p Fp(\)+)p Fo(q)q Fp(\()p Fo(ts;)g(s)1339 598 y Fl(\000)p Fn(1)1385 613 y Fp(\))p Fm(\000)p Fo(\027)s Fp(\()p Fo(t)p Fp(\))p Fo(q)q Fp(\()p Fo(s;)g(s)1597 598 y Fl(\000)p Fn(1)1642 613 y Fp(\)\)\))361 663 y Fo(\037)p Fp(\()p Fo(v)424 647 y Fn(2)443 663 y Fp(\()p Fo(\027)s Fp(\()p Fo(s)p Fp(\))p Fo(\014)r Fp(\()p Fo(s)594 647 y Fl(\000)p Fn(1)640 663 y Fo(ts)p Fp(\))j(+)g Fo(\027)s Fp(\()p Fo(t)p Fp(\))p Fo(\014)r Fp(\()p Fo(ts)p Fp(\))g Fm(\000)f Fo(\027)s Fp(\()p Fo(t)1010 647 y Fn(2)1028 663 y Fp(\))p Fo(\014)r Fp(\()p Fo(s)p Fp(\)\))p Fo(\013)p Fp(\()p Fo(s)p Fp(\)\))1230 647 y Fn(2)1251 663 y Fo(c)1269 670 y Fj(v)q(\027)r Fn(\()p Fj(s)p Fn(\))1358 663 y Fm(\012)h Fo(d)1422 670 y Fj(s)1438 662 y Fh(\000)p Fd(1)1476 670 y Fj(ts)308 746 y Fp(3.)20 b(\()p Fo(c)395 752 y Fj(u)426 746 y Fm(\012)10 b Fo(d)490 752 y Fj(s)507 746 y Fp(\))p Fo(])p Fp(\()p Fo(e)574 752 y Fj(v)604 746 y Fm(\012)f Fo(x)669 752 y Fj(t)684 746 y Fp(\))i(=)h Fo(\016)773 752 y Fj(v)q Fn(0)810 746 y Fo(\016)828 752 y Fj(s)p Fn(1)863 746 y Fo(c)881 753 y Fn(2)p Fj(u\014)q Fn(\()p Fj(t)p Fn(\))257 845 y Fq(Pro)q(of.)36 b Fp(Using)14 b(Lemma)d(1.13,)h(w)o(e)i(ha)o(v)o(e)g(for)g(the)g(left)g (coadjoin)o(t)f(action:)328 937 y Fm(h)p Fp(\()p Fo(c)378 943 y Fj(u)409 937 y Fm(\012)d Fo(d)473 943 y Fj(s)490 937 y Fp(\))i Fo(*)f Fp(\()p Fo(e)606 943 y Fj(v)635 937 y Fm(\012)f Fo(x)701 943 y Fj(t)715 937 y Fp(\))p Fo(;)d(c)768 943 y Fj(w)804 937 y Fm(\012)i Fo(d)867 943 y Fj(r)885 937 y Fm(i)901 943 y Fj(A)926 934 y Fh(\003)957 937 y Fp(=)j Fm(h)p Fo(e)1036 943 y Fj(v)1066 937 y Fm(\012)d Fo(x)1131 943 y Fj(t)1145 937 y Fo(;)e Fp(\()p Fo(c)1198 943 y Fj(w)1234 937 y Fm(\012)j Fo(d)1298 943 y Fj(r)1316 937 y Fp(\))h Fo(\))g Fp(\()p Fo(c)1430 943 y Fj(u)1461 937 y Fm(\012)f Fo(d)1525 943 y Fj(s)1542 937 y Fp(\))p Fm(i)1574 943 y Fj(A)1599 934 y Fh(\003)372 1004 y Fp(=)i Fo(\016)434 1010 y Fj(s)p Fn(1)468 1004 y Fo(\037)p Fp(\()p Fm(\000)p Fo(w)q(u\027)s Fp(\()p Fo(r)q Fp(\))p Fo(\014)r Fp(\()p Fo(r)q Fp(\))p Fo(\013)p Fp(\()p Fo(r)q Fp(\)\))845 987 y Fn(2)864 1004 y Fm(h)p Fo(e)899 1010 y Fj(v)929 1004 y Fm(\012)d Fo(x)994 1010 y Fj(t)1008 1004 y Fo(;)e(c)1045 1010 y Fj(w)1081 1004 y Fm(\012)j Fo(d)1145 1010 y Fj(r)1162 1004 y Fm(i)1178 1010 y Fj(A)1203 1002 y Fh(\003)372 1071 y Fp(=)i Fo(\016)434 1077 y Fj(s)p Fn(1)468 1071 y Fo(\037)p Fp(\()p Fm(\000)p Fo(w)q(u\027)s Fp(\()p Fo(r)q Fp(\))p Fo(\014)r Fp(\()p Fo(r)q Fp(\))p Fo(\013)p Fp(\()p Fo(r)q Fp(\)\))845 1054 y Fn(2)864 1071 y Fo(\021)q Fp(\()p Fo(v)q(q)q Fp(\()p Fo(t;)7 b(t)1008 1054 y Fl(\000)p Fn(1)1053 1071 y Fp(\)\))p Fo(\037)p Fp(\()p Fm(\000)p Fo(v)1180 1054 y Fn(2)1200 1071 y Fo(\014)r Fp(\()p Fo(t)p Fp(\))p Fo(\013)p Fp(\()p Fo(t)p Fp(\)\))1108 1134 y Fm(h)p Fo(e)1143 1141 y Fl(\000)p Fj(v)q(\027)r Fn(\()p Fj(t)p Fn(\))1255 1134 y Fm(\012)i Fo(x)1320 1141 y Fj(t)1333 1133 y Fh(\000)p Fd(1)1373 1134 y Fo(;)e(c)1410 1140 y Fj(w)1446 1134 y Fm(\012)j Fo(d)1510 1140 y Fj(r)1528 1134 y Fm(i)1544 1140 y Fj(A)372 1201 y Fp(=)i Fo(\016)434 1207 y Fj(s)p Fn(1)468 1201 y Fo(\037)p Fp(\()p Fo(v)q(\027)s Fp(\()p Fo(t)p Fp(\))p Fo(u\027)s Fp(\()p Fo(t)681 1184 y Fl(\000)p Fn(1)725 1201 y Fp(\))p Fo(\014)r Fp(\()p Fo(t)797 1184 y Fl(\000)p Fn(1)843 1201 y Fp(\))p Fo(\013)p Fp(\()p Fo(t)917 1184 y Fl(\000)p Fn(1)961 1201 y Fp(\)\))993 1184 y Fn(2)1012 1201 y Fo(\021)q Fp(\()p Fo(v)q(q)q Fp(\()p Fo(t;)7 b(t)1156 1184 y Fl(\000)p Fn(1)1201 1201 y Fp(\)\))p Fo(\037)p Fp(\()p Fm(\000)p Fo(v)1328 1184 y Fn(2)1348 1201 y Fo(\014)r Fp(\()p Fo(t)p Fp(\))p Fo(\013)p Fp(\()p Fo(t)p Fp(\)\))1108 1263 y Fm(h)p Fo(e)1143 1270 y Fl(\000)p Fj(v)q(\027)r Fn(\()p Fj(t)p Fn(\))1255 1263 y Fm(\012)i Fo(x)1320 1271 y Fj(t)1333 1263 y Fh(\000)p Fd(1)t Fo(;)e(c)1410 1269 y Fj(w)1446 1263 y Fm(\012)j Fo(d)1510 1269 y Fj(r)1528 1263 y Fm(i)1544 1269 y Fj(A)372 1331 y Fp(=)i Fo(\016)434 1337 y Fj(s)p Fn(1)468 1331 y Fo(\037)p Fp(\()p Fo(v)q(u\014)r Fp(\()p Fo(t)611 1314 y Fl(\000)p Fn(1)657 1331 y Fp(\))p Fo(\013)p Fp(\()p Fo(t)731 1314 y Fl(\000)p Fn(1)776 1331 y Fp(\)\))808 1314 y Fn(2)827 1331 y Fm(h)p Fo(e)862 1337 y Fj(v)891 1331 y Fm(\012)e Fo(x)957 1337 y Fj(t)971 1331 y Fo(;)d(c)1008 1337 y Fj(w)1044 1331 y Fm(\012)i Fo(d)1107 1337 y Fj(r)1125 1331 y Fm(i)1141 1337 y Fj(A)1166 1329 y Fh(\003)372 1398 y Fp(=)j Fo(\016)434 1404 y Fj(s)p Fn(1)468 1398 y Fo(\037)p Fp(\()p Fo(v)q(u\027)s Fp(\()p Fo(t)610 1381 y Fl(\000)p Fn(2)655 1398 y Fp(\))p Fo(\014)r Fp(\()p Fo(t)p Fp(\))p Fo(\013)p Fp(\()p Fo(t)p Fp(\)\))833 1381 y Fn(2)853 1398 y Fm(h)p Fo(e)888 1404 y Fj(v)917 1398 y Fm(\012)e Fo(x)983 1404 y Fj(t)997 1398 y Fo(;)d(c)1034 1404 y Fj(w)1069 1398 y Fm(\012)j Fo(d)1133 1404 y Fj(r)1151 1398 y Fm(i)1167 1404 y Fj(A)1192 1396 y Fh(\003)257 1489 y Fp(W)m(e)18 b(lea)o(v)o(e)g(the)g(v)o(eri\014cation)g(of)f(the)i (form)o(ula)c(for)j(the)g(righ)o(t)g(coadjoin)o(t)e(action)i(to)g(the) 257 1539 y(reader.)e(The)e(third)h(form)o(ula)d(follo)o(ws,)g(using)i (again)g(Lemma)e(1.13,)g(from)h(the)i(follo)o(wing)257 1589 y(calculation:)258 1680 y(\()p Fo(c)292 1686 y Fj(u)323 1680 y Fm(\012)10 b Fo(d)387 1686 y Fj(s)404 1680 y Fp(\))p Fo(])p Fp(\()p Fo(e)471 1686 y Fj(v)501 1680 y Fm(\012)f Fo(x)566 1686 y Fj(t)580 1680 y Fp(\))270 1756 y(=)357 1716 y Fg(X)313 1805 y Fj(w)q Fl(2)p Fj(R;r)q Fl(2)p Fj(G)467 1756 y Fo(\021)q Fp(\()p Fo(uq)q Fp(\()p Fo(r)o(;)e(r)622 1738 y Fl(\000)p Fn(1)666 1756 y Fo(s)p Fp(\)\))p Fo(\037)p Fp(\()p Fm(\000)p Fo(u)815 1738 y Fn(2)834 1756 y Fo(\027)s Fp(\()p Fo(r)q Fp(\))p Fo(\014)r Fp(\()p Fo(r)q Fp(\))p Fo(\013)p Fp(\()p Fo(r)1050 1738 y Fl(\000)p Fn(1)1095 1756 y Fo(s)p Fp(\)\))341 1870 y Fm(h)p Fo(e)376 1876 y Fj(v)q Fl(\000)p Fj(w)456 1870 y Fm(\012)j Fo(x)522 1876 y Fj(t)536 1870 y Fo(;)d Fp(\()p Fo(c)589 1877 y Fj(u\027)r Fn(\()p Fj(r)q Fn(\))680 1870 y Fm(\012)j Fo(d)744 1878 y Fj(r)760 1870 y Fh(\000)p Fd(1)798 1878 y Fj(s)816 1870 y Fp(\))832 1853 y Fn(\(1\))877 1870 y Fm(i)893 1876 y Fj(A)918 1868 y Fh(\003)937 1870 y Fp(\()p Fo(c)971 1877 y Fj(u\027)r Fn(\()p Fj(r)q Fn(\))1063 1870 y Fm(\012)g Fo(d)1127 1878 y Fj(r)1143 1870 y Fh(\000)p Fd(1)1181 1878 y Fj(s)1199 1870 y Fp(\))1215 1853 y Fn(\(2\))1260 1870 y Fp(\()p Fo(e)1295 1876 y Fj(w)1331 1870 y Fm(\012)g Fo(x)1397 1876 y Fj(t)1411 1870 y Fp(\))1427 1853 y Fn(\(1\))1171 1941 y Fm(h)p Fp(\()p Fo(e)1222 1947 y Fj(w)1259 1941 y Fm(\012)f Fo(x)1324 1947 y Fj(t)1339 1941 y Fp(\))1355 1924 y Fn(\(2\))1400 1941 y Fo(;)e(c)1437 1947 y Fj(u)1467 1941 y Fm(\012)i Fo(d)1530 1947 y Fj(r)1548 1941 y Fm(i)1564 1947 y Fj(A)270 2008 y Fp(=)i Fo(\021)q Fp(\()p Fo(uq)q Fp(\()p Fo(t;)c(t)460 1991 y Fl(\000)p Fn(1)505 2008 y Fo(s)p Fp(\)\))p Fo(\037)p Fp(\()p Fm(\000)p Fo(u)654 1991 y Fn(2)673 2008 y Fo(\027)s Fp(\()p Fo(t)p Fp(\))p Fo(\014)r Fp(\()p Fo(t)p Fp(\))p Fo(\013)p Fp(\()p Fo(t)874 1991 y Fl(\000)p Fn(1)919 2008 y Fo(s)p Fp(\)\))341 2071 y Fm(h)p Fo(e)376 2077 y Fj(v)q Fl(\000)p Fj(u)451 2071 y Fm(\012)j Fo(x)517 2077 y Fj(t)531 2071 y Fo(;)d(c)568 2078 y Fj(u\027)r Fn(\()p Fj(t)p Fn(\))655 2071 y Fm(\012)j Fo(d)719 2078 y Fj(t)732 2070 y Fh(\000)p Fd(1)770 2078 y Fj(s)787 2071 y Fm(i)803 2077 y Fj(A)828 2069 y Fh(\003)848 2071 y Fo(c)866 2078 y Fl(\000)p Fj(u\027)r Fn(\()p Fj(t)p Fn(\))p Fj(\014)q Fn(\()p Fj(t)1016 2070 y Fh(\000)p Fd(1)1053 2078 y Fj(s)p Fn(\))1084 2071 y Fo(c)1102 2078 y Fj(u\014)q Fn(\()p Fj(t)p Fn(\))270 2138 y Fp(=)h Fo(\021)q Fp(\()p Fo(uq)q Fp(\()p Fo(t;)c(t)460 2121 y Fl(\000)p Fn(1)505 2138 y Fo(s)p Fp(\)\))p Fo(\037)p Fp(\()p Fm(\000)p Fo(u)654 2121 y Fn(2)673 2138 y Fo(\027)s Fp(\()p Fo(t)p Fp(\))p Fo(\014)r Fp(\()p Fo(t)p Fp(\))p Fo(\013)p Fp(\()p Fo(t)874 2121 y Fl(\000)p Fn(1)919 2138 y Fo(s)p Fp(\)\))p Fo(\021)q Fp(\(\()p Fo(v)12 b Fm(\000)d Fo(u)p Fp(\))p Fo(q)q Fp(\()p Fo(t;)e(t)1222 2121 y Fl(\000)p Fn(1)1266 2138 y Fp(\)\))p Fo(\037)p Fp(\()p Fm(\000)p Fp(\()p Fo(v)12 b Fm(\000)e Fo(u)p Fp(\))1502 2121 y Fn(2)1520 2138 y Fo(\014)r Fp(\()p Fo(t)p Fp(\))p Fo(\013)p Fp(\()p Fo(t)p Fp(\)\))341 2200 y Fm(h)p Fo(e)376 2207 y Fn(\()p Fj(u)p Fl(\000)p Fj(v)q Fn(\))p Fj(\027)r Fn(\()p Fj(t)p Fn(\))534 2200 y Fm(\012)g Fo(x)600 2208 y Fj(t)613 2200 y Fh(\000)p Fd(1)t Fo(;)d(c)690 2207 y Fj(u\027)r Fn(\()p Fj(t)p Fn(\))777 2200 y Fm(\012)i Fo(d)840 2208 y Fj(t)853 2200 y Fh(\000)p Fd(1)891 2208 y Fj(s)909 2200 y Fm(i)925 2206 y Fj(A)952 2200 y Fo(c)970 2208 y Fl(\000)p Fj(u\027)r Fn(\()p Fj(t)p Fn(\))p Fj(\014)q Fn(\()p Fj(t)1120 2200 y Fh(\000)p Fd(1)1157 2208 y Fj(s)p Fn(\))1188 2200 y Fo(c)1206 2207 y Fj(u\014)q Fn(\()p Fj(t)p Fn(\))270 2267 y Fp(=)i Fo(\016)331 2273 y Fj(v)q Fn(0)368 2267 y Fo(\016)386 2273 y Fj(s)p Fn(1)421 2267 y Fo(c)439 2274 y Fn(2)p Fj(u\014)q Fn(\()p Fj(t)p Fn(\))1658 2267 y Fi(2)257 2402 y Fq(5.9)48 b Fp(The)16 b(second)h(construction)f (describ)q(ed)h(in)e([78)o(],)g(Sec.)h(3,)f(resp.)h([79)o(],)e(Sec.)i (4,)f(no)o(w)257 2452 y(enables)j(us)g(to)f(build)g(a)g(second)h (ordinary)f(Hopf)g(algebra)f(from)g(the)i(Y)m(etter-Drinfel'd)257 2502 y(Hopf)g(algebra)g Fo(A)g Fp(considered)h(in)f(P)o(aragraph)g (3.3.)f(The)h(underlying)g(v)o(ector)h(space)g(of)953 2628 y(76)p eop %%Page: 77 77 77 76 bop 257 262 a Fp(this)14 b(Hopf)e(algebra)h(is)g Fo(A)8 b Fm(\012)g Fo(H)j Fm(\012)d Fo(A)820 246 y Fl(\003)840 262 y Fp(;)k(it)h(con)o(tains)g(the)h(Radford)e(bipro)q(duct)i Fo(A)8 b Fm(\012)g Fo(H)16 b Fp(as)e(a)257 311 y(kind)g(of)f(Borel)h (subalgebra.)257 394 y(W)m(e)e(describ)q(e)i(the)f(Hopf)f(algebra)g (arising)g(from)e(the)j(second)h(construction)f(with)f(resp)q(ect)257 444 y(to)i(the)h(basis)654 493 y Fo(z)673 499 y Fj(uv)q(w)737 493 y Fp(\()p Fo(s;)7 b(t)p Fp(\))12 b(:=)f Fo(e)908 499 y Fj(u)939 493 y Fm(\012)f Fo(x)1005 499 y Fj(s)1032 493 y Fm(\012)f Fo(c)1091 499 y Fj(v)1120 493 y Fm(\012)g Fo(c)1179 499 y Fj(w)1216 493 y Fm(\012)g Fo(d)1279 499 y Fj(t)257 564 y Fp(of)14 b Fo(A)c Fm(\012)g Fo(H)j Fm(\012)d Fo(A)509 549 y Fl(\003)528 564 y Fp(,)k(where)i Fo(u;)7 b(v)q(;)g(w)13 b Fm(2)f Fo(R)i Fp(and)h Fo(s;)7 b(t)13 b Fm(2)f Fo(G)p Fp(.)i(With)g(resp)q(ect)j(to)d(this)h(basis,)f(the)257 614 y(structure)i(elemen)o(ts)e(of)f Fo(A)d Fm(\012)f Fo(H)j Fm(\012)e Fo(A)854 598 y Fl(\003)887 614 y Fp(tak)o(e)k(the)g (follo)o(wing)e(form:)257 700 y Fq(Prop)q(osition)308 750 y Fp(1.)20 b(Multiplication:)479 834 y Fo(z)498 840 y Fj(uv)q(w)562 834 y Fp(\()p Fo(s;)7 b(t)p Fp(\))p Fo(z)666 840 y Fj(u)686 832 y Fh(0)697 840 y Fj(v)715 832 y Fh(0)726 840 y Fj(w)751 832 y Fh(0)764 834 y Fp(\()p Fo(s)799 817 y Fl(0)812 834 y Fo(;)g(t)846 817 y Fl(0)857 834 y Fp(\))k(=)520 896 y Fo(\016)538 903 y Fj(u)558 895 y Fh(0)569 903 y Fl(\000)p Fj(u\027)r Fn(\()p Fj(s)p Fn(\))p Fj(;w)q Fl(\000)p Fj(w)q(\027)r Fn(\()p Fj(t)p Fn(\))820 896 y Fo(\016)838 902 y Fj(ts)867 894 y Fh(0)878 902 y Fj(;s)904 894 y Fh(0)914 902 y Fj(t)927 894 y Fh(0)520 963 y Fo(\021)q Fp(\()p Fo(uq)q Fp(\()p Fo(s;)c(s)675 946 y Fl(0)687 963 y Fp(\))j(+)f Fo(w)q Fp(\()p Fo(q)q Fp(\()p Fo(s)856 946 y Fl(0)868 963 y Fo(;)e(t)902 946 y Fl(0)914 963 y Fp(\))i(+)g Fo(q)q Fp(\()p Fo(s)1035 946 y Fl(0)1048 963 y Fo(t)1063 946 y Fl(0)1074 963 y Fo(;)e(s)1112 946 y Fl(0)r(\000)p Fn(1)1168 963 y Fp(\))j Fm(\000)f Fo(\027)s Fp(\()p Fo(t)p Fp(\))p Fo(q)q Fp(\()p Fo(s)1361 946 y Fl(0)1373 963 y Fo(;)e(s)1411 946 y Fl(0)r(\000)p Fn(1)1467 963 y Fp(\)\)\))520 1031 y Fo(\037)p Fp(\(2)p Fo(uw)q(\027)s Fp(\()p Fo(sts)731 1014 y Fl(0)q(\000)p Fn(2)787 1031 y Fp(\))p Fo(\014)r Fp(\()p Fo(s)863 1014 y Fl(0)876 1031 y Fp(\))p Fo(\013)p Fp(\()p Fo(s)954 1014 y Fl(0)966 1031 y Fp(\))j(+)f(2)p Fo(v)q(u\027)s Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)1236 1014 y Fl(0)1248 1031 y Fp(\))g(+)h(2)p Fo(v)1357 1014 y Fl(0)1369 1031 y Fo(w)q(\027)s Fp(\()p Fo(s)1459 1014 y Fl(0)1470 1031 y Fp(\))p Fo(\013)p Fp(\()p Fo(t)1544 1014 y Fl(0)1556 1031 y Fp(\))571 1098 y(+)f(2)p Fo(w)664 1081 y Fn(2)683 1098 y Fp(\()p Fo(\027)s Fp(\()p Fo(s)758 1081 y Fl(0)769 1098 y Fp(\))p Fo(\014)r Fp(\()p Fo(t)841 1081 y Fl(0)854 1098 y Fp(\))g(+)h Fo(\027)s Fp(\()p Fo(t)p Fp(\))p Fo(\014)r Fp(\()p Fo(t)p Fp(\)\))p Fo(\013)p Fp(\()p Fo(s)1142 1081 y Fl(0)1154 1098 y Fp(\))f(+)h Fo(u)1245 1081 y Fn(2)1263 1098 y Fo(\027)s Fp(\()p Fo(s)p Fp(\))p Fo(\014)r Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)1476 1081 y Fl(0)1490 1098 y Fp(\)\))645 1161 y Fo(z)664 1168 y Fj(u;v)q Fn(+)p Fj(v)755 1160 y Fh(0)766 1168 y Fn(+)p Fj(w)q(\014)q Fn(\()p Fj(s)865 1160 y Fh(0)876 1168 y Fn(\)+)p Fj(w)q(\027)r Fn(\()p Fj(t)p Fn(\))p Fj(\014)q Fn(\()p Fj(s)1046 1160 y Fh(0)1056 1168 y Fn(\))p Fj(;w)q(\027)r Fn(\()p Fj(s)1152 1160 y Fh(0)1163 1168 y Fn(\)+)p Fj(w)1226 1160 y Fh(0)1239 1161 y Fp(\()p Fo(ss)1293 1143 y Fl(0)1306 1161 y Fo(;)d(t)1340 1143 y Fl(0)1351 1161 y Fp(\))308 1260 y(2.)20 b(Unit:)13 b(1)f(=)544 1229 y Fg(P)588 1272 y Fj(u)p Fl(2)p Fj(R;s)p Fl(2)p Fj(G)738 1260 y Fo(z)757 1266 y Fj(u)p Fn(00)812 1260 y Fp(\(1)p Fo(;)7 b(s)p Fp(\))308 1340 y(3.)20 b(Com)o(ultiplicatio)o(n:)430 1424 y(\001\()p Fo(z)500 1430 y Fj(uv)q(w)564 1424 y Fp(\()p Fo(s;)7 b(t)p Fp(\)\))12 b(=)528 1460 y Fg(X)488 1549 y Fj(r)q Fl(2)p Fj(G;k)q Fl(2)p Fj(R)635 1499 y Fo(\021)q Fp(\()p Fo(w)q(q)q Fp(\()p Fo(r)o(;)7 b(r)797 1482 y Fl(\000)p Fn(1)841 1499 y Fo(t)p Fp(\)\))p Fo(\037)p Fp(\()p Fm(\000)p Fo(w)993 1482 y Fn(2)1012 1499 y Fo(\027)s Fp(\()p Fo(r)q Fp(\))p Fo(\014)r Fp(\()p Fo(r)q Fp(\))p Fo(\013)p Fp(\()p Fo(r)1228 1482 y Fl(\000)p Fn(1)1272 1499 y Fo(t)p Fp(\)\))689 1611 y Fo(z)708 1618 y Fj(u)p Fl(\000)p Fj(k)q(;v)q Fn(+)p Fj(k)q(\014)q Fn(\()p Fj(s)p Fn(\))p Fj(;w)q(\027)r Fn(\()p Fj(r)q Fn(\))1003 1611 y Fp(\()p Fo(s;)g(r)1077 1594 y Fl(\000)p Fn(1)1121 1611 y Fo(t)p Fp(\))j Fm(\012)f Fo(z)1222 1619 y Fj(k)q(;v)q Fl(\000)p Fj(w)q(\027)r Fn(\()p Fj(r)q Fn(\))p Fj(\014)q Fn(\()p Fj(r)1430 1611 y Fh(\000)p Fd(1)1468 1619 y Fj(t)p Fn(\))p Fj(;w)1530 1611 y Fp(\()p Fo(s;)e(r)q Fp(\))308 1710 y(4.)20 b(Counit:)13 b Fo(\017)p Fp(\()p Fo(z)563 1716 y Fj(uv)q(w)627 1710 y Fp(\()p Fo(s;)7 b(t)p Fp(\)\))12 b(=)g Fo(\016)802 1716 y Fj(u)p Fn(0)841 1710 y Fo(\016)859 1716 y Fj(t)p Fn(1)308 1790 y Fp(5.)20 b(An)o(tip)q(o)q(de:)366 1874 y Fo(S)r Fp(\()p Fo(z)428 1880 y Fj(uv)q(w)493 1874 y Fp(\()p Fo(s;)7 b(t)p Fp(\)\))12 b(=)366 1942 y Fo(\021)q Fp(\(\()p Fo(u)d Fm(\000)h Fo(w)q(\027)s Fp(\()p Fo(ts)600 1925 y Fl(\000)p Fn(1)644 1942 y Fp(\)\))p Fo(q)q Fp(\()p Fo(s;)d(s)769 1925 y Fl(\000)p Fn(1)814 1942 y Fp(\))i Fm(\000)h Fo(w)q(q)q Fp(\()p Fo(t;)d(t)997 1925 y Fl(\000)p Fn(1)1041 1942 y Fp(\))831 2009 y(+)j Fo(w)q(\027)s Fp(\()p Fo(ts)978 1992 y Fl(\000)p Fn(1)1022 2009 y Fp(\)\()p Fo(q)q Fp(\()p Fo(st)1124 1992 y Fl(\000)p Fn(1)1170 2009 y Fo(;)d(s)1208 1992 y Fl(\000)p Fn(1)1252 2009 y Fp(\))i(+)h Fo(q)q Fp(\()p Fo(s;)d(t)1408 1992 y Fl(\000)p Fn(1)1453 2009 y Fp(\)\)\))366 2077 y Fo(\037)p Fp(\(\()p Fm(\000)p Fo(u)480 2060 y Fn(2)508 2077 y Fp(+)i(2)p Fo(uw)q(\027)s Fp(\()p Fo(s)p Fp(\))g Fm(\000)g Fp(2)p Fo(w)802 2060 y Fn(2)821 2077 y Fo(\027)s Fp(\()p Fo(t)p Fp(\))f(+)i(2)p Fo(w)994 2060 y Fn(2)1021 2077 y Fp(+)g(2)p Fo(uw)q(\027)s Fp(\()p Fo(s)1198 2060 y Fl(\000)p Fn(1)1241 2077 y Fo(t)p Fp(\))g Fm(\000)f Fp(2)p Fo(uw)q(\027)s Fp(\()p Fo(s)1458 2060 y Fl(\000)p Fn(1)1502 2077 y Fp(\))416 2144 y(+)h(2)p Fo(w)510 2127 y Fn(2)528 2144 y Fo(\027)s Fp(\()p Fo(s)587 2127 y Fl(\000)p Fn(2)631 2144 y Fo(t)p Fp(\))g Fm(\000)f Fp(2)p Fo(w)765 2127 y Fn(2)783 2144 y Fo(\027)s Fp(\()p Fo(s)842 2127 y Fl(\000)p Fn(2)887 2144 y Fo(t)902 2127 y Fn(2)920 2144 y Fp(\)\))p Fo(\014)r Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)p Fp(\))j(+)d(4)p Fo(w)1211 2127 y Fn(2)1230 2144 y Fo(\027)s Fp(\()p Fo(s)1289 2127 y Fl(\000)p Fn(1)1333 2144 y Fp(\))p Fo(\014)r Fp(\()p Fo(t)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)p Fp(\))416 2211 y(+)h Fo(w)489 2194 y Fn(2)507 2211 y Fo(\014)r Fp(\()p Fo(t)p Fp(\))p Fo(\013)p Fp(\()p Fo(t)p Fp(\))h Fm(\000)e Fp(2\()p Fo(uv)i Fm(\000)e Fo(v)q(w)q(\027)s Fp(\()p Fo(s)949 2194 y Fl(\000)p Fn(1)994 2211 y Fo(t)p Fp(\))g(+)h Fo(v)q(w)q(\027)s Fp(\()p Fo(s)1187 2194 y Fl(\000)p Fn(1)1231 2211 y Fp(\)\))p Fo(\013)p Fp(\()p Fo(s)p Fp(\))g Fm(\000)g Fp(2)p Fo(v)q(w)q(\013)p Fp(\()p Fo(t)p Fp(\)\))366 2279 y Fo(z)385 2287 y Fl(\000)p Fj(u\027)r Fn(\()p Fj(s)p Fn(\)+)p Fj(w)q(\027)r Fn(\()p Fj(t)p Fn(\))p Fl(\000)p Fj(w)q(;w)q(\014)q Fn(\()p Fj(t)p Fn(\))p Fl(\000)p Fj(v)q Fl(\000)p Fj(u)o(\014)q Fn(\()p Fj(s)p Fn(\))p Fl(\000)p Fj(w)q(\027)r Fn(\()p Fj(t)o Fn(\))p Fj(\014)q Fn(\()p Fj(s)1053 2278 y Fh(\000)p Fd(1)1089 2287 y Fn(\))p Fl(\000)p Fj(w)q(\014)q Fn(\()p Fj(s)1202 2278 y Fh(\000)p Fd(1)1240 2287 y Fn(\))p Fj(;)p Fl(\000)p Fj(w)q(\027)r Fn(\()p Fj(ts)1375 2278 y Fh(\000)p Fd(1)1412 2287 y Fn(\))1427 2279 y Fp(\()p Fo(s)1462 2262 y Fl(\000)p Fn(1)1507 2279 y Fo(;)d(st)1560 2262 y Fl(\000)p Fn(1)1605 2279 y Fo(s)1624 2262 y Fl(\000)p Fn(1)1669 2279 y Fp(\))257 2374 y Fq(Pro)q(of.)36 b Fp(W)m(e)14 b(\014rst)g(establish)h(the)f(form)e(of)h(the)i(m)o (ultiplicatio)o(n.)c(W)m(e)i(ha)o(v)o(e:)332 2464 y(\(\001)383 2470 y Fj(A)419 2464 y Fm(\012)c Fp(id)495 2470 y Fj(A)522 2464 y Fp(\))g Fm(\016)g Fp(\001)612 2470 y Fj(A)639 2464 y Fp(\()p Fo(e)674 2470 y Fj(u)705 2464 y Fm(\012)h Fo(x)771 2470 y Fj(s)788 2464 y Fp(\))i(=)873 2425 y Fg(X)860 2514 y Fj(k)q(;l)p Fl(2)p Fj(R)946 2464 y Fp(\()p Fo(e)981 2470 y Fj(k)1011 2464 y Fm(\012)e Fo(x)1077 2470 y Fj(s)1094 2464 y Fp(\))g Fm(\012)f Fp(\()p Fo(e)1196 2470 y Fj(l)p Fl(\000)p Fj(k)1263 2464 y Fm(\012)g Fo(x)1328 2470 y Fj(s)1346 2464 y Fp(\))g Fm(\012)h Fp(\()p Fo(e)1448 2470 y Fj(u)p Fl(\000)p Fj(l)1516 2464 y Fm(\012)f Fo(x)1581 2470 y Fj(s)1599 2464 y Fp(\))953 2628 y(77)p eop %%Page: 78 78 78 77 bop 257 262 a Fp(Similarly)m(,)10 b(w)o(e)k(ha)o(v)o(e:)366 353 y(\(id)417 359 y Fj(A)442 351 y Fh(\003)468 353 y Fm(\012)p Fp(\001)535 359 y Fj(A)560 351 y Fh(\003)579 353 y Fp(\))c Fm(\016)f Fp(\001)670 359 y Fj(A)695 351 y Fh(\003)714 353 y Fp(\()p Fo(c)748 359 y Fj(w)784 353 y Fm(\012)g Fo(d)847 359 y Fj(t)862 353 y Fp(\))i(=)459 389 y Fg(X)443 478 y Fj(p;r)q Fl(2)p Fj(G)541 428 y Fo(\021)q Fp(\()p Fo(w)q Fp(\()p Fo(q)q Fp(\()p Fo(r)o(;)c(r)719 411 y Fl(\000)p Fn(1)763 428 y Fo(t)p Fp(\))i(+)h Fo(q)q Fp(\()p Fo(p;)d(p)942 411 y Fl(\000)p Fn(1)986 428 y Fo(r)q Fp(\)\)\))537 540 y Fo(\037)p Fp(\()j Fm(\000)f Fo(w)661 523 y Fn(2)680 540 y Fp(\()p Fo(\027)s Fp(\()p Fo(r)q Fp(\))p Fo(\014)r Fp(\()p Fo(r)q Fp(\))p Fo(\013)p Fp(\()p Fo(r)912 523 y Fl(\000)p Fn(1)956 540 y Fo(t)p Fp(\))h(+)f Fo(\027)s Fp(\()p Fo(p)p Fp(\))p Fo(\014)r Fp(\()p Fo(p)p Fp(\))p Fo(\013)p Fp(\()p Fo(p)1257 523 y Fl(\000)p Fn(1)1302 540 y Fo(r)q Fp(\)\)\))745 602 y(\()p Fo(c)779 609 y Fj(w)q(\027)r Fn(\()p Fj(r)q Fn(\))876 602 y Fm(\012)h Fo(d)940 610 y Fj(r)956 602 y Fh(\000)p Fd(1)995 610 y Fj(t)1009 602 y Fp(\))g Fm(\012)f Fp(\()p Fo(c)1110 609 y Fj(w)q(\027)r Fn(\()p Fj(p)p Fn(\))1208 602 y Fm(\012)g Fo(d)1271 610 y Fj(p)1288 602 y Fh(\000)p Fd(1)1327 610 y Fj(r)1345 602 y Fp(\))h Fm(\012)f Fp(\()p Fo(c)1446 608 y Fj(w)1482 602 y Fm(\012)h Fo(d)1546 608 y Fj(p)1565 602 y Fp(\))257 693 y(This)k(implies:)264 785 y(\()p Fo(e)299 791 y Fj(u)330 785 y Fm(\012)9 b Fo(x)395 791 y Fj(s)422 785 y Fm(\012)h Fo(c)482 791 y Fj(v)511 785 y Fm(\012)f Fo(c)570 791 y Fj(w)606 785 y Fm(\012)h Fo(d)670 791 y Fj(t)684 785 y Fp(\)\()p Fo(e)735 791 y Fj(u)755 783 y Fh(0)778 785 y Fm(\012)f Fo(x)843 791 y Fj(s)859 783 y Fh(0)881 785 y Fm(\012)h Fo(c)941 791 y Fj(v)959 783 y Fh(0)981 785 y Fm(\012)f Fo(c)1040 791 y Fj(w)1065 783 y Fh(0)1088 785 y Fm(\012)g Fo(d)1151 791 y Fj(t)1164 783 y Fh(0)1176 785 y Fp(\))291 860 y(=)399 820 y Fg(X)335 910 y Fj(k)q(;l)p Fl(2)p Fj(R;p;r)q Fl(2)p Fj(G)530 860 y Fo(\021)q Fp(\()p Fo(w)q Fp(\()p Fo(q)q Fp(\()p Fo(r)o(;)e(r)708 843 y Fl(\000)p Fn(1)752 860 y Fo(t)p Fp(\))i(+)g Fo(q)q Fp(\()p Fo(p;)e(p)930 843 y Fl(\000)p Fn(1)974 860 y Fo(r)q Fp(\)\)\))659 972 y Fo(\037)p Fp(\()p Fm(\000)p Fo(w)764 955 y Fn(2)783 972 y Fp(\()p Fo(\027)s Fp(\()p Fo(r)q Fp(\))p Fo(\014)r Fp(\()p Fo(r)q Fp(\))p Fo(\013)p Fp(\()p Fo(r)1015 955 y Fl(\000)p Fn(1)1060 972 y Fo(t)p Fp(\))i(+)h Fo(\027)s Fp(\()p Fo(p)p Fp(\))p Fo(\014)r Fp(\()p Fo(p)p Fp(\))p Fo(\013)p Fp(\()p Fo(p)1361 955 y Fl(\000)p Fn(1)1405 972 y Fo(r)q Fp(\)\)\))351 1043 y(\()p Fo(e)386 1049 y Fj(u)417 1043 y Fm(\012)g Fo(x)483 1049 y Fj(s)500 1043 y Fp(\)\()p Fo(c)550 1049 y Fj(v)582 1043 y Fm(!)h Fp([\()p Fo(c)681 1050 y Fj(w)q(\027)r Fn(\()p Fj(r)q Fn(\))777 1043 y Fm(\012)f Fo(d)841 1050 y Fj(r)857 1042 y Fh(\000)p Fd(1)895 1050 y Fj(t)910 1043 y Fp(\))926 1025 y Fn(\(1\))982 1043 y Fo(*)h Fp(\()p Fo(e)1070 1049 y Fj(k)1100 1043 y Fm(\012)f Fo(x)1166 1049 y Fj(s)1182 1041 y Fh(0)1195 1043 y Fp(\)]\))431 1113 y Fm(\012)g Fo(c)491 1119 y Fj(v)510 1113 y Fp(\()p Fo(c)544 1120 y Fj(w)q(\027)r Fn(\()p Fj(r)q Fn(\))642 1113 y Fm(\012)f Fo(d)705 1121 y Fj(r)721 1113 y Fh(\000)p Fd(1)760 1121 y Fj(t)774 1113 y Fp(\))790 1096 y Fn(\(2\))835 1113 y Fp([)p Fo(c)865 1120 y Fj(w)q(\027)r Fn(\()p Fj(p)p Fn(\))962 1113 y Fm(\012)h Fo(d)1026 1121 y Fj(p)1043 1113 y Fh(\000)p Fd(1)1081 1121 y Fj(r)1099 1113 y Fo(]e)1134 1119 y Fj(l)p Fl(\000)p Fj(k)1201 1113 y Fm(\012)g Fo(x)1267 1119 y Fj(s)1283 1111 y Fh(0)1295 1113 y Fp(]\()p Fo(e)1342 1119 y Fj(u)1362 1111 y Fh(0)1373 1119 y Fl(\000)p Fj(l)1421 1113 y Fm(\012)g Fo(x)1487 1119 y Fj(s)1503 1111 y Fh(0)1515 1113 y Fp(\))1531 1096 y Fn(\(1\))1576 1113 y Fo(c)1594 1119 y Fj(v)1612 1111 y Fh(0)502 1184 y Fm(\012)g Fp(\([\()p Fo(c)606 1190 y Fj(w)642 1184 y Fm(\012)f Fo(d)705 1190 y Fj(p)724 1184 y Fp(\))j Fo(\()f Fp(\()p Fo(e)840 1190 y Fj(u)860 1181 y Fh(0)871 1190 y Fl(\000)p Fj(l)919 1184 y Fm(\012)f Fo(x)985 1190 y Fj(s)1001 1181 y Fh(0)1013 1184 y Fp(\))1029 1166 y Fn(\(2\))1074 1184 y Fp(])h Fm( )g Fo(c)1168 1190 y Fj(v)1186 1181 y Fh(0)1199 1184 y Fp(\)\()p Fo(c)1249 1190 y Fj(w)1274 1181 y Fh(0)1297 1184 y Fm(\012)e Fo(d)1360 1190 y Fj(t)1373 1181 y Fh(0)1386 1184 y Fp(\))291 1259 y(=)399 1219 y Fg(X)335 1309 y Fj(k)q(;l)p Fl(2)p Fj(R;p;r)q Fl(2)p Fj(G)530 1259 y Fo(\021)q Fp(\()p Fo(w)q Fp(\()p Fo(q)q Fp(\()p Fo(r)o(;)e(r)708 1242 y Fl(\000)p Fn(1)752 1259 y Fo(t)p Fp(\))i(+)g Fo(q)q Fp(\()p Fo(p;)e(p)930 1242 y Fl(\000)p Fn(1)974 1259 y Fo(r)q Fp(\)\)\))659 1371 y Fo(\037)p Fp(\()p Fm(\000)p Fo(w)764 1354 y Fn(2)783 1371 y Fp(\()p Fo(\027)s Fp(\()p Fo(r)q Fp(\))p Fo(\014)r Fp(\()p Fo(r)q Fp(\))p Fo(\013)p Fp(\()p Fo(r)1015 1354 y Fl(\000)p Fn(1)1060 1371 y Fo(t)p Fp(\))i(+)h Fo(\027)s Fp(\()p Fo(p)p Fp(\))p Fo(\014)r Fp(\()p Fo(p)p Fp(\))p Fo(\013)p Fp(\()p Fo(p)1361 1354 y Fl(\000)p Fn(1)1405 1371 y Fo(r)q Fp(\)\)\))363 1438 y Fo(\016)381 1444 y Fj(r)q(t)412 1438 y Fo(\016)430 1445 y Fj(u)450 1437 y Fh(0)461 1445 y Fl(\000)p Fj(l;w)q Fl(\000)p Fj(w)q(\027)r Fn(\()p Fj(p)p Fn(\))647 1438 y Fo(\016)665 1444 y Fj(lk)697 1438 y Fo(\016)715 1444 y Fj(r)q(p)751 1438 y Fo(\037)p Fp(\()p Fo(k)q(w)q(\027)s Fp(\()p Fo(r)q Fp(\))p Fo(\027)s Fp(\()p Fo(s)982 1421 y Fl(0\000)p Fn(2)1037 1438 y Fp(\))p Fo(\014)r Fp(\()p Fo(s)1113 1421 y Fl(0)1126 1438 y Fp(\))p Fo(\013)p Fp(\()p Fo(s)1204 1421 y Fl(0)1216 1438 y Fp(\)\))1248 1421 y Fn(2)363 1506 y Fo(\021)q Fp(\()p Fo(w)q Fp(\()p Fo(q)q Fp(\()p Fo(s)503 1489 y Fl(0)515 1506 y Fo(;)d(s)553 1489 y Fl(0)r(\000)p Fn(1)609 1506 y Fo(ps)649 1489 y Fl(0)661 1506 y Fp(\))j(+)f Fo(q)q Fp(\()p Fo(ps)804 1489 y Fl(0)816 1506 y Fo(;)e(s)854 1489 y Fl(0)r(\000)p Fn(1)910 1506 y Fp(\))j Fm(\000)f Fo(\027)s Fp(\()p Fo(p)p Fp(\))p Fo(q)q Fp(\()p Fo(s)1109 1489 y Fl(0)1121 1506 y Fo(;)e(s)1159 1489 y Fl(0)q(\000)p Fn(1)1215 1506 y Fp(\)\)\))363 1573 y Fo(\037)p Fp(\()p Fo(w)436 1556 y Fn(2)454 1573 y Fp(\()p Fo(\027)s Fp(\()p Fo(s)529 1556 y Fl(0)541 1573 y Fp(\))p Fo(\014)r Fp(\()p Fo(s)617 1556 y Fl(0)s(\000)p Fn(1)674 1573 y Fo(ps)714 1556 y Fl(0)726 1573 y Fp(\))j(+)f Fo(\027)s Fp(\()p Fo(p)p Fp(\))p Fo(\014)r Fp(\()p Fo(ps)951 1556 y Fl(0)963 1573 y Fp(\))h Fm(\000)f Fo(\027)s Fp(\()p Fo(p)1091 1556 y Fn(2)1109 1573 y Fp(\))p Fo(\014)r Fp(\()p Fo(s)1185 1556 y Fl(0)1198 1573 y Fp(\)\))p Fo(\013)p Fp(\()p Fo(s)1292 1556 y Fl(0)1304 1573 y Fp(\)\))1336 1556 y Fn(2)351 1635 y Fp(\()p Fo(e)386 1641 y Fj(u)417 1635 y Fm(\012)h Fo(x)483 1641 y Fj(s)500 1635 y Fp(\)\()p Fo(c)550 1641 y Fj(v)582 1635 y Fm(!)h Fp(\()p Fo(e)670 1641 y Fj(k)700 1635 y Fm(\012)e Fo(x)765 1641 y Fj(s)781 1633 y Fh(0)794 1635 y Fp(\)\))h Fm(\012)f Fo(c)895 1641 y Fj(v)915 1635 y Fo(c)933 1643 y Fl(\000)p Fj(w)q(\027)r Fn(\()p Fj(r)q Fn(\))p Fj(\014)q Fn(\()p Fj(r)1094 1635 y Fh(\000)p Fd(1)1133 1643 y Fj(t)p Fn(\))1160 1635 y Fo(c)1178 1642 y Fn(2)p Fj(w)q(\027)r Fn(\()p Fj(p)p Fn(\))p Fj(\014)q Fn(\()p Fj(s)1331 1635 y Fh(0)1341 1642 y Fn(\))1356 1635 y Fo(c)1374 1642 y Fn(\()p Fj(u)1407 1635 y Fh(0)1418 1642 y Fl(\000)p Fj(l)p Fn(\))p Fj(\014)q Fn(\()p Fj(s)1516 1635 y Fh(0)1528 1642 y Fn(\))1543 1635 y Fo(c)1561 1641 y Fj(v)1579 1633 y Fh(0)977 1702 y Fm(\012)g Fp(\(\()p Fo(c)1068 1709 y Fj(w)q(\027)r Fn(\()p Fj(s)1141 1701 y Fh(0)1152 1709 y Fn(\))1176 1702 y Fm(\012)h Fo(d)1240 1710 y Fj(s)1256 1702 y Fh(0)q(\000)p Fd(1)1305 1710 y Fj(ps)1338 1702 y Fh(0)t Fp(\))i Fm( )f Fo(c)1450 1708 y Fj(v)1468 1700 y Fh(0)1481 1702 y Fp(\)\()p Fo(c)1531 1708 y Fj(w)1556 1700 y Fh(0)1578 1702 y Fm(\012)f Fo(d)1642 1708 y Fj(t)1655 1700 y Fh(0)1667 1702 y Fp(\))291 1769 y(=)i Fo(\037)p Fp(\(\()p Fo(u)417 1752 y Fl(0)438 1769 y Fm(\000)d Fo(w)i Fp(+)e Fo(w)q(\027)s Fp(\()p Fo(t)p Fp(\)\))p Fo(w)q(\027)s Fp(\()p Fo(t)p Fp(\))p Fo(\027)s Fp(\()p Fo(s)840 1752 y Fl(0\000)p Fn(2)895 1769 y Fp(\))p Fo(\014)r Fp(\()p Fo(s)971 1752 y Fl(0)984 1769 y Fp(\))p Fo(\013)p Fp(\()p Fo(s)1062 1752 y Fl(0)1074 1769 y Fp(\)\))1106 1752 y Fn(2)363 1837 y Fo(\021)q Fp(\()p Fo(w)q Fp(\()p Fo(q)q Fp(\()p Fo(s)503 1820 y Fl(0)515 1837 y Fo(;)e(s)553 1820 y Fl(0)r(\000)p Fn(1)609 1837 y Fo(ts)643 1820 y Fl(0)655 1837 y Fp(\))j(+)f Fo(q)q Fp(\()p Fo(ts)792 1820 y Fl(0)805 1837 y Fo(;)e(s)843 1820 y Fl(0)q(\000)p Fn(1)899 1837 y Fp(\))i Fm(\000)h Fo(\027)s Fp(\()p Fo(t)p Fp(\))p Fo(q)q Fp(\()p Fo(s)1092 1820 y Fl(0)1103 1837 y Fo(;)d(s)1141 1820 y Fl(0)r(\000)p Fn(1)1197 1837 y Fp(\)\)\))363 1904 y Fo(\037)p Fp(\()p Fo(w)436 1887 y Fn(2)454 1904 y Fp(\()p Fo(\027)s Fp(\()p Fo(s)529 1887 y Fl(0)541 1904 y Fp(\))p Fo(\014)r Fp(\()p Fo(s)617 1887 y Fl(0)s(\000)p Fn(1)674 1904 y Fo(ts)708 1887 y Fl(0)721 1904 y Fp(\))i(+)g Fo(\027)s Fp(\()p Fo(t)p Fp(\))p Fo(\014)r Fp(\()p Fo(ts)933 1887 y Fl(0)946 1904 y Fp(\))g Fm(\000)h Fo(\027)s Fp(\()p Fo(t)1068 1887 y Fn(2)1086 1904 y Fp(\))p Fo(\014)r Fp(\()p Fo(s)1162 1887 y Fl(0)1175 1904 y Fp(\)\))p Fo(\013)p Fp(\()p Fo(s)1269 1887 y Fl(0)1281 1904 y Fp(\)\))1313 1887 y Fn(2)363 1972 y Fo(\037)p Fp(\(\()p Fo(u)445 1954 y Fl(0)466 1972 y Fm(\000)f Fo(w)h Fp(+)g Fo(w)q(\027)s Fp(\()p Fo(t)p Fp(\)\))p Fo(v)q(\013)p Fp(\()p Fo(s)790 1954 y Fl(0)802 1972 y Fp(\)\))834 1954 y Fn(2)853 1972 y Fo(\037)p Fp(\()p Fo(v)916 1954 y Fl(0)928 1972 y Fo(w)q(\027)s Fp(\()p Fo(s)1018 1954 y Fl(0)1029 1972 y Fp(\))p Fo(\013)p Fp(\()p Fo(s)1107 1954 y Fl(0)r(\000)p Fn(1)1164 1972 y Fo(ts)1198 1954 y Fl(0)1210 1972 y Fp(\)\))1242 1954 y Fn(2)363 2034 y Fp(\()p Fo(e)398 2040 y Fj(u)429 2034 y Fm(\012)g Fo(x)495 2040 y Fj(s)512 2034 y Fp(\)\()p Fo(e)563 2041 y Fj(u)583 2033 y Fh(0)595 2041 y Fl(\000)p Fj(w)q Fn(+)p Fj(w)q(\027)r Fn(\()p Fj(t)p Fn(\))764 2034 y Fm(\012)f Fo(x)829 2040 y Fj(s)845 2032 y Fh(0)858 2034 y Fp(\))h Fm(\012)f Fo(c)943 2040 y Fj(v)963 2034 y Fo(c)981 2041 y Fn(2)p Fj(w)q(\027)r Fn(\()p Fj(t)p Fn(\))p Fj(\014)q Fn(\()p Fj(s)1130 2033 y Fh(0)1139 2041 y Fn(\))1154 2034 y Fo(c)1172 2041 y Fn(\()p Fj(w)q Fl(\000)p Fj(w)q(\027)r Fn(\()p Fj(t)p Fn(\)\))p Fj(\014)q Fn(\()p Fj(s)1381 2033 y Fh(0)1391 2041 y Fn(\))1406 2034 y Fo(c)1424 2040 y Fj(v)1442 2032 y Fh(0)977 2096 y Fm(\012)g Fp(\()p Fo(c)1052 2103 y Fj(w)q(\027)r Fn(\()p Fj(s)1125 2095 y Fh(0)1136 2103 y Fn(\))1160 2096 y Fm(\012)h Fo(d)1224 2104 y Fj(s)1240 2096 y Fh(0)q(\000)p Fd(1)1289 2104 y Fj(ts)1318 2096 y Fh(0)s Fp(\)\()p Fo(c)1380 2102 y Fj(w)1405 2094 y Fh(0)1428 2096 y Fm(\012)f Fo(d)1491 2102 y Fj(t)1504 2094 y Fh(0)1517 2096 y Fp(\))291 2164 y(=)j Fo(\016)353 2171 y Fj(u\027)r Fn(\()p Fj(s)p Fn(\))p Fj(;u)464 2163 y Fh(0)474 2171 y Fl(\000)p Fj(w)q Fn(+)p Fj(w)q(\027)r Fn(\()p Fj(t)p Fn(\))634 2164 y Fo(\016)652 2171 y Fj(s)668 2163 y Fh(0)s(\000)p Fd(1)718 2171 y Fj(ts)747 2163 y Fh(0)758 2171 y Fj(;t)781 2163 y Fh(0)793 2164 y Fo(\037)p Fp(\()p Fo(u\027)s Fp(\()p Fo(s)p Fp(\))p Fo(w)q(\027)s Fp(\()p Fo(t)p Fp(\))p Fo(\027)s Fp(\()p Fo(s)1095 2146 y Fl(0)q(\000)p Fn(2)1150 2164 y Fp(\))p Fo(\014)r Fp(\()p Fo(s)1226 2146 y Fl(0)1239 2164 y Fp(\))p Fo(\013)p Fp(\()p Fo(s)1317 2146 y Fl(0)1330 2164 y Fp(\)\))1362 2146 y Fn(2)363 2231 y Fo(\021)q Fp(\()p Fo(w)q Fp(\()p Fo(q)q Fp(\()p Fo(s)503 2214 y Fl(0)515 2231 y Fo(;)7 b(t)549 2214 y Fl(0)560 2231 y Fp(\))j(+)f Fo(q)q Fp(\()p Fo(s)682 2214 y Fl(0)694 2231 y Fo(t)709 2214 y Fl(0)721 2231 y Fo(;)e(s)759 2214 y Fl(0)r(\000)p Fn(1)815 2231 y Fp(\))j Fm(\000)f Fo(\027)s Fp(\()p Fo(t)p Fp(\))p Fo(q)q Fp(\()p Fo(s)1008 2214 y Fl(0)1020 2231 y Fo(;)e(s)1058 2214 y Fl(0)q(\000)p Fn(1)1114 2231 y Fp(\)\)\))363 2298 y Fo(\037)p Fp(\()p Fo(w)436 2281 y Fn(2)454 2298 y Fp(\()p Fo(\027)s Fp(\()p Fo(s)529 2281 y Fl(0)541 2298 y Fp(\))p Fo(\014)r Fp(\()p Fo(t)613 2281 y Fl(0)625 2298 y Fp(\))j(+)f Fo(\027)s Fp(\()p Fo(t)p Fp(\))p Fo(\014)r Fp(\()p Fo(s)823 2281 y Fl(0)836 2298 y Fo(t)851 2281 y Fl(0)862 2298 y Fp(\))h Fm(\000)f Fo(\027)s Fp(\()p Fo(t)984 2281 y Fn(2)1002 2298 y Fp(\))p Fo(\014)r Fp(\()p Fo(s)1078 2281 y Fl(0)1091 2298 y Fp(\)\))p Fo(\013)p Fp(\()p Fo(s)1185 2281 y Fl(0)1197 2298 y Fp(\)\))1229 2281 y Fn(2)363 2366 y Fo(\037)p Fp(\()p Fo(v)q(u\027)s Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)587 2349 y Fl(0)599 2366 y Fp(\)\))631 2349 y Fn(2)650 2366 y Fo(\037)p Fp(\()p Fo(v)713 2349 y Fl(0)725 2366 y Fo(w)q(\027)s Fp(\()p Fo(s)815 2349 y Fl(0)827 2366 y Fp(\))p Fo(\013)p Fp(\()p Fo(t)901 2349 y Fl(0)912 2366 y Fp(\)\))944 2349 y Fn(2)963 2366 y Fo(\021)q Fp(\()p Fo(uq)q Fp(\()p Fo(s;)e(s)1118 2349 y Fl(0)1130 2366 y Fp(\)\))p Fo(\037)p Fp(\()p Fo(u)1228 2349 y Fn(2)1247 2366 y Fo(\027)s Fp(\()p Fo(s)p Fp(\))p Fo(\014)r Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)1460 2349 y Fl(0)1473 2366 y Fp(\)\))363 2428 y Fo(e)382 2434 y Fj(u)413 2428 y Fm(\012)i Fo(x)478 2434 y Fj(ss)510 2426 y Fh(0)532 2428 y Fm(\012)h Fo(c)592 2435 y Fj(v)q Fn(+)p Fj(v)653 2427 y Fh(0)664 2435 y Fn(+)p Fj(w)q(\014)q Fn(\()p Fj(s)763 2427 y Fh(0)774 2435 y Fn(\)+)p Fj(w)q(\027)r Fn(\()p Fj(t)p Fn(\))p Fj(\014)q Fn(\()p Fj(s)944 2427 y Fh(0)955 2435 y Fn(\))979 2428 y Fm(\012)f Fo(c)1038 2435 y Fj(w)q(\027)r Fn(\()p Fj(s)1111 2427 y Fh(0)1122 2435 y Fn(\)+)p Fj(w)1185 2427 y Fh(0)1207 2428 y Fm(\012)h Fo(d)1271 2434 y Fj(t)1284 2426 y Fh(0)953 2628 y Fp(78)p eop %%Page: 79 79 79 78 bop 291 262 a Fp(=)12 b Fo(\016)353 269 y Fj(u)373 261 y Fh(0)384 269 y Fl(\000)p Fj(u\027)r Fn(\()p Fj(s)p Fn(\))p Fj(;w)q Fl(\000)p Fj(w)q(\027)r Fn(\()p Fj(t)p Fn(\))635 262 y Fo(\016)653 268 y Fj(ts)682 259 y Fh(0)693 268 y Fj(;s)719 259 y Fh(0)729 268 y Fj(t)742 259 y Fh(0)363 329 y Fo(\021)q Fp(\()p Fo(uq)q Fp(\()p Fo(s;)7 b(s)518 312 y Fl(0)530 329 y Fp(\))i(+)h Fo(w)q Fp(\()p Fo(q)q Fp(\()p Fo(s)699 312 y Fl(0)711 329 y Fo(;)d(t)745 312 y Fl(0)756 329 y Fp(\))i(+)h Fo(q)q Fp(\()p Fo(s)878 312 y Fl(0)890 329 y Fo(t)905 312 y Fl(0)917 329 y Fo(;)d(s)955 312 y Fl(0)q(\000)p Fn(1)1011 329 y Fp(\))i Fm(\000)h Fo(\027)s Fp(\()p Fo(t)p Fp(\))p Fo(q)q Fp(\()p Fo(s)1204 312 y Fl(0)1216 329 y Fo(;)d(s)1254 312 y Fl(0)q(\000)p Fn(1)1310 329 y Fp(\)\)\))363 396 y Fo(\037)p Fp(\()p Fo(uw)q(\027)s Fp(\()p Fo(sts)553 379 y Fl(0)q(\000)p Fn(2)609 396 y Fp(\))p Fo(\014)r Fp(\()p Fo(s)685 379 y Fl(0)698 396 y Fp(\))p Fo(\013)p Fp(\()p Fo(s)776 379 y Fl(0)788 396 y Fp(\))i(+)h Fo(v)q(u\027)s Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)1037 379 y Fl(0)1049 396 y Fp(\))f(+)h Fo(v)1137 379 y Fl(0)1149 396 y Fo(w)q(\027)s Fp(\()p Fo(s)1239 379 y Fl(0)1250 396 y Fp(\))p Fo(\013)p Fp(\()p Fo(t)1324 379 y Fl(0)1336 396 y Fp(\)\))1368 379 y Fn(2)363 464 y Fo(\037)p Fp(\()p Fo(w)436 447 y Fn(2)454 464 y Fp(\()p Fo(\027)s Fp(\()p Fo(s)529 447 y Fl(0)541 464 y Fp(\))p Fo(\014)r Fp(\()p Fo(t)613 447 y Fl(0)625 464 y Fp(\))g(+)f Fo(\027)s Fp(\()p Fo(t)p Fp(\))p Fo(\014)r Fp(\()p Fo(s)823 447 y Fl(0)836 464 y Fo(t)851 447 y Fl(0)862 464 y Fp(\))h Fm(\000)f Fo(\027)s Fp(\()p Fo(t)984 447 y Fn(2)1002 464 y Fp(\))p Fo(\014)r Fp(\()p Fo(s)1078 447 y Fl(0)1091 464 y Fp(\)\))p Fo(\013)p Fp(\()p Fo(s)1185 447 y Fl(0)1197 464 y Fp(\)\))1229 447 y Fn(2)1248 464 y Fo(\037)p Fp(\()p Fo(u)1314 447 y Fn(2)1333 464 y Fo(\027)s Fp(\()p Fo(s)p Fp(\))p Fo(\014)r Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)1546 447 y Fl(0)1559 464 y Fp(\)\))363 526 y Fo(e)382 532 y Fj(u)413 526 y Fm(\012)g Fo(x)478 532 y Fj(ss)510 524 y Fh(0)532 526 y Fm(\012)h Fo(c)592 533 y Fj(v)q Fn(+)p Fj(v)653 525 y Fh(0)664 533 y Fn(+)p Fj(w)q(\014)q Fn(\()p Fj(s)763 525 y Fh(0)774 533 y Fn(\)+)p Fj(w)q(\027)r Fn(\()p Fj(t)p Fn(\))p Fj(\014)q Fn(\()p Fj(s)944 525 y Fh(0)955 533 y Fn(\))979 526 y Fm(\012)f Fo(c)1038 533 y Fj(w)q(\027)r Fn(\()p Fj(s)1111 525 y Fh(0)1122 533 y Fn(\)+)p Fj(w)1185 525 y Fh(0)1207 526 y Fm(\012)h Fo(d)1271 532 y Fj(t)1284 524 y Fh(0)257 614 y Fp(If)k Fo(s)318 599 y Fl(0)330 614 y Fo(t)345 599 y Fl(0)368 614 y Fp(=)e Fo(ts)446 599 y Fl(0)458 614 y Fp(,)h(w)o(e)i(get)f(from)e(the)i(co)q(cycle)h (iden)o(tit)o(y)f(that:)437 702 y Fo(\027)s Fp(\()p Fo(t)p Fp(\))p Fo(\014)r Fp(\()p Fo(s)568 685 y Fl(0)581 702 y Fo(t)596 685 y Fl(0)607 702 y Fp(\))c Fm(\000)f Fo(\027)s Fp(\()p Fo(t)729 685 y Fn(2)747 702 y Fp(\))p Fo(\014)r Fp(\()p Fo(s)823 685 y Fl(0)836 702 y Fp(\))j(=)g Fo(\027)s Fp(\()p Fo(t)p Fp(\))p Fo(\014)r Fp(\()p Fo(ts)1054 685 y Fl(0)1066 702 y Fp(\))d Fm(\000)h Fo(\027)s Fp(\()p Fo(t)1188 685 y Fn(2)1206 702 y Fp(\))p Fo(\014)r Fp(\()p Fo(s)1282 685 y Fl(0)1295 702 y Fp(\))h(=)h Fo(\027)s Fp(\()p Fo(t)p Fp(\))p Fo(\014)r Fp(\()p Fo(t)p Fp(\))257 790 y(Therefore,)j(the)f(second)h(argumen)o(t)e(of)g Fo(\037)g Fp(in)h(the)g(ab)q(o)o(v)o(e)f(expression)i(for)f(the)g(pro)q (duct)h(is)257 840 y(equal)e(to)g Fo(w)447 825 y Fn(2)466 840 y Fp(\()p Fo(\027)s Fp(\()p Fo(s)541 825 y Fl(0)552 840 y Fp(\))p Fo(\014)r Fp(\()p Fo(t)624 825 y Fl(0)637 840 y Fp(\))8 b(+)g Fo(\027)s Fp(\()p Fo(t)p Fp(\))p Fo(\014)r Fp(\()p Fo(t)p Fp(\)\))p Fo(\013)p Fp(\()p Fo(s)922 825 y Fl(0)934 840 y Fp(\).)13 b(This)g(implies)e(the)j(form)d (of)i(the)h(m)o(ultipli)o(ca-)257 890 y(tion)g(stated)g(ab)q(o)o(v)o (e.)257 974 y(The)21 b(form)e(of)g(the)i(com)o(ultiplication)c(follo)o (ws)i(easily)h(from)e(the)j(de\014nition)f(\(cf.)g([79)o(],)257 1024 y(Subsec.)15 b(3.2,)e(p.)g(39\):)337 1112 y(\001)o(\()p Fo(e)406 1118 y Fj(u)438 1112 y Fm(\012)c Fo(x)503 1118 y Fj(s)530 1112 y Fm(\012)h Fo(c)590 1118 y Fj(v)619 1112 y Fm(\012)f Fo(c)678 1118 y Fj(w)714 1112 y Fm(\012)h Fo(d)778 1118 y Fj(t)792 1112 y Fp(\))h(=)371 1183 y(\()p Fo(e)406 1189 y Fj(u)438 1183 y Fm(\012)e Fo(x)503 1189 y Fj(s)521 1183 y Fp(\))537 1190 y Fn(\(1\))591 1183 y Fm(\012)g Fp(\()p Fo(e)667 1189 y Fj(u)699 1183 y Fm(\012)g Fo(x)764 1189 y Fj(s)782 1183 y Fp(\))798 1190 y Fn(\(2\))842 1165 y(\(1\))887 1183 y Fo(c)905 1189 y Fj(v)934 1183 y Fm(\012)g Fp(\()p Fo(c)1009 1189 y Fj(w)1046 1183 y Fm(\012)g Fo(d)1109 1189 y Fj(t)1123 1183 y Fp(\))1139 1190 y Fn(\(1\))1184 1165 y(\(1\))713 1253 y Fm(\012)g Fp(\()p Fo(e)789 1259 y Fj(u)821 1253 y Fm(\012)g Fo(x)886 1259 y Fj(s)904 1253 y Fp(\))920 1260 y Fn(\(2\))964 1236 y(\(2\))1018 1253 y Fm(\012)h Fo(c)1078 1259 y Fj(v)1097 1253 y Fp(\()p Fo(c)1131 1259 y Fj(w)1168 1253 y Fm(\012)f Fo(d)1231 1259 y Fj(t)1245 1253 y Fp(\))1261 1260 y Fn(\(1\))1306 1236 y(\(2\))1360 1253 y Fm(\012)g Fp(\()p Fo(c)1435 1259 y Fj(w)1472 1253 y Fm(\012)g Fo(d)1535 1259 y Fj(t)1549 1253 y Fp(\))1565 1260 y Fn(\(2\))419 1289 y Fg(X)378 1378 y Fj(r)q Fl(2)p Fj(G;k)q Fl(2)p Fj(R)526 1328 y Fo(\021)q Fp(\()p Fo(w)q(q)q Fp(\()p Fo(r)o(;)e(r)688 1311 y Fl(\000)p Fn(1)731 1328 y Fo(t)p Fp(\)\))p Fo(\037)p Fp(\()p Fm(\000)p Fo(w)883 1311 y Fn(2)902 1328 y Fo(\027)s Fp(\()p Fo(r)q Fp(\))p Fo(\014)r Fp(\()p Fo(r)q Fp(\))p Fo(\013)p Fp(\()p Fo(r)1118 1311 y Fl(\000)p Fn(1)1163 1328 y Fo(t)p Fp(\)\))538 1443 y(\()p Fo(e)573 1449 y Fj(u)p Fl(\000)p Fj(k)648 1443 y Fm(\012)j Fo(x)714 1449 y Fj(s)731 1443 y Fp(\))g Fm(\012)f Fp(\()p Fo(e)833 1449 y Fj(k)863 1443 y Fm(\012)h Fo(x)929 1449 y Fj(s)946 1443 y Fp(\))962 1426 y Fn(\(1\))1007 1443 y Fo(c)1025 1449 y Fj(v)1054 1443 y Fm(\012)f Fp(\()p Fo(c)1129 1450 y Fj(w)q(\027)r Fn(\()p Fj(r)q Fn(\))1226 1443 y Fm(\012)h Fo(d)1290 1451 y Fj(r)1306 1443 y Fh(\000)p Fd(1)1344 1451 y Fj(t)1359 1443 y Fp(\))1375 1426 y Fn(\(1\))713 1514 y Fm(\012)f Fp(\()p Fo(e)789 1520 y Fj(k)819 1514 y Fm(\012)h Fo(x)885 1520 y Fj(s)902 1514 y Fp(\))918 1497 y Fn(\(2\))972 1514 y Fm(\012)g Fo(c)1032 1520 y Fj(v)1052 1514 y Fp(\()p Fo(c)1086 1521 y Fj(w)q(\027)r Fn(\()p Fj(r)q Fn(\))1183 1514 y Fm(\012)f Fo(d)1246 1522 y Fj(r)1262 1513 y Fh(\000)p Fd(1)1301 1522 y Fj(t)1315 1514 y Fp(\))1331 1497 y Fn(\(2\))1385 1514 y Fm(\012)h Fp(\()p Fo(c)1461 1520 y Fj(w)1497 1514 y Fm(\012)f Fo(d)1560 1520 y Fj(r)1578 1514 y Fp(\))419 1550 y Fg(X)378 1639 y Fj(r)q Fl(2)p Fj(G;k)q Fl(2)p Fj(R)526 1589 y Fo(\021)q Fp(\()p Fo(w)q(q)q Fp(\()p Fo(r)o(;)e(r)688 1572 y Fl(\000)p Fn(1)731 1589 y Fo(t)p Fp(\)\))p Fo(\037)p Fp(\()p Fm(\000)p Fo(w)883 1572 y Fn(2)902 1589 y Fo(\027)s Fp(\()p Fo(r)q Fp(\))p Fo(\014)r Fp(\()p Fo(r)q Fp(\))p Fo(\013)p Fp(\()p Fo(r)1118 1572 y Fl(\000)p Fn(1)1163 1589 y Fo(t)p Fp(\)\))538 1700 y(\()p Fo(e)573 1706 y Fj(u)p Fl(\000)p Fj(k)648 1700 y Fm(\012)j Fo(x)714 1706 y Fj(s)740 1700 y Fm(\012)g Fo(c)800 1707 y Fj(v)q Fn(+)p Fj(k)q(\014)q Fn(\()p Fj(s)p Fn(\))935 1700 y Fm(\012)f Fo(c)994 1707 y Fj(w)q(\027)r Fn(\()p Fj(r)q Fn(\))1091 1700 y Fm(\012)h Fo(d)1155 1708 y Fj(r)1171 1700 y Fh(\000)p Fd(1)1209 1708 y Fj(t)1224 1700 y Fp(\))713 1763 y Fm(\012)f Fp(\()p Fo(e)789 1769 y Fj(k)819 1763 y Fm(\012)h Fo(x)885 1769 y Fj(s)912 1763 y Fm(\012)f Fo(c)971 1770 y Fj(v)q Fl(\000)p Fj(w)q(\027)r Fn(\()p Fj(r)q Fn(\))p Fj(\014)q Fn(\()p Fj(r)1150 1762 y Fh(\000)p Fd(1)1189 1770 y Fj(t)p Fn(\))1225 1763 y Fm(\012)h Fo(c)1285 1769 y Fj(w)1321 1763 y Fm(\012)f Fo(d)1384 1769 y Fj(r)1402 1763 y Fp(\))257 1885 y(F)m(or)14 b(the)g(an)o(tip)q(o)q(de,)g(w)o(e)g(ha)o(v)o(e:)264 1973 y Fo(S)r Fp(\()p Fo(e)326 1979 y Fj(u)358 1973 y Fm(\012)c Fo(x)424 1979 y Fj(s)451 1973 y Fm(\012)f Fo(c)510 1979 y Fj(v)539 1973 y Fm(\012)g Fo(c)598 1979 y Fj(w)635 1973 y Fm(\012)g Fo(d)698 1979 y Fj(t)712 1973 y Fp(\))276 2044 y(=)i(\(1)356 2050 y Fj(A)393 2044 y Fm(\012)e Fp(1)455 2050 y Fj(H)496 2044 y Fm(\012)g Fo(S)562 2050 y Fj(A)587 2042 y Fh(\003)607 2044 y Fp(\(\()p Fo(c)657 2050 y Fj(w)693 2044 y Fm(\012)h Fo(d)757 2050 y Fj(t)771 2044 y Fp(\))787 2026 y Fn(\(1\))832 2044 y Fp(\)\)\(1)901 2050 y Fj(A)937 2044 y Fm(\012)g Fo(S)1004 2050 y Fj(H)1036 2044 y Fp(\(\()p Fo(e)1087 2050 y Fj(u)1118 2044 y Fm(\012)g Fo(x)1184 2050 y Fj(s)1201 2044 y Fp(\))1217 2026 y Fn(\(1\))1262 2044 y Fo(c)1280 2050 y Fj(v)1300 2044 y Fp(\()p Fo(c)1334 2050 y Fj(w)1370 2044 y Fm(\012)f Fo(d)1433 2050 y Fj(t)1447 2044 y Fp(\))1463 2026 y Fn(\(2\))1508 2044 y Fp(\))h Fm(\012)f Fp(1)1596 2050 y Fj(A)1621 2042 y Fh(\003)1640 2044 y Fp(\))347 2114 y(\()p Fo(S)388 2120 y Fj(A)416 2114 y Fp(\(\()p Fo(e)467 2120 y Fj(u)498 2114 y Fm(\012)h Fo(x)564 2120 y Fj(s)581 2114 y Fp(\))597 2097 y Fn(\(2\))642 2114 y Fp(\))f Fm(\012)h Fp(1)730 2120 y Fj(H)770 2114 y Fm(\012)g Fp(1)833 2120 y Fj(A)858 2112 y Fh(\003)877 2114 y Fp(\))276 2181 y(=)h Fo(\021)q Fp(\()p Fm(\000)p Fo(w)q(q)q Fp(\()p Fo(t;)c(t)505 2164 y Fl(\000)p Fn(1)550 2181 y Fp(\)\))p Fo(\037)p Fp(\()p Fm(\000)p Fo(w)687 2164 y Fn(2)706 2181 y Fo(\014)r Fp(\()p Fo(t)p Fp(\))p Fo(\013)p Fp(\()p Fo(t)p Fp(\)\))p Fo(\021)q Fp(\()p Fo(uq)q Fp(\()p Fo(s;)g(s)1023 2164 y Fl(\000)p Fn(1)1069 2181 y Fp(\)\))p Fo(\037)p Fp(\()p Fo(u)1167 2164 y Fn(2)1186 2181 y Fo(\014)r Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)p Fp(\)\))347 2244 y(\(1)384 2250 y Fj(A)420 2244 y Fm(\012)j Fp(1)483 2250 y Fj(H)523 2244 y Fm(\012)g Fo(c)583 2251 y Fl(\000)p Fj(w)q(\027)r Fn(\()p Fj(t)p Fn(\))702 2244 y Fm(\012)f Fo(d)765 2252 y Fj(t)778 2243 y Fh(\000)p Fd(1)818 2244 y Fp(\)\(1)871 2250 y Fj(A)908 2244 y Fm(\012)g Fo(c)967 2251 y Fj(w)q(\014)q Fn(\()p Fj(t)p Fn(\))p Fl(\000)p Fj(v)q Fl(\000)p Fj(u\014)q Fn(\()p Fj(s)p Fn(\))1213 2244 y Fm(\012)h Fp(1)1276 2250 y Fj(A)1301 2242 y Fh(\003)1320 2244 y Fp(\))347 2306 y(\()p Fo(e)382 2313 y Fl(\000)p Fj(u\027)r Fn(\()p Fj(s)p Fn(\))500 2306 y Fm(\012)f Fo(x)565 2314 y Fj(s)581 2306 y Fh(\000)p Fd(1)14 b Fm(\012)9 b Fp(1)693 2312 y Fj(H)734 2306 y Fm(\012)g Fp(1)796 2312 y Fj(A)821 2304 y Fh(\003)840 2306 y Fp(\))276 2373 y(=)i Fo(\021)q Fp(\()p Fo(uq)q Fp(\()p Fo(s;)c(s)474 2356 y Fl(\000)p Fn(1)520 2373 y Fp(\))i Fm(\000)g Fo(w)q(q)q Fp(\()p Fo(t;)e(t)702 2356 y Fl(\000)p Fn(1)746 2373 y Fp(\)\))p Fo(\037)p Fp(\()p Fo(u)844 2356 y Fn(2)863 2373 y Fo(\014)r Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)p Fp(\))k Fm(\000)f Fo(w)1101 2356 y Fn(2)1119 2373 y Fo(\014)r Fp(\()p Fo(t)p Fp(\))p Fo(\013)p Fp(\()p Fo(t)p Fp(\)\))347 2441 y Fo(\037)p Fp(\()p Fm(\000)p Fo(u\027)s Fp(\()p Fo(s)p Fp(\)\()p Fo(w)q(\014)r Fp(\()p Fo(t)p Fp(\))g Fm(\000)g Fo(v)h Fm(\000)e Fo(u\014)r Fp(\()p Fo(s)p Fp(\)\))p Fo(\013)p Fp(\()p Fo(s)941 2424 y Fl(\000)p Fn(1)987 2441 y Fp(\)\))1019 2424 y Fn(2)347 2503 y Fp(\(1)384 2509 y Fj(A)420 2503 y Fm(\012)h Fp(1)483 2509 y Fj(H)523 2503 y Fm(\012)g Fo(c)583 2510 y Fl(\000)p Fj(w)q(\027)r Fn(\()p Fj(t)p Fn(\))702 2503 y Fm(\012)f Fo(d)765 2511 y Fj(t)778 2503 y Fh(\000)p Fd(1)t Fp(\)\()p Fo(e)869 2510 y Fl(\000)p Fj(u\027)r Fn(\()p Fj(s)p Fn(\))987 2503 y Fm(\012)g Fo(x)1052 2511 y Fj(s)1068 2503 y Fh(\000)p Fd(1)14 b Fm(\012)9 b Fo(c)1177 2510 y Fj(w)q(\014)q Fn(\()p Fj(t)p Fn(\))p Fl(\000)p Fj(v)q Fl(\000)p Fj(u\014)q Fn(\()p Fj(s)p Fn(\))1423 2503 y Fm(\012)h Fp(1)1486 2509 y Fj(A)1511 2501 y Fh(\003)1530 2503 y Fp(\))953 2628 y(79)p eop %%Page: 80 80 80 79 bop 276 264 a Fp(=)360 224 y Fg(X)319 314 y Fj(k)q Fl(2)p Fj(R;r)q Fl(2)p Fj(G)467 264 y Fo(\021)q Fp(\()p Fo(uq)q Fp(\()p Fo(s;)7 b(s)622 246 y Fl(\000)p Fn(1)667 264 y Fp(\))i Fm(\000)h Fo(w)q(q)q Fp(\()p Fo(t;)d(t)850 246 y Fl(\000)p Fn(1)894 264 y Fp(\)\))p Fo(\037)p Fp(\()p Fo(u)992 246 y Fn(2)1010 264 y Fo(\014)r Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)p Fp(\))k Fm(\000)f Fo(w)1248 246 y Fn(2)1266 264 y Fo(\014)r Fp(\()p Fo(t)p Fp(\))p Fo(\013)p Fp(\()p Fo(t)p Fp(\)\))347 376 y Fo(\037)p Fp(\()p Fm(\000)p Fo(u\027)s Fp(\()p Fo(s)p Fp(\)\()p Fo(w)q(\014)r Fp(\()p Fo(t)p Fp(\))g Fm(\000)g Fo(v)h Fm(\000)e Fo(u\014)r Fp(\()p Fo(s)p Fp(\)\))p Fo(\013)p Fp(\()p Fo(s)941 359 y Fl(\000)p Fn(1)987 376 y Fp(\)\))1019 359 y Fn(2)347 438 y Fp(\()p Fo(e)382 444 y Fj(k)412 438 y Fm(\012)h Fo(x)478 444 y Fn(1)505 438 y Fm(\012)g Fo(c)565 444 y Fn(0)593 438 y Fm(\012)f Fo(c)652 445 y Fl(\000)p Fj(w)q(\027)r Fn(\()p Fj(t)p Fn(\))771 438 y Fm(\012)h Fo(d)835 446 y Fj(t)848 438 y Fh(\000)p Fd(1)t Fp(\)\()p Fo(e)939 445 y Fl(\000)p Fj(u\027)r Fn(\()p Fj(s)p Fn(\))1056 438 y Fm(\012)g Fo(x)1122 446 y Fj(s)1138 438 y Fh(\000)p Fd(1)j Fm(\012)d Fo(c)1247 445 y Fj(w)q(\014)q Fn(\()p Fj(t)p Fn(\))p Fl(\000)p Fj(v)q Fl(\000)p Fj(u\014)q Fn(\()p Fj(s)p Fn(\))1493 438 y Fm(\012)f Fo(c)1552 444 y Fn(0)1580 438 y Fm(\012)h Fo(d)1644 444 y Fj(r)1662 438 y Fp(\))276 505 y(=)h Fo(\021)q Fp(\()p Fo(uq)q Fp(\()p Fo(s;)c(s)474 488 y Fl(\000)p Fn(1)520 505 y Fp(\))i Fm(\000)g Fo(w)q(q)q Fp(\()p Fo(t;)e(t)702 488 y Fl(\000)p Fn(1)746 505 y Fp(\)\))p Fo(\037)p Fp(\()p Fo(u)844 488 y Fn(2)863 505 y Fo(\014)r Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)p Fp(\))k Fm(\000)f Fo(w)1101 488 y Fn(2)1119 505 y Fo(\014)r Fp(\()p Fo(t)p Fp(\))p Fo(\013)p Fp(\()p Fo(t)p Fp(\)\))347 573 y Fo(\037)p Fp(\()p Fm(\000)p Fo(u\027)s Fp(\()p Fo(s)p Fp(\)\()p Fo(w)q(\014)r Fp(\()p Fo(t)p Fp(\))g Fm(\000)g Fo(v)h Fm(\000)e Fo(u\014)r Fp(\()p Fo(s)p Fp(\)\))p Fo(\013)p Fp(\()p Fo(s)941 556 y Fl(\000)p Fn(1)987 573 y Fp(\)\))1019 556 y Fn(2)347 640 y Fo(\021)q Fp(\()p Fm(\000)p Fo(w)q(\027)s Fp(\()p Fo(t)p Fp(\)\()p Fo(q)q Fp(\()p Fo(s)590 623 y Fl(\000)p Fn(1)635 640 y Fo(;)e(st)688 623 y Fl(\000)p Fn(1)733 640 y Fo(s)752 623 y Fl(\000)p Fn(1)797 640 y Fp(\))i(+)h Fo(q)q Fp(\()p Fo(t)915 623 y Fl(\000)p Fn(1)959 640 y Fo(s)978 623 y Fl(\000)p Fn(1)1023 640 y Fo(;)d(s)p Fp(\))j Fm(\000)f Fo(\027)s Fp(\()p Fo(t)1183 623 y Fl(\000)p Fn(1)1227 640 y Fp(\))p Fo(q)q Fp(\()p Fo(s)1298 623 y Fl(\000)p Fn(1)1343 640 y Fo(;)e(s)p Fp(\)\)\))347 708 y Fo(\037)p Fp(\()p Fo(w)q(\027)s Fp(\()p Fo(t)p Fp(\)\()p Fo(u\027)s Fp(\()p Fo(s)p Fp(\))i Fm(\000)g Fo(w)q(\027)s Fp(\()p Fo(t)p Fp(\))g(+)g Fo(w)q Fp(\))p Fo(\027)s Fp(\()p Fo(t)910 691 y Fl(\000)p Fn(1)954 708 y Fo(s)973 691 y Fn(2)992 708 y Fp(\))p Fo(\014)r Fp(\()p Fo(s)1068 691 y Fl(\000)p Fn(1)1114 708 y Fp(\))p Fo(\013)p Fp(\()p Fo(s)1192 691 y Fl(\000)p Fn(1)1237 708 y Fp(\))605 775 y Fm(\000)h Fp(\()p Fo(w)q(\014)r Fp(\()p Fo(t)p Fp(\))g Fm(\000)g Fo(v)g Fm(\000)g Fo(u\014)r Fp(\()p Fo(s)p Fp(\)\))p Fo(w)q(\027)s Fp(\()p Fo(t)p Fp(\))p Fo(\027)s Fp(\()p Fo(s)1167 758 y Fl(\000)p Fn(1)1212 775 y Fp(\))p Fo(\013)p Fp(\()p Fo(st)1305 758 y Fl(\000)p Fn(1)1350 775 y Fo(s)1369 758 y Fl(\000)p Fn(1)1414 775 y Fp(\)\))1446 758 y Fn(2)347 843 y Fo(\037)p Fp(\()p Fo(w)420 825 y Fn(2)439 843 y Fo(\027)s Fp(\()p Fo(t)p Fp(\))510 825 y Fn(2)528 843 y Fp(\()p Fo(\027)s Fp(\()p Fo(s)603 825 y Fl(\000)p Fn(1)647 843 y Fp(\))p Fo(\014)r Fp(\()p Fo(st)738 825 y Fl(\000)p Fn(1)784 843 y Fo(s)803 825 y Fl(\000)p Fn(1)848 843 y Fp(\))g(+)f Fo(\027)s Fp(\()p Fo(t)970 825 y Fl(\000)p Fn(1)1014 843 y Fp(\))p Fo(\014)r Fp(\()p Fo(t)1086 825 y Fl(\000)p Fn(1)1131 843 y Fo(s)1150 825 y Fl(\000)p Fn(1)1195 843 y Fp(\))h Fm(\000)f Fo(\027)s Fp(\()p Fo(t)1317 825 y Fl(\000)p Fn(2)1361 843 y Fp(\))p Fo(\014)r Fp(\()p Fo(s)1437 825 y Fl(\000)p Fn(1)1483 843 y Fp(\)\))p Fo(\013)p Fp(\()p Fo(s)1577 825 y Fl(\000)p Fn(1)1622 843 y Fp(\)\))1654 825 y Fn(2)347 905 y Fo(e)366 912 y Fl(\000)p Fj(u\027)r Fn(\()p Fj(s)p Fn(\)+)p Fj(w)q(\027)r Fn(\()p Fj(t)p Fn(\))p Fl(\000)p Fj(w)642 905 y Fm(\012)g Fo(x)707 913 y Fj(s)723 904 y Fh(\000)p Fd(1)772 905 y Fm(\012)h Fo(c)832 913 y Fj(w)q(\014)q Fn(\()p Fj(t)p Fn(\))p Fl(\000)p Fj(v)q Fl(\000)p Fj(u\014)q Fn(\()p Fj(s)p Fn(\))p Fl(\000)p Fj(w)q(\027)r Fn(\()p Fj(t)p Fn(\))p Fj(\014)q Fn(\()p Fj(s)1226 904 y Fh(\000)p Fd(1)1262 913 y Fn(\))p Fl(\000)p Fj(w)q(\014)q Fn(\()p Fj(s)1375 904 y Fh(\000)p Fd(1)1413 913 y Fn(\))1438 905 y Fm(\012)1260 971 y Fo(c)1278 979 y Fl(\000)p Fj(w)q(\027)r Fn(\()p Fj(t)p Fn(\))p Fj(\027)r Fn(\()p Fj(s)1435 971 y Fh(\000)p Fd(1)1472 979 y Fn(\))1496 971 y Fm(\012)g Fo(d)1560 979 y Fj(st)1589 971 y Fh(\000)p Fd(1)1626 979 y Fj(s)1642 971 y Fh(\000)p Fd(1)257 1148 y Fp(T)m(o)16 b(simplify)d(this)j(expression,)h(w)o(e)f(treat)h(the)f (argumen)o(ts)g(of)f Fo(\037)h Fp(and)g Fo(\021)h Fp(separately)m(.)f (As)257 1198 y(already)i(p)q(oin)o(ted)f(out)h(in)f(P)o(aragraph)g (3.3,)f Fo(\037)i Fp(v)n(anishes)g(b)o(y)f(assumption)f(on)i(the)g(t)o (w)o(o-)257 1248 y(sided)d(ideal)e Fo(I)k Fp(generated)e(b)o(y)f(the)g (additiv)o(e)g(comm)o(utators)d Fo(uv)g Fm(\000)e Fo(v)q(u)14 b Fp(and)g(the)h(elemen)o(ts)257 1298 y(of)f(the)i(form)d Fo(\014)r Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(t)p Fp(\))e Fm(\000)f Fo(\014)r Fp(\()p Fo(t)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)p Fp(\),)15 b(for)g Fo(u;)7 b(v)13 b Fm(2)g Fo(R)i Fp(and)f Fo(s;)7 b(t)13 b Fm(2)f Fo(G)p Fp(.)i(Therefore,)i (denoting)257 1348 y(congruence)g(mo)q(dulo)c Fo(I)17 b Fp(b)o(y)d Fm(\021)p Fp(,)f(w)o(e)h(can)h(rewrite)f(the)h(argumen)o (t)e(of)g Fo(\037)h Fp(as)g(follo)o(ws:)275 1439 y Fo(u)299 1422 y Fn(2)317 1439 y Fo(\014)r Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)p Fp(\))d Fm(\000)f Fo(w)555 1422 y Fn(2)573 1439 y Fo(\014)r Fp(\()p Fo(t)p Fp(\))p Fo(\013)p Fp(\()p Fo(t)p Fp(\))h Fm(\000)e Fp(2)p Fo(u\027)s Fp(\()p Fo(s)p Fp(\)\()p Fo(w)q(\014)r Fp(\()p Fo(t)p Fp(\))h Fm(\000)f Fo(v)i Fm(\000)e Fo(u\014)r Fp(\()p Fo(s)p Fp(\)\))p Fo(\013)p Fp(\()p Fo(s)1311 1422 y Fl(\000)p Fn(1)1358 1439 y Fp(\))326 1506 y(+)g(2)p Fo(w)q(\027)s Fp(\()p Fo(t)p Fp(\)\()p Fo(u\027)s Fp(\()p Fo(s)p Fp(\))f Fm(\000)i Fo(w)q(\027)s Fp(\()p Fo(t)p Fp(\))e(+)i Fo(w)q Fp(\))p Fo(\027)s Fp(\()p Fo(t)909 1489 y Fl(\000)p Fn(1)953 1506 y Fo(s)972 1489 y Fn(2)991 1506 y Fp(\))p Fo(\014)r Fp(\()p Fo(s)1067 1489 y Fl(\000)p Fn(1)1113 1506 y Fp(\))p Fo(\013)p Fp(\()p Fo(s)1191 1489 y Fl(\000)p Fn(1)1236 1506 y Fp(\))326 1574 y Fm(\000)f Fp(2\()p Fo(w)q(\014)r Fp(\()p Fo(t)p Fp(\))h Fm(\000)g Fo(v)g Fm(\000)g Fo(u\014)r Fp(\()p Fo(s)p Fp(\)\))p Fo(w)q(\027)s Fp(\()p Fo(t)p Fp(\))p Fo(\027)s Fp(\()p Fo(s)908 1557 y Fl(\000)p Fn(1)953 1574 y Fp(\))p Fo(\013)p Fp(\()p Fo(st)1046 1557 y Fl(\000)p Fn(1)1091 1574 y Fo(s)1110 1557 y Fl(\000)p Fn(1)1155 1574 y Fp(\))326 1641 y(+)f(2)p Fo(w)419 1624 y Fn(2)437 1641 y Fo(\027)s Fp(\()p Fo(t)p Fp(\))508 1624 y Fn(2)526 1641 y Fp(\()p Fo(\027)s Fp(\()p Fo(s)601 1624 y Fl(\000)p Fn(1)646 1641 y Fp(\))p Fo(\014)r Fp(\()p Fo(st)737 1624 y Fl(\000)p Fn(1)783 1641 y Fo(s)802 1624 y Fl(\000)p Fn(1)847 1641 y Fp(\))g(+)h Fo(\027)s Fp(\()p Fo(t)969 1624 y Fl(\000)p Fn(1)1013 1641 y Fp(\))p Fo(\014)r Fp(\()p Fo(t)1085 1624 y Fl(\000)p Fn(1)1130 1641 y Fo(s)1149 1624 y Fl(\000)p Fn(1)1194 1641 y Fp(\))f Fm(\000)h Fo(\027)s Fp(\()p Fo(t)1316 1624 y Fl(\000)p Fn(2)1360 1641 y Fp(\))p Fo(\014)r Fp(\()p Fo(s)1436 1624 y Fl(\000)p Fn(1)1482 1641 y Fp(\)\))p Fo(\013)p Fp(\()p Fo(s)1576 1624 y Fl(\000)p Fn(1)1621 1641 y Fp(\))286 1709 y Fm(\021)i Fo(u)354 1691 y Fn(2)373 1709 y Fo(\014)r Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)p Fp(\))f Fm(\000)e Fo(w)610 1691 y Fn(2)629 1709 y Fo(\014)r Fp(\()p Fo(t)p Fp(\))p Fo(\013)p Fp(\()p Fo(t)p Fp(\))h(+)f(2)p Fo(u)p Fp(\()p Fo(w)q(\014)r Fp(\()p Fo(t)p Fp(\))h Fm(\000)g Fo(v)g Fm(\000)g Fo(u\014)r Fp(\()p Fo(s)p Fp(\)\))p Fo(\013)p Fp(\()p Fo(s)p Fp(\))326 1776 y(+)f(2)p Fo(w)q Fp(\()p Fo(u\027)s Fp(\()p Fo(s)p Fp(\))g Fm(\000)g Fo(w)q(\027)s Fp(\()p Fo(t)p Fp(\))g(+)g Fo(w)q Fp(\))p Fo(\014)r Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)p Fp(\))i Fm(\000)f Fp(2\()p Fo(w)q(\014)r Fp(\()p Fo(t)p Fp(\))g Fm(\000)f Fo(v)i Fm(\000)f Fo(u\014)r Fp(\()p Fo(s)p Fp(\)\))p Fo(w)q(\027)s Fp(\()p Fo(s)1460 1759 y Fl(\000)p Fn(1)1505 1776 y Fo(t)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)p Fp(\))326 1843 y Fm(\000)f Fp(2\()p Fo(w)q(\014)r Fp(\()p Fo(t)p Fp(\))h Fm(\000)g Fo(v)g Fm(\000)g Fo(u\014)r Fp(\()p Fo(s)p Fp(\)\))p Fo(w)q(\027)s Fp(\()p Fo(t)p Fp(\))p Fo(\013)p Fp(\()p Fo(t)907 1826 y Fl(\000)p Fn(1)952 1843 y Fp(\))f(+)h(2\()p Fo(w)q(\014)r Fp(\()p Fo(t)p Fp(\))g Fm(\000)f Fo(v)i Fm(\000)f Fo(u\014)r Fp(\()p Fo(s)p Fp(\)\))p Fo(w)q(\027)s Fp(\()p Fo(s)1489 1826 y Fl(\000)p Fn(1)1534 1843 y Fp(\))p Fo(\013)p Fp(\()p Fo(s)p Fp(\))326 1911 y(+)f(2\()p Fo(w)435 1894 y Fn(2)454 1911 y Fo(\027)s Fp(\()p Fo(s)513 1894 y Fl(\000)p Fn(2)557 1911 y Fo(t)p Fp(\))g Fm(\000)h Fo(w)670 1894 y Fn(2)688 1911 y Fo(\027)s Fp(\()p Fo(s)747 1894 y Fl(\000)p Fn(2)791 1911 y Fo(t)806 1894 y Fn(2)825 1911 y Fp(\)\))p Fo(\014)r Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)p Fp(\))h(+)f(2\()p Fo(w)1132 1894 y Fn(2)1150 1911 y Fo(\027)s Fp(\()p Fo(s)1209 1894 y Fl(\000)p Fn(1)1254 1911 y Fp(\))f(+)h Fo(w)1352 1894 y Fn(2)1370 1911 y Fo(\027)s Fp(\()p Fo(s)1429 1894 y Fl(\000)p Fn(1)1473 1911 y Fo(t)p Fp(\)\))p Fo(\014)r Fp(\()p Fo(t)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)p Fp(\))286 1978 y Fm(\021)i Fp(\()p Fm(\000)p Fo(u)402 1961 y Fn(2)430 1978 y Fp(+)e(2)p Fo(uw)q(\027)s Fp(\()p Fo(s)p Fp(\))e Fm(\000)i Fp(2)p Fo(w)725 1961 y Fn(2)743 1978 y Fo(\027)s Fp(\()p Fo(t)p Fp(\))f(+)g(2)p Fo(w)916 1961 y Fn(2)944 1978 y Fp(+)g(2)p Fo(uw)q(\027)s Fp(\()p Fo(s)1120 1961 y Fl(\000)p Fn(1)1164 1978 y Fo(t)p Fp(\))g Fm(\000)h Fp(2)p Fo(uw)q(\027)s Fp(\()p Fo(s)1381 1961 y Fl(\000)p Fn(1)1424 1978 y Fp(\))948 2046 y(+)g(2)p Fo(w)1042 2028 y Fn(2)1060 2046 y Fo(\027)s Fp(\()p Fo(s)1119 2028 y Fl(\000)p Fn(2)1163 2046 y Fo(t)p Fp(\))g Fm(\000)f Fp(2)p Fo(w)1297 2028 y Fn(2)1315 2046 y Fo(\027)s Fp(\()p Fo(s)1374 2028 y Fl(\000)p Fn(2)1419 2046 y Fo(t)1434 2028 y Fn(2)1452 2046 y Fp(\)\))p Fo(\014)r Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)p Fp(\))326 2113 y(+)g(4)p Fo(w)419 2096 y Fn(2)437 2113 y Fo(\027)s Fp(\()p Fo(s)496 2096 y Fl(\000)p Fn(1)541 2113 y Fp(\))p Fo(\014)r Fp(\()p Fo(t)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)p Fp(\))h(+)g Fo(w)790 2096 y Fn(2)808 2113 y Fo(\014)r Fp(\()p Fo(t)p Fp(\))p Fo(\013)p Fp(\()p Fo(t)p Fp(\))h Fm(\000)e Fp(2\()p Fo(uv)i Fm(\000)e Fo(v)q(w)q(\027)s Fp(\()p Fo(s)1250 2096 y Fl(\000)p Fn(1)1295 2113 y Fo(t)p Fp(\))g(+)h Fo(v)q(w)q(\027)s Fp(\()p Fo(s)1488 2096 y Fl(\000)p Fn(1)1532 2113 y Fp(\)\))p Fo(\013)p Fp(\()p Fo(s)p Fp(\))326 2175 y Fm(\000)f Fp(2)p Fo(v)q(w)q(\013)p Fp(\()p Fo(t)p Fp(\))257 2267 y(Here)15 b(w)o(e)g(ha)o(v)o(e)e(used)i(the)g(equalit)o(y)628 2358 y Fo(\013)p Fp(\()p Fo(sts)724 2341 y Fl(\000)p Fn(1)770 2358 y Fp(\))c Fm(\021)h Fo(\013)p Fp(\()p Fo(s)p Fp(\))e(+)f Fo(\027)s Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(t)p Fp(\))g Fm(\000)h Fo(\027)s Fp(\()p Fo(t)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)p Fp(\))257 2449 y(whic)o(h)k(follo)o(ws)e(easily)i(from)e (the)i(de\014nition)g(of)f(a)h(1-co)q(cycle)g(and)g(Lemma)d(1.13,)h (and,)953 2628 y(80)p eop %%Page: 81 81 81 80 bop 257 262 a Fp(based)15 b(on)f(this)g(equalit)o(y)m(,)e(the)i (expansion)293 353 y(2)p Fo(w)345 336 y Fn(2)364 353 y Fo(\027)s Fp(\()p Fo(t)p Fp(\))435 336 y Fn(2)453 353 y Fp(\()p Fo(\027)s Fp(\()p Fo(s)528 336 y Fl(\000)p Fn(1)572 353 y Fp(\))p Fo(\014)r Fp(\()p Fo(st)663 336 y Fl(\000)p Fn(1)709 353 y Fo(s)728 336 y Fl(\000)p Fn(1)773 353 y Fp(\))9 b(+)h Fo(\027)s Fp(\()p Fo(t)895 336 y Fl(\000)p Fn(1)939 353 y Fp(\))p Fo(\014)r Fp(\()p Fo(t)1011 336 y Fl(\000)p Fn(1)1056 353 y Fo(s)1075 336 y Fl(\000)p Fn(1)1120 353 y Fp(\))g Fm(\000)f Fo(\027)s Fp(\()p Fo(t)1242 336 y Fl(\000)p Fn(2)1286 353 y Fp(\))p Fo(\014)r Fp(\()p Fo(s)1362 336 y Fl(\000)p Fn(1)1408 353 y Fp(\)\))p Fo(\013)p Fp(\()p Fo(s)1502 336 y Fl(\000)p Fn(1)1547 353 y Fp(\))305 420 y Fm(\021)j Fp(2\()p Fo(w)417 403 y Fn(2)435 420 y Fo(\027)s Fp(\()p Fo(s)494 403 y Fl(\000)p Fn(2)538 420 y Fo(t)p Fp(\))e Fm(\000)f Fo(w)651 403 y Fn(2)670 420 y Fo(\027)s Fp(\()p Fo(s)729 403 y Fl(\000)p Fn(2)773 420 y Fo(t)788 403 y Fn(2)807 420 y Fp(\)\))p Fo(\014)r Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)p Fp(\))i(+)f(2\()p Fo(w)1114 403 y Fn(2)1132 420 y Fo(\027)s Fp(\()p Fo(s)1191 403 y Fl(\000)p Fn(1)1235 420 y Fp(\))g(+)f Fo(w)1333 403 y Fn(2)1352 420 y Fo(\027)s Fp(\()p Fo(s)1411 403 y Fl(\000)p Fn(1)1455 420 y Fo(t)p Fp(\)\))p Fo(\014)r Fp(\()p Fo(t)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)p Fp(\))257 512 y(for)14 b(the)g(last)g(summand)d(in)j(the)g(\014rst)h(expression)g (ab)q(o)o(v)o(e.)257 597 y(T)m(o)e(simplify)d(the)k(argumen)o(t)e(of)g Fo(\021)q Fp(,)h(w)o(e)g(\014rst)h(note)g(that)f(w)o(e)g(ha)o(v)o(e)g (from)f(the)h(de\014nition)g(of)257 647 y(a)h(normalized)e(2-co)q (cycle)j(that:)266 738 y Fo(q)q Fp(\()p Fo(s)321 721 y Fl(\000)p Fn(1)367 738 y Fo(;)7 b(st)420 721 y Fl(\000)p Fn(1)464 738 y Fo(s)483 721 y Fl(\000)p Fn(1)528 738 y Fp(\))j(+)f Fo(q)q Fp(\()p Fo(t)646 721 y Fl(\000)p Fn(1)691 738 y Fo(s)710 721 y Fl(\000)p Fn(1)755 738 y Fo(;)e(s)p Fp(\))k(=)h Fo(\027)s Fp(\()p Fo(s)923 721 y Fl(\000)p Fn(1)967 738 y Fp(\))p Fo(q)q Fp(\()p Fo(st)1053 721 y Fl(\000)p Fn(1)1099 738 y Fo(s)1118 721 y Fl(\000)p Fn(1)1163 738 y Fo(;)7 b(s)p Fp(\))i(+)h Fo(q)q Fp(\()p Fo(s)1323 721 y Fl(\000)p Fn(1)1368 738 y Fo(;)d(st)1421 721 y Fl(\000)p Fn(1)1465 738 y Fp(\))320 806 y(=)k Fo(\027)s Fp(\()p Fo(s)422 789 y Fl(\000)p Fn(1)467 806 y Fp(\))p Fo(q)q Fp(\()p Fo(st)553 789 y Fl(\000)p Fn(1)598 806 y Fo(s)617 789 y Fl(\000)p Fn(1)662 806 y Fo(;)c(s)p Fp(\))i Fm(\000)h Fo(\027)s Fp(\()p Fo(s)826 789 y Fl(\000)p Fn(1)870 806 y Fp(\))p Fo(q)q Fp(\()p Fo(st)956 789 y Fl(\000)p Fn(1)1001 806 y Fo(;)d(s)1039 789 y Fl(\000)p Fn(1)1084 806 y Fo(s)p Fp(\))j(+)f Fo(q)q Fp(\()p Fo(s)1225 789 y Fl(\000)p Fn(1)1270 806 y Fo(;)e(st)1323 789 y Fl(\000)p Fn(1)1368 806 y Fp(\))i Fm(\000)h Fo(q)q Fp(\()p Fo(s)1490 789 y Fl(\000)p Fn(1)1535 806 y Fo(s;)d(t)1588 789 y Fl(\000)p Fn(1)1632 806 y Fp(\))320 873 y(=)k Fo(\027)s Fp(\()p Fo(s)422 856 y Fl(\000)p Fn(1)467 873 y Fp(\))p Fo(\027)s Fp(\()p Fo(st)557 856 y Fl(\000)p Fn(1)601 873 y Fp(\))p Fo(q)q Fp(\()p Fo(s)672 856 y Fl(\000)p Fn(1)717 873 y Fo(;)c(s)p Fp(\))j Fm(\000)f Fo(\027)s Fp(\()p Fo(s)881 856 y Fl(\000)p Fn(1)926 873 y Fp(\))p Fo(q)q Fp(\()p Fo(st)1012 856 y Fl(\000)p Fn(1)1057 873 y Fo(;)e(s)1095 856 y Fl(\000)p Fn(1)1139 873 y Fp(\))j(+)f Fo(q)q Fp(\()p Fo(s)1261 856 y Fl(\000)p Fn(1)1306 873 y Fo(;)e(s)p Fp(\))j Fm(\000)f Fo(\027)s Fp(\()p Fo(s)1470 856 y Fl(\000)p Fn(1)1514 873 y Fp(\))p Fo(q)q Fp(\()p Fo(s;)e(t)1619 856 y Fl(\000)p Fn(1)1664 873 y Fp(\))320 941 y(=)k Fo(\027)s Fp(\()p Fo(t)418 923 y Fl(\000)p Fn(1)462 941 y Fp(\))p Fo(q)q Fp(\()p Fo(s)533 923 y Fl(\000)p Fn(1)579 941 y Fo(;)c(s)p Fp(\))i(+)g Fo(q)q Fp(\()p Fo(s)738 923 y Fl(\000)p Fn(1)783 941 y Fo(;)e(s)p Fp(\))j Fm(\000)f Fo(\027)s Fp(\()p Fo(s)947 923 y Fl(\000)p Fn(1)992 941 y Fp(\))p Fo(q)q Fp(\()p Fo(st)1078 923 y Fl(\000)p Fn(1)1123 941 y Fo(;)e(s)1161 923 y Fl(\000)p Fn(1)1205 941 y Fp(\))j Fm(\000)f Fo(\027)s Fp(\()p Fo(s)1331 923 y Fl(\000)p Fn(1)1376 941 y Fp(\))p Fo(q)q Fp(\()p Fo(s;)e(t)1481 923 y Fl(\000)p Fn(1)1525 941 y Fp(\))257 1032 y(Therefore,)15 b(the)g(argumen)o(t)d(of)i Fo(\021)h Fp(in)e(the)h(ab)q(o)o(v)o(e)g(expression)h(is)f(giv)o(en)f(b)o(y)h (the)g(form)o(ula:)288 1123 y Fo(uq)q Fp(\()p Fo(s;)7 b(s)405 1106 y Fl(\000)p Fn(1)450 1123 y Fp(\))i Fm(\000)h Fo(w)q(q)q Fp(\()p Fo(t;)d(t)633 1106 y Fl(\000)p Fn(1)677 1123 y Fp(\))463 1191 y Fm(\000)j Fo(w)q(\027)s Fp(\()p Fo(t)p Fp(\)\()p Fo(q)q Fp(\()p Fo(s)678 1173 y Fl(\000)p Fn(1)722 1191 y Fo(;)d(st)775 1173 y Fl(\000)p Fn(1)820 1191 y Fo(s)839 1173 y Fl(\000)p Fn(1)884 1191 y Fp(\))i(+)h Fo(q)q Fp(\()p Fo(t)1002 1173 y Fl(\000)p Fn(1)1046 1191 y Fo(s)1065 1173 y Fl(\000)p Fn(1)1110 1191 y Fo(;)d(s)p Fp(\))j Fm(\000)f Fo(\027)s Fp(\()p Fo(t)1270 1173 y Fl(\000)p Fn(1)1314 1191 y Fp(\))p Fo(q)q Fp(\()p Fo(s)1385 1173 y Fl(\000)p Fn(1)1430 1191 y Fo(;)e(s)p Fp(\)\))299 1258 y(=)12 b Fo(uq)q Fp(\()p Fo(s;)7 b(s)460 1241 y Fl(\000)p Fn(1)505 1258 y Fp(\))i Fm(\000)h Fo(w)q(q)q Fp(\()p Fo(t;)d(t)688 1241 y Fl(\000)p Fn(1)732 1258 y Fp(\))463 1325 y Fm(\000)j Fo(w)q(\027)s Fp(\()p Fo(t)p Fp(\)\()p Fo(q)q Fp(\()p Fo(s)678 1308 y Fl(\000)p Fn(1)722 1325 y Fo(;)d(s)p Fp(\))j Fm(\000)f Fo(\027)s Fp(\()p Fo(s)886 1308 y Fl(\000)p Fn(1)930 1325 y Fp(\))p Fo(q)q Fp(\()p Fo(st)1016 1308 y Fl(\000)p Fn(1)1062 1325 y Fo(;)e(s)1100 1308 y Fl(\000)p Fn(1)1144 1325 y Fp(\))j Fm(\000)f Fo(\027)s Fp(\()p Fo(s)1270 1308 y Fl(\000)p Fn(1)1314 1325 y Fp(\))p Fo(q)q Fp(\()p Fo(s;)e(t)1419 1308 y Fl(\000)p Fn(1)1464 1325 y Fp(\)\))299 1393 y(=)12 b(\()p Fo(u)d Fm(\000)h Fo(w)q(\027)s Fp(\()p Fo(ts)539 1376 y Fl(\000)p Fn(1)583 1393 y Fp(\)\))p Fo(q)q Fp(\()p Fo(s;)d(s)708 1376 y Fl(\000)p Fn(1)753 1393 y Fp(\))j Fm(\000)f Fo(w)q(q)q Fp(\()p Fo(t;)e(t)936 1376 y Fl(\000)p Fn(1)980 1393 y Fp(\))j(+)f Fo(w)q(\027)s Fp(\()p Fo(ts)1152 1376 y Fl(\000)p Fn(1)1196 1393 y Fp(\)\()p Fo(q)q Fp(\()p Fo(st)1298 1376 y Fl(\000)p Fn(1)1344 1393 y Fo(;)e(s)1382 1376 y Fl(\000)p Fn(1)1426 1393 y Fp(\))j(+)f Fo(q)q Fp(\()p Fo(s;)e(t)1582 1376 y Fl(\000)p Fn(1)1627 1393 y Fp(\)\))257 1484 y(This)14 b(implies)e(the)j(asserted)g(form)d(of)i (the)g(an)o(tip)q(o)q(de.)257 1570 y(The)g(form)o(ulas)e(for)h(the)h (unit)g(and)f(the)h(counit)g(follo)o(w)e(imm)o(ediately)f(from)g(their) j(resp)q(ec-)257 1620 y(tiv)o(e)g(de\014nition.)f Fi(2)257 1738 y Fp(The)e(reader)g(is)f(in)o(vited)g(to)f(c)o(hec)o(k)i(directly) g(that)f(the)h(ab)q(o)o(v)o(e)e(structures)k(mak)o(e)8 b Fo(A)r Fm(\012)r Fo(H)d Fm(\012)r Fo(A)1671 1723 y Fl(\003)257 1788 y Fp(in)o(to)14 b(a)f(Hopf)h(algebra,)f(whic)o(h)g(is) h(not)g(en)o(tirely)g(ob)o(vious.)257 1923 y Fq(5.10)48 b Fp(W)m(e)14 b(no)o(w)f(w)o(an)o(t)h(to)g(determine)g(when)g(the)h(ab) q(o)o(v)o(e)e(Hopf)h(algebra)g(is)f(semisimple.)257 1973 y(As)h(in)e(P)o(aragraph)g(5.3,)g(w)o(e)h(approac)o(h)g(this)f(problem) g(b)o(y)g(explicitly)g(exhibiting)g(an)h(in)o(te-)257 2023 y(gral.)257 2114 y Fq(Prop)q(osition)f(1)308 2164 y Fp(1.)20 b(\003)12 b(:=)457 2133 y Fg(P)501 2176 y Fj(s)p Fl(2)p Fj(G;u;v)q Fl(2)p Fj(R)678 2164 y Fo(z)697 2170 y Fn(0)p Fj(uv)753 2164 y Fp(\()p Fo(s;)7 b Fp(1\))14 b(is)g(a)f(t)o(w)o(o-sided)h(in)o(tegral)f(in)h Fo(A)9 b Fm(\012)h Fo(H)i Fm(\012)d Fo(A)1520 2149 y Fl(\003)1539 2164 y Fp(.)308 2247 y(2.)20 b Fo(A)11 b Fm(\012)h Fo(H)i Fm(\012)e Fo(A)571 2232 y Fl(\003)607 2247 y Fp(is)k(semisimple)f(if)h (and)g(only)g(if)g(the)i(c)o(haracteristic)g(of)e Fo(K)21 b Fp(neither)361 2297 y(divides)14 b(the)g(cardinalit)o(y)f(of)g Fo(R)h Fp(nor)g(the)g(cardinalit)o(y)f(of)g Fo(G)p Fp(.)257 2396 y Fq(Pro)q(of.)36 b Fp(W)m(e)14 b(ha)o(v)o(e)h(seen)h(in)e(Prop)q (osition)g(5.3.1)f(that)h(\003)1175 2402 y Fj(A)1215 2396 y Fp(:=)1271 2365 y Fg(P)1315 2409 y Fj(s)p Fl(2)p Fj(G)1388 2396 y Fo(e)1407 2402 y Fn(0)1436 2396 y Fm(\012)9 b Fo(x)1501 2402 y Fj(s)1533 2396 y Fp(is)15 b(a)f(t)o(w)o(o-)257 2446 y(sided)21 b(in)o(tegral)e(of)g Fo(A)h Fp(that)g(is)g(in)o(v)n (arian)o(t)f(and)g(coin)o(v)n(arian)o(t.)f(The)j(in)o(tegral)e(c)o (haracter)257 2496 y(therefore)h(coincides)f(with)e(the)i(counit,)f (and)f(the)i(in)o(tegral)e(group)h(elemen)o(t)g(coincides)953 2628 y(81)p eop %%Page: 82 82 82 81 bop 257 262 a Fp(with)18 b(the)g(unit)f(\(cf.)g([81)o(],)g(Prop.) g(2.10,)f(p.)h(15\).)g(Since,)g(as)h(an)f(algebra,)g Fo(A)1482 246 y Fl(\003)1518 262 y Fp(is)h(the)g(or-)257 311 y(dinary)e(tensor)h(pro)q(duct)g(of)f(a)f(group)h(ring)g(and)g(a)g (dual)f(group)h(ring,)f(it)h(is)g(easy)g(to)g(see)257 361 y(that)j(\003)381 367 y Fj(A)406 359 y Fh(\003)444 361 y Fp(:=)506 330 y Fg(P)550 374 y Fj(v)q Fl(2)p Fj(R)624 361 y Fo(c)642 367 y Fj(v)674 361 y Fm(\012)13 b Fo(d)741 367 y Fn(1)777 361 y Fp(is)18 b(a)g(t)o(w)o(o-sided)g(in)o(tegral)g(of) f Fo(A)1289 346 y Fl(\003)1309 361 y Fp(,)g(whic)o(h)h(is,)g(as)g(a)g (con-)257 411 y(sequence)k(of)e(Lemma)d(1.13,)h(also)h(in)o(v)n(arian)o (t)f(and)i(coin)o(v)n(arian)o(t.)e(If)i(\003)1412 417 y Fj(H)1465 411 y Fp(:=)1530 380 y Fg(P)1574 423 y Fj(u)p Fl(2)p Fj(R)1650 411 y Fo(c)1668 417 y Fj(u)257 461 y Fp(denotes)h(the)g(in)o(tegral)e(of)g(the)h(group)g(ring)f Fo(H)s Fp(,)g(w)o(e)h(ha)o(v)o(e)g(b)o(y)g([81)o(],)e(Thm.)g(5.4,)h(p.) g(62)257 511 y(that)d(\003)11 b(=)h(\003)462 517 y Fj(A)498 511 y Fm(\012)d Fp(\003)568 517 y Fj(H)609 511 y Fm(\012)g Fp(\003)679 517 y Fj(A)704 509 y Fh(\003)739 511 y Fp(is)15 b(a)g(t)o(w)o(o-sided)g(in)o(tegral)f(of)h Fo(A)10 b Fm(\012)h Fo(H)i Fm(\012)d Fo(A)1410 496 y Fl(\003)1429 511 y Fp(.)15 b(Since)h Fo(\017)p Fp(\(\003\))d(=)257 560 y(card)q(\()p Fo(G)p Fp(\))7 b(card\()p Fo(R)p Fp(\))550 545 y Fn(2)569 560 y Fp(,)14 b(the)i(second)g(assertion)g(follo)o(ws)d (from)g(Masc)o(hk)o(e's)j(theorem)e(for)h(or-)257 610 y(dinary)f(Hopf)f(algebras)h(\(cf.)g([57)o(],)f(Thm.)f(2.2.1,)f(p.)j (20\).)f Fi(2)257 725 y Fp(W)m(e)k(no)o(w)f(turn)i(to)f(the)g(question) g(of)g(cosemisimpli)o(cit)o(y)m(.)c(In)k(general,)g(w)o(e)g(obtain)f(a) h(left)257 775 y(in)o(tegral)10 b Fo(\025)g Fp(on)g(the)g(Hopf)g (algebra)g Fo(A)r Fm(\012)r Fo(H)5 b Fm(\012)r Fo(A)972 760 y Fl(\003)1000 775 y Fp(emerging)j(from)h(the)h(second)h (construction)257 825 y(b)o(y)j(the)h(form)o(ula:)651 875 y Fo(\025)p Fp(\()p Fo(a)10 b Fm(\012)f Fo(h)h Fm(\012)f Fo(b)p Fp(\))j(=)f Fo(\025)952 881 y Fj(A)980 875 y Fp(\()p Fo(a)p Fp(\))p Fo(\025)1058 881 y Fj(H)1090 875 y Fp(\()p Fo(g)1126 881 y Fj(A)1153 875 y Fo(h)p Fp(\))p Fo(\025)1217 881 y Fj(B)1245 875 y Fp(\()p Fo(b)p Fp(\))257 946 y(where)18 b Fo(\025)404 952 y Fj(A)446 946 y Fm(2)d Fo(A)520 931 y Fl(\003)540 946 y Fp(,)g Fo(\025)591 952 y Fj(H)638 946 y Fm(2)g Fo(H)719 931 y Fl(\003)738 946 y Fp(,)h(and)g Fo(\025)873 952 y Fj(B)917 946 y Fm(2)f Fo(B)993 931 y Fl(\003)1029 946 y Fp(are)i(left)f(in)o(tegrals)f(and)i Fo(g)1449 952 y Fj(A)1491 946 y Fm(2)e Fo(H)k Fp(is)d(the)257 996 y(in)o(tegral)f(group)f(elemen)o(t)h(of)f Fo(A)p Fp(.)g(A)h(righ)o(t)g(in)o(tegral)f Fo(\032)h Fp(on)g Fo(A)10 b Fm(\012)g Fo(H)j Fm(\012)d Fo(A)1387 981 y Fl(\003)1421 996 y Fp(is)15 b(giv)o(en)f(b)o(y)h(the)257 1046 y(form)o(ula:)647 1095 y Fo(\032)p Fp(\()p Fo(a)10 b Fm(\012)g Fo(h)f Fm(\012)g Fo(b)p Fp(\))j(=)g Fo(\032)943 1101 y Fj(A)970 1095 y Fp(\()p Fo(a)p Fp(\))p Fo(\032)1045 1101 y Fj(H)1077 1095 y Fp(\()p Fo(hg)1138 1078 y Fl(\000)p Fn(1)1137 1108 y Fj(A)1183 1095 y Fp(\))p Fo(\032)1220 1101 y Fj(B)1249 1095 y Fp(\()p Fo(b)p Fp(\))257 1167 y(where)18 b Fo(\032)401 1173 y Fj(A)445 1167 y Fm(2)e Fo(A)520 1152 y Fl(\003)539 1167 y Fp(,)h Fo(\032)589 1173 y Fj(H)637 1167 y Fm(2)f Fo(H)719 1152 y Fl(\003)738 1167 y Fp(,)g(and)h Fo(\032)871 1173 y Fj(B)916 1167 y Fm(2)f Fo(B)993 1152 y Fl(\003)1030 1167 y Fp(are)h(righ)o(t)f(in)o (tegrals.)g(In)h(our)g(situation,)257 1217 y(this)d(yields)g(the)g (follo)o(wing:)257 1304 y Fq(Prop)q(osition)e(2)308 1354 y Fp(1.)20 b(The)14 b(linear)g(form)e Fo(\025)g Fp(:)f Fo(A)e Fm(\012)h Fo(H)i Fm(\012)d Fo(A)920 1339 y Fl(\003)951 1354 y Fm(!)i Fo(K)17 b Fp(determined)d(b)o(y)800 1440 y Fo(\025)p Fp(\()p Fo(z)859 1446 y Fj(uv)q(w)924 1440 y Fp(\()p Fo(s;)7 b(t)p Fp(\)\))12 b(=)g Fo(\016)1099 1446 y Fj(s)p Fn(1)1134 1440 y Fo(\016)1152 1446 y Fj(v)q Fn(0)1188 1440 y Fo(\016)1206 1446 y Fj(w)q Fn(0)361 1526 y Fp(is)i(a)f(t)o(w)o(o-sided)h(in)o(tegral)f(on)h Fo(A)9 b Fm(\012)h Fo(H)i Fm(\012)e Fo(A)1033 1511 y Fl(\003)1052 1526 y Fp(.)308 1607 y(2.)20 b Fo(A)10 b Fm(\012)f Fo(H)j Fm(\012)e Fo(A)563 1592 y Fl(\003)596 1607 y Fp(is)k(cosemisimple)e(if)h(and)h(only)f(if)g(the)i(c)o (haracteristic)g(of)e Fo(K)18 b Fp(neither)361 1657 y(divides)c(the)g (cardinalit)o(y)f(of)g Fo(R)h Fp(nor)g(the)g(cardinalit)o(y)f(of)g Fo(G)p Fp(.)257 1753 y Fq(Pro)q(of.)36 b Fp(W)m(e)16 b(ha)o(v)o(e)h(seen)g(in)f(Prop)q(osition)g(5.3.2)f(that)i(the)g (linear)f(form)e Fo(\025)1476 1759 y Fj(A)1520 1753 y Fp(:)h Fo(A)h Fm(!)f Fo(K)257 1803 y Fp(determined)f(b)o(y)811 1853 y Fo(\025)835 1859 y Fj(A)862 1853 y Fp(\()p Fo(e)897 1859 y Fj(u)929 1853 y Fm(\012)9 b Fo(x)994 1859 y Fj(s)1012 1853 y Fp(\))i(=)h Fo(\016)1101 1859 y Fj(s)p Fn(1)257 1925 y Fp(is)20 b(a)f(t)o(w)o(o-sided)g(in)o(tegral)f(of)h Fo(A)775 1909 y Fl(\003)794 1925 y Fp(.)g(If)g(\003)901 1931 y Fj(A)948 1925 y Fp(:=)1013 1893 y Fg(P)1057 1937 y Fj(s)p Fl(2)p Fj(G)1129 1925 y Fo(e)1148 1931 y Fn(0)1180 1925 y Fm(\012)13 b Fo(x)1249 1931 y Fj(s)1286 1925 y Fp(denotes)21 b(the)f(t)o(w)o(o-sided)257 1974 y(in)o(tegral)14 b(of)g Fo(A)g Fp(obtained)g(in)g(Prop)q(osition)g(5.3.1,)e(w)o(e)i(kno) o(w)g(from)e([81)o(],)i(Lem.)e(5.4,)h(p.)h(61)257 2024 y(that)g(the)h(linear)e(form)720 2110 y Fo(\025)744 2116 y Fj(A)769 2108 y Fh(\003)800 2110 y Fp(:)e Fo(A)854 2093 y Fl(\003)885 2110 y Fm(!)g Fo(K)q(;)c(b)j Fm(7!)h(h)p Fp(\003)1119 2116 y Fj(A)1146 2110 y Fo(;)c(b)p Fm(i)1199 2116 y Fj(B)257 2196 y Fp(is)14 b(a)f(t)o(w)o(o-sided)g(in)o(tegral)f (on)i Fo(A)756 2181 y Fl(\003)775 2196 y Fp(.)f(Since)h(the)g(an)o(tip) q(o)q(de)f(\014xes)h(\003)1272 2202 y Fj(A)1299 2196 y Fp(,)f(it)g(is)g(easy)h(to)f(see)h(that)257 2246 y(w)o(e)g(ha)o(v)o (e:)802 2296 y Fo(\025)826 2302 y Fj(A)851 2294 y Fh(\003)871 2296 y Fp(\()p Fo(c)905 2302 y Fj(u)936 2296 y Fm(\012)9 b Fo(d)999 2302 y Fj(s)1017 2296 y Fp(\))i(=)h Fo(\016)1106 2302 y Fj(u)p Fn(0)257 2367 y Fp(As)h(already)g(noted)g(in)f(the)h(pro) q(of)f(of)g(Prop)q(osition)g(1,)g(the)h(in)o(tegral)f(group)g(elemen)o (t)g Fo(g)1622 2373 y Fj(A)1662 2367 y Fp(is)257 2417 y(equal)i(to)g(one.)f(Using)h(the)g(t)o(w)o(o-sided)g(in)o(tegral)769 2503 y Fo(\025)793 2509 y Fj(H)837 2503 y Fp(:)d Fo(H)j Fm(!)d Fo(K)q(;)c(c)1035 2509 y Fj(u)1068 2503 y Fm(7!)k Fo(\016)1139 2509 y Fj(u)p Fn(0)953 2628 y Fp(82)p eop %%Page: 83 83 83 82 bop 257 262 a Fp(on)15 b(the)g(group)g(ring)f Fo(H)s Fp(,)h(w)o(e)g(therefore)h(get)f(from)e(the)i(form)o(ulas)e(ab)q(o)o(v) o(e)i(that)f(the)i(linear)257 311 y(form)d Fo(\025)g Fp(:=)g Fo(\025)474 317 y Fj(A)511 311 y Fm(\012)d Fo(\025)577 317 y Fj(H)618 311 y Fm(\012)g Fo(\025)684 317 y Fj(A)709 309 y Fh(\003)743 311 y Fp(is)15 b(a)f(t)o(w)o(o-sided)g(in)o(tegral)g (on)g Fo(A)c Fm(\012)g Fo(H)j Fm(\012)d Fo(A)1420 296 y Fl(\003)1439 311 y Fp(.)k(It)h(is)f(ob)o(vious)257 361 y(that)g(this)g(in)o(tegral)f(satis\014es:)749 439 y Fo(\025)p Fp(\()p Fo(z)808 445 y Fj(uv)q(w)872 439 y Fp(\()p Fo(s;)7 b(t)p Fp(\)\))12 b(=)g Fo(\016)1047 445 y Fj(s)p Fn(1)1082 439 y Fo(\016)1100 445 y Fj(v)q 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Fl(0)510 475 y Fp(\))j(+)f Fo(w)q Fp(\()p Fo(q)q Fp(\()p Fo(s)679 458 y Fl(0)691 475 y Fo(;)e(s)729 458 y Fl(0)r(\000)p Fn(1)785 475 y Fo(t)800 458 y Fl(0)812 475 y Fo(s)831 458 y Fl(0)843 475 y Fp(\))j(+)f Fo(q)q Fp(\()p Fo(t)961 458 y Fl(0)973 475 y Fo(s)992 458 y Fl(0)1004 475 y Fo(;)e(s)1042 458 y Fl(0)r(\000)p Fn(1)1098 475 y Fp(\))j Fm(\000)f Fo(\027)s Fp(\()p Fo(s)1224 458 y Fl(\000)p Fn(1)1268 475 y Fo(ts)p Fp(\))p Fo(q)q Fp(\()p Fo(s)1373 458 y Fl(0)1386 475 y Fo(;)e(s)1424 458 y Fl(0)r(\000)p Fn(1)1480 475 y Fp(\)\)\))343 542 y Fo(\037)p Fp(\(2)p Fo(uw)q(\027)s Fp(\()p Fo(tss)554 525 y Fl(0)q(\000)p Fn(2)610 542 y Fp(\))p Fo(\014)r Fp(\()p Fo(s)686 525 y Fl(0)699 542 y Fp(\))p Fo(\013)p Fp(\()p Fo(s)777 525 y Fl(0)789 542 y Fp(\))i(+)h(2)p Fo(v)q(u\027)s Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)1059 525 y Fl(0)1071 542 y Fp(\))f(+)h(2)p Fo(w)q(\014)r Fp(\()p Fo(s)1250 525 y Fl(\000)p Fn(1)1295 542 y Fo(ts)p Fp(\))p Fo(u\027)s Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)1506 525 y Fl(0)1518 542 y Fp(\))352 610 y(+)g(2)p Fo(v)436 593 y Fl(0)448 610 y Fo(w)q(\027)s Fp(\()p Fo(s)538 593 y Fl(0)549 610 y Fp(\))p Fo(\013)p Fp(\()p Fo(s)627 593 y Fl(0)r(\000)p Fn(1)684 610 y Fo(t)699 593 y Fl(0)710 610 y Fo(s)729 593 y Fl(0)741 610 y Fp(\))g(+)f(2)p Fo(w)q(w)891 593 y Fl(0)902 610 y Fo(\027)s Fp(\()p Fo(s)961 593 y Fl(0)973 610 y Fp(\))p Fo(\014)r Fp(\()p Fo(s)1049 593 y Fl(0)s(\000)p Fn(1)1106 610 y Fo(t)1121 593 y Fl(0)1133 610 y Fo(s)1152 593 y Fl(0)1164 610 y Fp(\))p Fo(\013)p Fp(\()p Fo(s)1242 593 y Fl(0)r(\000)p Fn(1)1299 610 y Fo(t)1314 593 y Fl(0)1325 610 y Fo(s)1344 593 y Fl(0)1357 610 y Fp(\))352 677 y(+)h(2)p Fo(w)446 660 y Fn(2)464 677 y Fp(\()p Fo(\027)s Fp(\()p Fo(s)539 660 y Fl(0)551 677 y Fp(\))p Fo(\014)r Fp(\()p Fo(s)627 660 y Fl(0)s(\000)p Fn(1)684 677 y Fo(t)699 660 y Fl(0)711 677 y Fo(s)730 660 y Fl(0)742 677 y Fp(\))f(+)h Fo(\027)s Fp(\()p Fo(s)868 660 y Fl(\000)p Fn(1)912 677 y Fo(ts)p Fp(\))p Fo(\014)r Fp(\()p Fo(s)1022 660 y Fl(\000)p Fn(1)1069 677 y Fo(ts)p Fp(\)\))p Fo(\013)p Fp(\()p Fo(s)1197 660 y Fl(0)1209 677 y Fp(\))g(+)f Fo(u)1300 660 y Fn(2)1319 677 y Fo(\027)s Fp(\()p Fo(s)p Fp(\))p Fo(\014)r Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)1532 660 y Fl(0)1545 677 y Fp(\)\))343 745 y Fo(z)362 752 y Fj(u;v)q Fn(+)p Fj(v)453 744 y Fh(0)464 752 y Fn(+)p Fj(w)q(\014)q Fn(\()p Fj(s)563 744 y Fh(\000)p Fd(1)602 752 y Fj(ts)p Fn(\)+)p Fj(w)694 744 y Fh(0)704 752 y Fj(\014)q Fn(\()p Fj(s)753 744 y Fh(0)s(\000)p Fd(1)803 752 y Fj(t)816 744 y Fh(0)827 752 y Fj(s)843 744 y Fh(0)854 752 y Fn(\)+)p Fj(w)q(\014)q Fn(\()p Fj(s)966 744 y Fh(0)977 752 y Fn(\)+)p Fj(w)q(\027)r Fn(\()p Fj(s)1088 744 y Fh(\000)p Fd(1)1127 752 y Fj(ts)p Fn(\))p Fj(\014)q Fn(\()p Fj(s)1218 744 y Fh(0)1228 752 y Fn(\))p Fj(;w)q(\027)r Fn(\()p Fj(s)1324 744 y Fh(0)1334 752 y Fn(\)+)p Fj(w)1397 744 y Fh(0)1411 745 y Fp(\()p Fo(ss)1465 727 y Fl(0)1477 745 y Fo(;)e(s)1515 727 y Fl(0)r(\000)p Fn(1)1571 745 y Fo(t)1586 727 y Fl(0)1598 745 y Fo(s)1617 727 y Fl(0)1629 745 y Fp(\))257 828 y(W)m(e)14 b(w)o(an)o(t)f(to)h(rewrite)h(this)f (form)o(ula)d(sligh)o(tly)m(.)h(First,)i(since)402 911 y Fo(\014)r Fp(\()p Fo(t)p Fp(\))c(+)g Fo(\014)r Fp(\()p Fo(s)p Fp(\))h(+)e Fo(\027)s Fp(\()p Fo(t)p Fp(\))o Fo(\014)r Fp(\()p Fo(s)p Fp(\))i Fm(\000)f Fo(\027)s Fp(\()p Fo(s)p Fp(\))p Fo(\014)r Fp(\()p Fo(s)988 894 y Fl(\000)p Fn(1)1033 911 y Fo(ts)p Fp(\))736 979 y(=)i Fo(\014)r Fp(\()p Fo(s)p Fp(\))e(+)g Fo(\014)r Fp(\()p Fo(ts)p Fp(\))h Fm(\000)e Fo(\027)s Fp(\()p Fo(s)p Fp(\))p Fo(\014)r Fp(\()p Fo(s)1186 962 y Fl(\000)p Fn(1)1232 979 y Fp(\))g Fm(\000)h Fo(\014)r Fp(\()p Fo(ts)p Fp(\))j(=)e(2)p Fo(\014)r Fp(\()p Fo(s)p Fp(\))257 1062 y(b)o(y)j(Lemma)d(1.13,)h(w)o(e)i(ha)o(v)o(e)g(for)g Fo(t)798 1047 y Fl(0)821 1062 y Fp(=)e Fo(s)884 1047 y Fl(\000)p Fn(1)929 1062 y Fo(ts)p Fp(:)314 1146 y Fo(z)333 1154 y Fj(u;v)q Fn(+)p Fj(v)424 1145 y Fh(0)435 1154 y Fn(+)p Fj(w)q(\014)q Fn(\()p Fj(s)534 1145 y Fh(\000)p Fd(1)573 1154 y Fj(ts)p Fn(\)+)p Fj(w)665 1145 y Fh(0)675 1154 y Fj(\014)q Fn(\()p Fj(s)724 1145 y Fh(0)s(\000)p Fd(1)774 1154 y Fj(t)787 1145 y Fh(0)798 1154 y Fj(s)814 1145 y Fh(0)825 1154 y Fn(\)+)p Fj(w)q(\014)q Fn(\()p Fj(s)937 1145 y Fh(0)948 1154 y Fn(\)+)p Fj(w)q(\027)r Fn(\()p Fj(s)1059 1145 y Fh(\000)p Fd(1)1098 1154 y Fj(ts)p Fn(\))p Fj(\014)q Fn(\()p Fj(s)1189 1145 y Fh(0)1199 1154 y Fn(\))p Fj(;w)q(\027)r Fn(\()p Fj(s)1295 1145 y Fh(0)1305 1154 y Fn(\)+)p Fj(w)1368 1145 y Fh(0)1382 1146 y Fp(\()p Fo(ss)1436 1129 y Fl(0)1448 1146 y Fo(;)7 b(s)1486 1129 y Fl(0)r(\000)p Fn(1)1542 1146 y Fo(t)1557 1129 y Fl(0)1569 1146 y Fo(s)1588 1129 y Fl(0)1600 1146 y Fp(\))367 1213 y(=)12 b Fo(z)430 1221 y Fj(u;v)q Fn(+)p Fj(v)521 1213 y Fh(0)532 1221 y Fn(+)p Fj(w)q(\014)q Fn(\()p Fj(t)628 1213 y Fh(0)639 1221 y Fn(\)+)p Fj(w)702 1213 y Fh(0)713 1221 y Fj(\014)q Fn(\()p Fj(s)762 1213 y Fh(0)s(\000)p Fd(1)812 1221 y Fj(t)825 1213 y Fh(0)836 1221 y Fj(s)852 1213 y Fh(0)863 1221 y Fn(\)+)p Fj(w)q(\014)q Fn(\()p Fj(s)975 1213 y Fh(0)986 1221 y Fn(\)+)p Fj(w)q(\027)r Fn(\()p Fj(t)1094 1213 y Fh(0)1105 1221 y Fn(\))p Fj(\014)q Fn(\()p Fj(s)1167 1213 y Fh(0)1178 1221 y Fn(\))p Fj(;w)q(\027)r Fn(\()p Fj(s)1274 1213 y Fh(0)1284 1221 y Fn(\)+)p Fj(w)1347 1213 y Fh(0)5 b Fp(\()p Fo(ss)1415 1196 y Fl(0)1428 1213 y Fo(;)i(s)1466 1196 y Fl(0)q(\000)p Fn(1)1522 1213 y Fo(t)1537 1196 y Fl(0)1548 1213 y Fo(s)1567 1196 y Fl(0)1579 1213 y Fp(\))367 1281 y(=)12 b Fo(z)432 1264 y Fl(0)430 1293 y Fj(u;v)q Fn(+)p Fj(v)521 1284 y Fh(0)532 1293 y Fn(+)p Fj(w)q(\014)q Fn(\()p Fj(t)628 1284 y Fh(0)639 1293 y Fn(\)+)p Fj(w)q(\014)q Fn(\()p Fj(s)751 1284 y Fh(0)762 1293 y Fn(\)+)p Fj(w)q(\027)r Fn(\()p Fj(t)870 1284 y Fh(0)881 1293 y Fn(\))p Fj(\014)q Fn(\()p Fj(s)943 1284 y Fh(0)954 1293 y Fn(\))p Fl(\000)p Fj(w)q(\027)r Fn(\()p Fj(s)1066 1284 y Fh(0)1076 1293 y Fn(\))p Fj(\014)q Fn(\()p Fj(s)1138 1284 y Fh(0)r(\000)p Fd(1)1188 1293 y Fj(t)1201 1284 y Fh(0)1212 1293 y Fj(s)1228 1284 y Fh(0)1239 1293 y Fn(\))p Fj(;w)q(\027)r Fn(\()p Fj(s)1335 1284 y Fh(0)1345 1293 y Fn(\)+)p Fj(w)1408 1284 y Fh(0)1421 1281 y Fp(\()p Fo(ss)1475 1264 y Fl(0)1488 1281 y Fo(;)7 b(st)1541 1264 y Fl(0)1553 1281 y Fo(s)1572 1264 y Fl(\000)p Fn(1)1617 1281 y Fp(\))367 1351 y(=)12 b Fo(z)432 1334 y Fl(0)430 1363 y Fj(u;v)q Fn(+)p Fj(v)521 1354 y Fh(0)532 1363 y Fn(+2)p Fj(w)q(\014)q Fn(\()p Fj(s)648 1354 y Fh(0)659 1363 y Fn(\))p Fj(;w)q(\027)r Fn(\()p Fj(s)755 1354 y Fh(0)765 1363 y Fn(\)+)p Fj(w)828 1354 y Fh(0)841 1351 y Fp(\()p Fo(ss)895 1334 y Fl(0)908 1351 y Fo(;)7 b(t)p Fp(\))257 1435 y(W)m(e)14 b(also)f(ha)o(v)o(e:)443 1518 y Fo(q)q Fp(\()p Fo(s;)7 b(s)536 1501 y Fl(\000)p Fn(1)581 1518 y Fo(ts)p Fp(\))i(+)h Fo(q)q Fp(\()p Fo(ts;)d(s)790 1501 y Fl(\000)p Fn(1)835 1518 y Fp(\))i Fm(\000)h Fo(\027)s Fp(\()p Fo(t)p Fp(\))p Fo(q)q Fp(\()p Fo(s;)d(s)1066 1501 y Fl(\000)p Fn(1)1110 1518 y Fp(\))537 1586 y(=)12 b Fo(q)q Fp(\()p Fo(s;)7 b(s)674 1569 y Fl(\000)p Fn(1)719 1586 y Fo(ts)p Fp(\))j Fm(\000)f Fo(q)q Fp(\()p Fo(ss)894 1569 y Fl(\000)p Fn(1)940 1586 y Fo(;)e(ts)p Fp(\))i(+)h Fo(q)q Fp(\()p Fo(ts;)d(s)1168 1569 y Fl(\000)p Fn(1)1213 1586 y Fp(\))i Fm(\000)h Fo(\027)s Fp(\()p Fo(t)p Fp(\))p Fo(q)q Fp(\()p Fo(s;)d(s)1444 1569 y Fl(\000)p Fn(1)1488 1586 y Fp(\))537 1653 y(=)12 b Fo(q)q Fp(\()p Fo(s;)7 b(s)674 1636 y Fl(\000)p Fn(1)719 1653 y Fp(\))i Fm(\000)h Fo(\027)s Fp(\()p Fo(s)p Fp(\))p Fo(q)q Fp(\()p Fo(s)916 1636 y Fl(\000)p Fn(1)961 1653 y Fo(;)d(ts)p Fp(\))i(+)h Fo(q)q Fp(\()p Fo(t;)d(ss)1189 1636 y Fl(\000)p Fn(1)1234 1653 y Fp(\))i Fm(\000)h Fo(q)q Fp(\()p Fo(t;)d(s)p Fp(\))257 1737 y(and)17 b(therefore,)i(if)d Fo(t)587 1722 y Fl(0)616 1737 y Fp(=)h Fo(s)684 1722 y Fl(\000)p Fn(1)729 1737 y Fo(ts)p Fp(,)h(the)g(argumen)o(t)e(of)g Fo(\021)j Fp(in)e(the)g(ab)q (o)o(v)o(e)g(equation)g(can)h(b)q(e)257 1786 y(written)d(in)e(the)i (form:)346 1870 y Fo(uq)q Fp(\()p Fo(s;)7 b(s)463 1853 y Fl(0)475 1870 y Fp(\))i(+)h Fo(w)q Fp(\()p Fo(q)q Fp(\()p Fo(s)644 1853 y Fl(0)656 1870 y Fo(;)d(s)694 1853 y Fl(0)q(\000)p Fn(1)750 1870 y Fo(t)765 1853 y Fl(0)777 1870 y Fo(s)796 1853 y Fl(0)808 1870 y Fp(\))i(+)h Fo(q)q Fp(\()p Fo(t)926 1853 y Fl(0)937 1870 y Fo(s)956 1853 y Fl(0)968 1870 y Fo(;)d(s)1006 1853 y Fl(0)r(\000)p Fn(1)1063 1870 y Fp(\))i Fm(\000)g Fo(\027)s Fp(\()p Fo(s)1188 1853 y Fl(\000)p Fn(1)1233 1870 y Fo(ts)p Fp(\))p Fo(q)q Fp(\()p Fo(s)1338 1853 y Fl(0)1351 1870 y Fo(;)e(s)1389 1853 y Fl(0)q(\000)p Fn(1)1445 1870 y Fp(\)\))600 1937 y(=)12 b Fo(uq)q Fp(\()p Fo(s;)7 b(s)761 1920 y Fl(0)773 1937 y Fp(\))i(+)h Fo(w)q Fp(\()p Fo(q)q Fp(\()p Fo(s)942 1920 y Fl(0)954 1937 y Fo(;)d(s)992 1920 y Fl(0)r(\000)p Fn(1)1048 1937 y Fp(\))i Fm(\000)h Fo(\027)s Fp(\()p Fo(s)1174 1920 y Fl(0)1185 1937 y Fp(\))p Fo(q)q Fp(\()p Fo(s)1256 1920 y Fl(0)s(\000)p Fn(1)1313 1937 y Fo(;)d(t)1347 1920 y Fl(0)1358 1937 y Fo(s)1377 1920 y Fl(0)1390 1937 y Fp(\))i Fm(\000)g Fo(q)q Fp(\()p Fo(t)1507 1920 y Fl(0)1519 1937 y Fo(;)e(s)1557 1920 y Fl(0)1569 1937 y Fp(\)\))257 2021 y(W)m(e)12 b(no)o(w)g(turn)g(to)g(the)g(argumen)o(t)f(of)h Fo(\037)p Fp(.)f(As)i(p)q(oin)o(ted)f(out)g(in)f(P)o(aragraph)h(5.9,)e Fo(\037)i Fp(v)n(anishes)257 2071 y(on)17 b(the)g(t)o(w)o(o-sided)f (ideal)g Fo(I)k Fp(generated)e(b)o(y)f(the)g(additiv)o(e)f(comm)o (utators)e Fo(uv)e Fm(\000)g Fo(v)q(u)17 b Fp(and)257 2120 y(the)g(elemen)o(ts)e(of)h(the)g(form)e Fo(\014)r Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(t)p Fp(\))e Fm(\000)f Fo(\014)r Fp(\()p Fo(t)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)p Fp(\),)16 b(for)g Fo(u;)7 b(v)15 b Fm(2)g Fo(R)g Fp(and)h Fo(s;)7 b(t)15 b Fm(2)f Fo(G)p Fp(.)h(Since)257 2170 y(w)o(e)f(ha)o(v)o(e,)g(using)f(Lemma)f(1.13,)g(that:)257 2254 y Fo(\014)r Fp(\()p Fo(s)317 2237 y Fl(\000)p Fn(1)363 2254 y Fo(ts)p Fp(\))g(=)g Fo(\027)s Fp(\()p Fo(s)528 2237 y Fl(\000)p Fn(1)572 2254 y Fp(\)\()p Fm(\000)p Fo(\014)r Fp(\()p Fo(s)p Fp(\))5 b(+)g Fo(\014)s Fp(\()p Fo(t)p Fp(\))g(+)g Fo(\027)s Fp(\()p Fo(t)p Fp(\))p Fo(\014)s Fp(\()p Fo(s)p Fp(\)\))15 b(=)d Fo(\027)s Fp(\()p Fo(s)1151 2237 y Fl(\000)p Fn(1)1195 2254 y Fp(\)\()p Fo(\027)s Fp(\()p Fo(t)p Fp(\))5 b Fm(\000)g Fp(1\))p Fo(\014)r Fp(\()p Fo(s)p Fp(\))g(+)g Fo(\027)s Fp(\()p Fo(s)1554 2237 y Fl(\000)p Fn(1)1600 2254 y Fp(\))p Fo(\014)r Fp(\()p Fo(t)p Fp(\))257 2337 y(w)o(e)14 b(get,)g(denoting)g(congruence)h(mo)q (dulo)d Fo(I)18 b Fp(b)o(y)13 b Fm(\021)p Fp(,)h(that:)372 2421 y Fo(\014)r Fp(\()p Fo(s)432 2404 y Fl(\000)p Fn(1)478 2421 y Fo(ts)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)590 2404 y Fl(\000)p Fn(1)635 2421 y Fo(ts)p Fp(\))e Fm(\021)g Fo(\027)s Fp(\()p Fo(s)800 2404 y Fl(\000)p Fn(2)845 2421 y Fp(\)\(1)d Fm(\000)g Fp(2)p Fo(\027)s Fp(\()p Fo(t)p Fp(\))g(+)g Fo(\027)s Fp(\()p Fo(t)1145 2404 y Fn(2)1163 2421 y Fp(\)\))p Fo(\014)r Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)p Fp(\))750 2488 y(+)h(2)p Fo(\027)s Fp(\()p Fo(s)872 2471 y Fl(\000)p Fn(2)916 2488 y Fp(\)\()p Fo(\027)s Fp(\()p Fo(t)p Fp(\))f Fm(\000)g Fp(1\))p Fo(\014)r Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(t)p Fp(\))i(+)f Fo(\027)s Fp(\()p Fo(s)1368 2471 y Fl(\000)p Fn(2)1412 2488 y Fp(\))p Fo(\014)r Fp(\()p Fo(t)p Fp(\))p Fo(\013)p Fp(\()p Fo(t)p Fp(\))953 2628 y(85)p eop %%Page: 86 86 86 85 bop 257 262 a Fp(Applying)13 b(this)g(to)h(the)g(primed)e (argumen)o(ts)h(instead)h(and)f(collecting)g(terms,)g(the)h(ab)q(o)o(v) o(e)257 311 y(form)o(ula)e(can)i(b)q(e)g(rewritten)h(in)f(the)g(form:) 331 403 y Fo(z)352 386 y Fl(0)350 413 y Fj(uv)q(w)414 403 y Fp(\()p Fo(s;)7 b(t)p Fp(\))p Fo(z)520 386 y Fl(0)518 413 y Fj(u)538 405 y Fh(0)550 413 y Fj(v)568 405 y Fh(0)578 413 y Fj(w)603 405 y Fh(0)617 403 y Fp(\()p Fo(s)652 386 y Fl(0)664 403 y Fo(;)g(t)698 386 y Fl(0)709 403 y Fp(\))12 b(=)372 465 y Fo(\016)390 472 y Fj(u)410 464 y Fh(0)422 472 y Fl(\000)p Fj(u\027)r Fn(\()p Fj(s)p Fn(\))p Fj(;w)q Fl(\000)p Fj(w)q(\027)r Fn(\()p Fj(t)660 464 y Fh(0)668 472 y Fn(\))683 465 y Fo(\016)701 473 y Fj(s)717 464 y Fh(\000)p Fd(1)756 473 y Fj(ts;t)808 464 y Fh(0)372 532 y Fo(\021)q Fp(\()p Fo(uq)q Fp(\()p Fo(s;)7 b(s)527 515 y Fl(0)539 532 y Fp(\))j(+)f Fo(w)q Fp(\()p Fo(q)q Fp(\()p Fo(s)708 515 y Fl(0)721 532 y Fo(;)e(s)759 515 y Fl(0)q(\000)p Fn(1)815 532 y Fp(\))i Fm(\000)h Fo(\027)s Fp(\()p Fo(s)941 515 y Fl(0)952 532 y Fp(\))p Fo(q)q Fp(\()p Fo(s)1023 515 y Fl(0)r(\000)p Fn(1)1080 532 y Fo(;)d(t)1114 515 y Fl(0)1125 532 y Fo(s)1144 515 y Fl(0)1156 532 y Fp(\))j Fm(\000)f Fo(q)q Fp(\()p Fo(t)1274 515 y Fl(0)1286 532 y Fo(;)e(s)1324 515 y Fl(0)1335 532 y Fp(\)\)\))372 600 y Fo(\037)p Fp(\([2)p Fo(u\027)s Fp(\()p Fo(tss)564 583 y Fl(0)q(\000)p Fn(2)620 600 y Fp(\))i(+)h(2)p Fo(w)739 583 y Fl(0)750 600 y Fo(\027)s Fp(\()p Fo(s)809 583 y Fl(0)q(\000)p Fn(1)865 600 y Fp(\))f Fm(\000)h Fp(4)p Fo(w)984 583 y Fl(0)995 600 y Fo(\027)s Fp(\()p Fo(s)1054 583 y Fl(0)q(\000)p Fn(1)1110 600 y Fo(t)1125 583 y Fl(0)1137 600 y Fp(\))f(+)h(2)p Fo(w)1256 583 y Fl(0)1267 600 y Fo(\027)s Fp(\()p Fo(s)1326 583 y Fl(0)q(\000)p Fn(1)1382 600 y Fo(t)1397 583 y Fl(0)q Fn(2)1427 600 y Fp(\))423 667 y(+)g(2)p Fo(w)q(\027)s Fp(\()p Fo(t)572 650 y Fl(0)582 667 y Fp(\))f Fm(\000)h Fp(2)p Fo(w)q Fp(])p Fo(w)q(\014)r Fp(\()p Fo(s)804 650 y Fl(0)816 667 y Fp(\))p Fo(\013)p Fp(\()p Fo(s)894 650 y Fl(0)906 667 y Fp(\))f(+)h(2)p Fo(w)q(w)1056 650 y Fl(0)1067 667 y Fo(\027)s Fp(\()p Fo(s)1126 650 y Fl(0)q(\000)p Fn(1)1182 667 y Fp(\))p Fo(\014)r Fp(\()p Fo(t)1254 650 y Fl(0)1266 667 y Fp(\))p Fo(\013)p Fp(\()p Fo(t)1340 650 y Fl(0)1352 667 y Fp(\))423 735 y(+)g([2)p Fo(u\027)s Fp(\()p Fo(s)p Fp(\))e(+)h(4)p Fo(w)698 717 y Fl(0)709 735 y Fo(\027)s Fp(\()p Fo(s)768 717 y Fl(0)r(\000)p Fn(1)824 735 y Fo(t)839 717 y Fl(0)851 735 y Fp(\))g Fm(\000)h Fp(4)p Fo(w)970 717 y Fl(0)981 735 y Fo(\027)s Fp(\()p Fo(s)1040 717 y Fl(0)r(\000)p Fn(1)1096 735 y Fp(\))g(+)f(2)p Fo(w)h Fp(+)f(2)p Fo(w)q(\027)s Fp(\()p Fo(t)1372 717 y Fl(0)1383 735 y Fp(\)])p Fo(w)q(\014)r Fp(\()p Fo(s)1502 717 y Fl(0)1514 735 y Fp(\))p Fo(\013)p Fp(\()p Fo(t)1588 717 y Fl(0)1600 735 y Fp(\))423 802 y(+)h(2)p Fo(v)q(u\027)s Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)668 785 y Fl(0)680 802 y Fp(\))f(+)g(2)p Fo(v)788 785 y Fl(0)800 802 y Fo(w)q(\027)s Fp(\()p Fo(s)890 785 y Fl(0)902 802 y Fp(\))p Fo(\013)p Fp(\()p Fo(s)980 785 y Fl(0)r(\000)p Fn(1)1036 802 y Fo(t)1051 785 y Fl(0)1063 802 y Fo(s)1082 785 y Fl(0)1094 802 y Fp(\))g(+)h Fo(u)1185 785 y Fn(2)1203 802 y Fo(\027)s Fp(\()p Fo(s)p Fp(\))p Fo(\014)r Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)1416 785 y Fl(0)1430 802 y Fp(\)\))372 864 y Fo(z)393 847 y Fl(0)391 876 y Fj(u;v)q Fn(+)p Fj(v)482 867 y Fh(0)493 876 y Fn(+2)p Fj(w)q(\014)q Fn(\()p Fj(s)609 867 y Fh(0)620 876 y Fn(\))p Fj(;w)q(\027)r Fn(\()p Fj(s)716 867 y Fh(0)726 876 y Fn(\)+)p Fj(w)789 867 y Fh(0)803 864 y Fp(\()p Fo(ss)857 847 y Fl(0)869 864 y Fo(;)d(t)p Fp(\))257 991 y(W)m(e)k(no)o(w)f(can)h(determine)f(the)i(co)q(cycle)f(and)g(the)g (corresp)q(onding)h(action)e(for)g Fo(B)j Fp(explicitly)m(.)257 1041 y(Consider)i Fo(K)469 1026 y Fj(R)505 1041 y Fm(\012)10 b Fo(K)585 1026 y Fj(G)611 1030 y Fd(op)659 1041 y Fp(as)k(a)f Fo(T)6 b Fp({mo)q(dule)12 b(via:)479 1132 y(\()p Fo(s;)7 b(v)q(;)g(w)q Fp(\))p Fo(:)p Fp(\()p Fo(e)667 1138 y Fj(u)698 1132 y Fm(\012)j Fo(d)762 1138 y Fj(t)776 1132 y Fp(\))i(:=)f Fo(e)878 1140 y Fj(u\027)r Fn(\()p Fj(s)946 1132 y Fh(\000)p Fd(1)984 1140 y Fn(\))p Fl(\000)p Fj(w)q(\027)r Fn(\()p Fj(s)1096 1132 y Fh(\000)p Fd(1)1134 1140 y Fn(\)+)p Fj(w)q(\027)r Fn(\()p Fj(ts)1258 1132 y Fh(\000)p Fd(1)1295 1140 y Fn(\))1320 1132 y Fm(\012)e Fo(d)1383 1140 y Fj(sts)1428 1132 y Fh(\000)p Fd(1)257 1224 y Fp(W)m(e)17 b(extend)h(this)f(action)f (to)h(a)g Fo(K)s Fp([)p Fo(T)6 b Fp(]{mo)q(dule)14 b(structure)19 b(b)o(y)e(linearit)o(y)m(.)e(F)m(urthermore,)257 1274 y(w)o(e)f(de\014ne)h(for)f Fo(s;)7 b(s)559 1259 y Fl(0)571 1274 y Fo(;)g(t)k Fm(2)g Fo(G)j Fp(and)f Fo(u;)7 b(v)q(;)g(w)q(;)g(v) 936 1259 y Fl(0)947 1274 y Fo(;)g(w)997 1259 y Fl(0)1020 1274 y Fm(2)k Fo(R)p Fp(:)265 1365 y Fo(\032)286 1371 y Fj(ut)320 1365 y Fp(\()p Fo(s;)c(v)q(;)g(w)q Fp(;)g Fo(s)483 1348 y Fl(0)495 1365 y Fo(;)g(v)535 1348 y Fl(0)546 1365 y Fo(;)g(w)596 1348 y Fl(0)607 1365 y Fp(\))12 b(:=)265 1432 y Fo(\021)q Fp(\()p Fo(uq)q Fp(\()p Fo(s;)7 b(s)420 1415 y Fl(0)432 1432 y Fp(\))i(+)h Fo(w)q Fp(\()p Fo(q)q Fp(\()p Fo(s)601 1415 y Fl(0)613 1432 y Fo(;)d(s)651 1415 y Fl(0)q(\000)p Fn(1)707 1432 y Fp(\))i Fm(\000)h Fo(\027)s Fp(\()p Fo(s)833 1415 y Fl(0)844 1432 y Fp(\))p Fo(q)q Fp(\()p Fo(s)915 1415 y Fl(0)s(\000)p Fn(1)972 1432 y Fo(;)d(s)1010 1415 y Fl(\000)p Fn(1)1055 1432 y Fo(tss)1108 1415 y Fl(0)1120 1432 y Fp(\))j Fm(\000)f Fo(q)q Fp(\()p Fo(s)1242 1415 y Fl(\000)p Fn(1)1287 1432 y Fo(ts;)e(s)1359 1415 y Fl(0)1371 1432 y Fp(\)\)\))265 1500 y Fo(\037)p Fp(\([2)p Fo(u\027)s Fp(\()p Fo(tss)457 1483 y Fl(0)q(\000)p Fn(2)512 1500 y Fp(\))i(+)h(2)p Fo(w)631 1483 y Fl(0)642 1500 y Fo(\027)s Fp(\()p Fo(s)701 1483 y Fl(0)r(\000)p Fn(1)757 1500 y Fp(\))g Fm(\000)f Fp(4)p Fo(w)876 1483 y Fl(0)887 1500 y Fo(\027)s Fp(\()p Fo(s)946 1483 y Fl(0)r(\000)p Fn(1)1002 1500 y Fo(s)1021 1483 y Fl(\000)p Fn(1)1066 1500 y Fo(ts)p Fp(\))h(+)g(2)p Fo(w)1220 1483 y Fl(0)1231 1500 y Fo(\027)s Fp(\()p Fo(s)1290 1483 y Fl(0)q(\000)p Fn(1)1346 1500 y Fo(s)1365 1483 y Fl(\000)p Fn(1)1410 1500 y Fo(t)1425 1483 y Fn(2)1443 1500 y Fo(s)p Fp(\))274 1567 y(+)f(2)p Fo(w)q(\027)s Fp(\()p Fo(s)426 1550 y Fl(\000)p Fn(1)470 1567 y Fo(ts)p Fp(\))h Fm(\000)g Fp(2)p Fo(w)q Fp(])p Fo(w)q(\014)r Fp(\()p Fo(s)727 1550 y Fl(0)738 1567 y Fp(\))p Fo(\013)p Fp(\()p Fo(s)816 1550 y Fl(0)828 1567 y Fp(\))g(+)f(2)p Fo(w)q(w)978 1550 y Fl(0)989 1567 y Fo(\027)s Fp(\()p Fo(s)1048 1550 y Fl(0)r(\000)p Fn(1)1104 1567 y Fp(\))p Fo(\014)r Fp(\()p Fo(s)1180 1550 y Fl(\000)p Fn(1)1226 1567 y Fo(ts)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)1338 1550 y Fl(\000)p Fn(1)1384 1567 y Fo(ts)p Fp(\))274 1635 y(+)g([2)p Fo(u\027)s Fp(\()p Fo(s)p Fp(\))f(+)i(4)p Fo(w)549 1617 y Fl(0)560 1635 y Fo(\027)s Fp(\()p Fo(s)619 1617 y Fl(0)r(\000)p Fn(1)675 1635 y Fo(s)694 1617 y Fl(\000)p Fn(1)739 1635 y Fo(ts)p Fp(\))g Fm(\000)g Fp(4)p Fo(w)893 1617 y Fl(0)904 1635 y Fo(\027)s Fp(\()p Fo(s)963 1617 y Fl(0)q(\000)p Fn(1)1019 1635 y Fp(\))f(+)h(2)p Fo(w)f Fp(+)h(2)p Fo(w)q(\027)s Fp(\()p Fo(s)1299 1617 y Fl(\000)p Fn(1)1343 1635 y Fo(ts)p Fp(\)])p Fo(w)q(\014)r Fp(\()p Fo(s)1496 1617 y Fl(0)1509 1635 y Fp(\))p Fo(\013)p Fp(\()p Fo(s)1587 1617 y Fl(\000)p Fn(1)1632 1635 y Fo(ts)p Fp(\))274 1702 y(+)f(2)p Fo(v)q(u\027)s Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)518 1685 y Fl(0)530 1702 y Fp(\))h(+)f(2)p Fo(v)639 1685 y Fl(0)651 1702 y Fo(w)q(\027)s Fp(\()p Fo(s)741 1685 y Fl(0)752 1702 y Fp(\))p Fo(\013)p Fp(\()p Fo(s)830 1685 y Fl(0)r(\000)p Fn(1)887 1702 y Fo(s)906 1685 y Fl(\000)p Fn(1)951 1702 y Fo(tss)1004 1685 y Fl(0)1017 1702 y Fp(\))g(+)h Fo(u)1108 1685 y Fn(2)1126 1702 y Fo(\027)s Fp(\()p Fo(s)p Fp(\))p Fo(\014)r Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(s)1339 1685 y Fl(0)1352 1702 y Fp(\)\))257 1856 y(and)k(set:)462 1947 y Fo(\032)p Fp(\()p Fo(s;)7 b(v)q(;)g(w)q Fp(;)g Fo(s)646 1930 y Fl(0)658 1947 y Fo(;)g(v)698 1930 y Fl(0)709 1947 y Fo(;)g(w)759 1930 y Fl(0)770 1947 y Fp(\))12 b(:=)892 1907 y Fg(X)853 1997 y Fj(u)p Fl(2)p Fj(R;t)p Fl(2)p Fj(G)998 1947 y Fo(\032)1019 1953 y Fj(ut)1054 1947 y Fp(\()p Fo(s;)7 b(v)q(;)g(w)q Fp(;)g Fo(s)1217 1930 y Fl(0)1228 1947 y Fo(;)g(v)1268 1930 y Fl(0)1280 1947 y Fo(;)g(w)1330 1930 y Fl(0)1341 1947 y Fp(\))p Fo(e)1376 1953 y Fj(u)1407 1947 y Fm(\012)j Fo(d)1471 1953 y Fj(t)257 2089 y Fq(Prop)q(osition)33 b Fp(The)15 b(map)473 2180 y Fo(K)511 2163 y Fj(R)547 2180 y Fm(\012)10 b Fo(K)627 2163 y Fj(G)653 2167 y Fd(op)696 2180 y Fm(\012)g Fo(K)s Fp([)p Fo(T)c Fp(])k Fm(!)i Fo(B)r(;)7 b(e)965 2186 y Fj(u)996 2180 y Fm(\012)i Fo(d)1059 2186 y Fj(t)1083 2180 y Fm(\012)g Fo(y)1144 2186 y Fj(v)q(w)1189 2180 y Fp(\()p Fo(s)p Fp(\))j Fm(7!)f Fo(z)1326 2163 y Fl(0)1324 2191 y Fj(uv)q(w)1389 2180 y Fp(\()p Fo(s;)c(t)p Fp(\))257 2272 y(is)k(an)g(isomorphism)d(b)q(et)o(w)o(een)k Fo(B)i Fp(and)c(the)i(crossed)h(pro)q(duct)e(of)g Fo(T)17 b Fp(and)10 b Fo(K)1433 2257 y Fj(R)1464 2272 y Fm(\012)s Fo(K)1537 2257 y Fj(G)1563 2261 y Fd(op)1609 2272 y Fp(with)257 2322 y(resp)q(ect)17 b(to)e(the)g(2-co)q(cycle)h Fo(\032)f Fp(and)g(the)g(sp)q(eci\014ed)h(action.)e Fo(z)1219 2307 y Fl(0)1217 2332 y Fj(uv)q(w)1282 2322 y Fp(\()p Fo(s;)7 b(t)p Fp(\))15 b(is)f(a)h(normal)d(basis)257 2371 y(of)17 b Fo(B)k Fp(in)c(the)h(sense)h(that)f(this)g(mapping)d(is)i Fo(K)1033 2356 y Fj(R)1073 2371 y Fm(\012)12 b Fo(K)1155 2356 y Fj(G)1181 2360 y Fd(op)1215 2371 y Fp(-linear)17 b(and)g Fo(K)s Fp([)p Fo(T)6 b Fp(]-colinear,)257 2421 y(where)15 b Fo(B)i Fp(is)d(a)f Fo(K)539 2406 y Fj(R)576 2421 y Fm(\012)c Fo(K)655 2406 y Fj(G)681 2410 y Fd(op)715 2421 y Fp(-mo)q(dule)k(via)g Fo(\023)g Fp(and)h(a)f Fo(K)s Fp([)p Fo(T)6 b Fp(]-como)q(dule)12 b(via)h Fo(\031)q Fp(.)953 2628 y(86)p eop %%Page: 87 87 87 86 bop 257 262 a Fq(Pro)q(of.)36 b Fp(In)14 b(this)g(crossed)i(pro)q (duct,)e(t)o(w)o(o)f(basis)h(elemen)o(ts)g(are)g(m)o(ultiplied)e(as)h (follo)o(ws:)310 353 y(\()p 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Fp(\))363 477 y(=)i(\()p Fo(e)442 483 y Fj(u)473 477 y Fm(\012)e Fo(d)537 483 y Fj(t)551 477 y Fp(\)\()p Fo(e)602 485 y Fj(u)622 477 y Fh(0)633 485 y Fj(\027)r Fn(\()p Fj(s)681 477 y Fh(\000)p Fd(1)719 485 y Fn(\))p Fl(\000)p Fj(w)q(\027)r Fn(\()p Fj(s)831 477 y Fh(\000)p Fd(1)869 485 y Fn(\)+)p Fj(w)q(\027)r Fn(\()p Fj(t)977 477 y Fh(0)988 485 y Fj(s)1004 477 y Fh(\000)p Fd(1)1042 485 y Fn(\))1066 477 y Fm(\012)g Fo(d)1130 485 y Fj(st)1159 477 y Fh(0)1169 485 y Fj(s)1185 477 y Fh(\000)p Fd(1)t Fp(\))p Fo(\032)1262 483 y Fj(ut)1297 477 y Fp(\()p Fo(s;)d(v)q(;)g(w)q Fp(;)g Fo(s)1460 460 y Fl(0)1472 477 y Fo(;)g(v)1512 460 y Fl(0)1523 477 y Fo(;)g(w)1573 460 y Fl(0)1584 477 y Fp(\))1078 544 y Fm(\012)j Fo(y)1140 551 y Fj(v)q Fn(+)p Fj(v)1201 543 y Fh(0)1212 551 y Fn(+2)p Fj(w)q(\014)q Fn(\()p Fj(s)1328 543 y Fh(0)1339 551 y Fn(\))p Fj(;w)q(\027)r Fn(\()p Fj(s)1435 543 y Fh(0)1445 551 y Fn(\)+)p Fj(w)1508 543 y Fh(0)1522 544 y Fp(\()p Fo(ss)1576 527 y Fl(0)1588 544 y Fp(\))363 606 y(=)i 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Fp(is)f(ev)o(en)g(an)f(ideal)g(of)g Fo(A)p Fp(.)g(Therefore,)i Fo(V)1218 1331 y Fl(0)1245 1346 y Fp(is)f(a)f Fo(B)r Fp(-submo)q(dule)g(of)g(the)257 1396 y Fo(B)r Fp(-mo)q(dule)e Fo(A)p Fp(.)257 1481 y(No)o(w)i(c)o(ho)q (ose)i(a)e(nonzero)h(righ)o(t)f(in)o(tegral)f Fo(\032)954 1487 y Fj(A)996 1481 y Fm(2)g Fo(A)1069 1466 y Fl(\003)1103 1481 y Fp(and)h(de\014ne,)h(as)g(in)f(P)o(aragraph)g(1.9,)257 1530 y(the)g(bilinear)e(form:)607 1619 y Fm(h\001)p Fo(;)7 b Fm(\001i)k Fp(:)g Fo(A)e Fm(\002)h Fo(A)h Fm(!)g Fo(K)q(;)c Fp(\()p Fo(a;)g(a)1027 1602 y Fl(0)1038 1619 y Fp(\))k Fm(7!)g Fo(\032)1139 1625 y Fj(A)1167 1619 y Fp(\()p Fo(S)1208 1625 y Fj(A)1236 1619 y Fp(\()p Fo(a)p Fp(\))p Fo(a)1312 1602 y Fl(0)1323 1619 y Fp(\))257 1707 y(Since)j Fo(A)g Fp(is)f(a)h(F)m(rob)q(enius)f(algebra)g(with)h(F)m(rob)q(enius)f (homomorphism)c Fo(\032)1420 1713 y Fj(A)1448 1707 y Fp(,)k(this)g(bilinear)257 1757 y(form)f(is)i(nondegenerate)i(and)e (restricts)h(to)f(a)g(nondegenerate)h(bilinear)e(form)803 1846 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Fp(\))f(is)g Fo( )q Fp(-in)o(v)n(arian)o(t.)308 2445 y(2.)20 b(If)c Fo(\024)p Fp(\()p Fo(V)9 b Fp(\))17 b(consists)g(of)e(\014xed)i(p)q (oin)o(ts)e(of)h Fo( )q Fp(,)g(then)h Fo(\024)p Fp(\()p Fo(V)1220 2430 y Fl(\003)1239 2445 y Fp(\))g(consists)g(of)e(\014xed)i (p)q(oin)o(ts)361 2495 y(of)c Fo( )q Fp(.)953 2628 y(89)p eop %%Page: 90 90 90 89 bop 257 262 a Fq(Pro)q(of.)36 b Fp(W)m(e)14 b(consider)h(the)g (case)g(where)g Fo(V)24 b Fp(is)14 b(purely)g(unstable)h(\014rst.)f (Since)h Fo(S)1553 268 y Fj(A)1594 262 y Fp(is)f(co-)257 311 y(linear,)j(it)g(comm)o(utes)f(with)h Fo( )q Fp(,)g(and)g (therefore)i(Span\()p Fo(\024)p Fp(\()p Fo(V)10 b Fp(\)\))18 b(=)f Fo(S)1353 317 y Fj(A)1381 311 y Fp(\(Span\()p Fo(\024)p Fp(\()p Fo(V)1577 296 y Fl(\003)1596 311 y Fp(\)\)\))h(is)257 361 y Fo( )q Fp(-in)o(v)n(arian)o(t)11 b(if)g(and)h(only)f(if)h(Span\() p Fo(\024)p Fp(\()p Fo(V)892 346 y Fl(\003)911 361 y Fp(\)\))h(is)f Fo( )q Fp(-in)o(v)n(arian)o(t,)e(whic)o(h)i(is)g(ob)o (viously)e(equiv)n(a-)257 411 y(len)o(t)g(to)g(the)g Fo( )q Fp(-in)o(v)n(ariance)g(of)f Fo(\024)p Fp(\()p Fo(V)800 396 y Fl(\003)819 411 y Fp(\).)h(Also,)f(Span\()p Fo(\024)p Fp(\()p Fo(V)g Fp(\)\))j(=)g Fo(S)1250 417 y Fj(A)1277 411 y Fp(\(Span)q(\()p Fo(\024)p Fp(\()p Fo(V)1473 396 y Fl(\003)1492 411 y Fp(\)\)\))e(consists)257 461 y(of)k Fo( )q Fp(-\014xed)g(p)q(oin)o(ts)g(if)f(and)h(only)f(if)g (Span\()p Fo(\024)p Fp(\()p Fo(V)1000 446 y Fl(\003)1019 461 y Fp(\)\))h(consists)h(of)e Fo( )q Fp(-\014xed)i(p)q(oin)o(ts.)257 546 y(W)m(e)k(no)o(w)g(consider)h(the)f(case)h(where)h Fo(V)28 b Fp(is)19 b(stable.)g(In)g(this)g(case,)h(the)f(c)o(haracter)i Fo(\037)1661 552 y Fj(V)257 596 y Fp(of)14 b Fo(V)23 b Fp(is)13 b(an)h(algebra)f(homomorphism)c(to)14 b(the)g(base)h (\014eld,)e(and)h(therefore)h(has)f(the)g(form)257 646 y Fo(\037)283 652 y Fj(V)329 646 y Fp(=)j Fo(\021)c Fm(\012)f Fo(\015)19 b Fp(for)e(some)f Fo(H)s Fp(-linear)h(algebra)f (homomorphism)d Fo(\021)18 b Fp(:)f Fo(A)g Fm(!)f Fo(K)k Fp(and)d(some)257 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Fp(.)c(Dualizing)f (this,)h(w)o(e)h(get)g(for)f(the)h(c)o(haracter)257 1960 y(of)19 b Fo(V)29 b Fp(that)19 b(\()p Fo(\017)491 1966 y Fj(A)531 1960 y Fm(\012)13 b Fo(\015)599 1945 y Fl(\000)p Fn(1)645 1960 y Fp(\))p Fo(\037)687 1966 y Fj(V)736 1960 y Fp(=)21 b Fo(\037)815 1966 y Fj(V)844 1960 y Fp(.)e(Applying)g Fo(\024)p Fp(,)f(the)i(assertion)g(no)o(w)f(follo)o(ws)f(from)257 2010 y(Prop)q(osition)c(2.7.)e Fi(2)257 2145 y Fq(6.5)48 b Fp(In)19 b(P)o(aragraph)f(1.7,)g(w)o(e)h(ha)o(v)o(e)f(already)h(in)o (v)o(estigated)f(mo)q(dules)g(o)o(v)o(er)h(the)g(alge-)257 2195 y(bra)e Fo(A)371 2189 y Fp(^)365 2195 y Fm(\012)q Fo(A)p Fp(.)g(No)o(w)f(w)o(e)i(w)o(an)o(t)e(to)h(determine,)g(in)f(our) i(situation,)d(all)h(mo)q(dules)g(o)o(v)o(er)h(this)257 2245 y(algebra.)10 b(W)m(e)g(shall)f(ac)o(hiev)o(e)i(this)f(b)o(y)g (describing)h(explicitly)e(the)i(decomp)q(osition)e(of)h Fo(A)1632 2239 y Fp(^)1626 2245 y Fm(\012)p Fo(A)257 2295 y Fp(in)o(to)k(simple)e(t)o(w)o(o-sided)i(ideals.)257 2380 y(Let)g Fo(O)363 2386 y Fn(1)382 2380 y Fo(;)7 b(:)g(:)g(:)k(;)c (O)513 2386 y Fj(l)538 2380 y Fp(denote)15 b(the)f Fo(C)s Fp(-orbits)f(in)f(the)i(set)h Fo(E)g Fp(of)e(primitiv)o(e)e(idemp)q (oten)o(ts.)h(Sim-)257 2430 y(ilarly)m(,)g(w)o(e)i(denote)h(the)g (orbits)f(of)f Fo( )j Fp(b)o(y)e Fo(O)939 2415 y Fl(0)938 2441 y Fn(1)956 2430 y Fo(;)7 b(:)g(:)g(:)12 b(;)7 b(O)1089 2415 y Fl(0)1088 2441 y Fj(m)1118 2430 y Fp(.)14 b(W)m(e)f(de\014ne)j Fo(V)1360 2436 y Fj(i)1385 2430 y Fp(:=)c(Span\()p Fo(O)1579 2436 y Fj(i)1592 2430 y Fp(\))j(and)953 2628 y(90)p eop %%Page: 91 91 91 90 bop 257 262 a Fo(W)296 268 y Fj(j)326 262 y Fp(:=)11 b(Span\()p Fo(O)520 246 y Fl(0)519 272 y Fj(j)536 262 y Fp(\).)f(Then)i Fo(V)704 268 y Fj(i)728 262 y Fp(is)f(an)f Fo(H)s Fp(-submo)q(dule)g(of)g Fo(A)h Fp(and)f Fo(W)1281 268 y Fj(j)1309 262 y Fp(is)h(an)f Fo(H)s Fp(-sub)q(como)q(dule)257 311 y(of)k Fo(A)p Fp(.)f(It)h(is)g(easy)g(to)g(see)h(that)792 443 y Fo(A)829 437 y Fp(^)823 443 y Fm(\012)p Fo(A)d Fp(=)956 389 y Fj(l;m)950 404 y Fg(M)942 492 y Fj(i;j)r Fn(=1)1028 443 y Fo(V)1052 449 y Fj(i)1072 437 y Fp(^)1066 443 y Fm(\012)p Fo(W)1137 449 y Fj(j)257 583 y Fp(is)j(a)f(decomp)q (osition)g(of)g Fo(A)695 577 y Fp(^)689 583 y Fm(\012)p Fo(A)h Fp(in)o(to)f(t)o(w)o(o-sided)h(ideals.)f(Ho)o(w)o(ev)o(er,)h (not)f(all)g(of)g(these)i(are)257 633 y(simple.)c(There)j(are)g(four)e (p)q(ossibilities:)257 732 y Fq(Prop)q(osition)33 b Fp(Supp)q(ose)18 b(that)g Fo(O)g Fp(is)f(an)g(orbit)g(of)f Fo(\036)h Fp(and)g(that)g Fo(O)1359 717 y Fl(0)1388 732 y Fp(is)g(an)g(orbit)g(of)f Fo( )q Fp(.)257 782 y(De\014ne)f Fo(V)21 b Fp(:=)11 b(Span\()p Fo(O)q Fp(\))j(and)g Fo(W)j Fp(:=)12 b(Span\()p Fo(O)987 767 y Fl(0)998 782 y Fp(\).)308 901 y(1.)20 b(If)15 b Fo(O)g Fp(=)g Fm(f)p Fo(e)p Fm(g)g Fp(and)h Fo(O)690 886 y Fl(0)716 901 y Fp(=)e Fm(f)p Fo(e)802 886 y Fl(0)814 901 y Fm(g)h Fp(are)h(b)q(oth)g(orbits)g(of)f(length)g(one,)g Fo(V)1447 895 y Fp(^)1441 901 y Fm(\012)p Fo(W)22 b Fp(is)15 b(a)g(one-)361 951 y(dimensional)c(t)o(w)o(o-sided)i(ideal)f(and)h Fo(e)8 b Fm(\012)g Fo(e)1037 936 y Fl(0)1062 951 y Fp(is)13 b(a)g(cen)o(trally)g(primitiv)o(e)e(idemp)q(oten)o(t)361 1001 y(of)i Fo(A)445 995 y Fp(^)439 1001 y Fm(\012)q Fo(A)p Fp(.)308 1084 y(2.)20 b(If)h Fo(O)j Fp(=)f Fm(f)p Fo(e)p Fm(g)e Fp(is)f(an)h(orbit)f(of)h(length)f(one,)h(i.)f(e.,)g(a)h (stable)g(orbit,)f(and)h Fo(O)1623 1068 y Fl(0)1657 1084 y Fp(=)361 1133 y Fm(f)p Fo(e)401 1118 y Fl(0)401 1144 y Fn(0)420 1133 y Fo(;)7 b(:)g(:)g(:)12 b(;)7 b(e)539 1118 y Fl(0)539 1144 y Fj(p)p Fl(\000)p Fn(1)600 1133 y Fm(g)i Fp(is)h(an)f(orbit)h(of)f(length)g Fo(p)p Fp(,)g(i.)g(e.,)g(a) g(purely)h(unstable)g(orbit,)f Fo(V)1574 1127 y Fp(^)1568 1133 y Fm(\012)q Fo(W)17 b Fp(=)361 1162 y Fg(L)407 1172 y Fj(p)p Fl(\000)p Fn(1)407 1205 y Fj(j)r Fn(=0)476 1193 y Fo(K)s(e)10 b Fm(\012)f Fo(e)603 1178 y Fl(0)603 1204 y Fj(j)635 1193 y Fp(is)14 b(a)f(decomp)q(osition)g(of)g Fo(V)1071 1187 y Fp(^)1065 1193 y Fm(\012)q Fo(W)19 b Fp(in)o(to)14 b(simple)e(t)o(w)o(o-sided)i(ideals.)308 1276 y(3.)20 b(If)12 b Fo(O)h Fp(=)f Fm(f)p Fo(e)530 1282 y Fn(0)548 1276 y Fo(;)7 b(:)g(:)g(:)12 b(;)7 b(e)667 1282 y Fj(p)p Fl(\000)p Fn(1)728 1276 y Fm(g)13 b Fp(is)f(an)h(orbit)f (of)g(length)g Fo(p)p Fp(,)g(i.)g(e.,)g(a)g(purely)h(unstable)g(orbit,) 361 1332 y(and)19 b Fo(O)480 1317 y Fl(0)511 1332 y Fp(=)h Fm(f)p Fo(e)603 1317 y Fl(0)615 1332 y Fm(g)e Fp(is)h(an)f(orbit)h(of)f (length)h(one,)g Fo(V)1186 1326 y Fp(^)1180 1332 y Fm(\012)p Fo(W)26 b Fp(=)1329 1301 y Fg(L)1375 1311 y Fj(p)p Fl(\000)p Fn(1)1375 1344 y Fj(i)p Fn(=0)1444 1332 y Fo(K)s(e)1501 1338 y Fj(i)1528 1332 y Fm(\012)12 b Fo(e)1591 1317 y Fl(0)1622 1332 y Fp(is)19 b(a)361 1381 y(decomp)q(osition)13 b(of)g Fo(V)721 1375 y Fp(^)715 1381 y Fm(\012)q Fo(W)19 b Fp(in)o(to)14 b(simple)e(t)o(w)o(o-sided)i(ideals.)308 1465 y(4.)20 b(If)11 b Fo(O)i Fp(=)f Fm(f)p Fo(e)529 1471 y Fn(0)547 1465 y Fo(;)7 b(:)g(:)g(:)12 b(;)7 b(e)666 1471 y Fj(p)p Fl(\000)p Fn(1)727 1465 y Fm(g)12 b Fp(and)f Fo(O)871 1449 y Fl(0)894 1465 y Fp(=)h Fm(f)p Fo(e)978 1449 y Fl(0)978 1475 y Fn(0)997 1465 y Fo(;)7 b(:)g(:)g(:)k(;)c(e)1115 1449 y Fl(0)1115 1475 y Fj(p)p Fl(\000)p Fn(1)1177 1465 y Fm(g)k Fp(are)h(b)q(oth)g(orbits)g(of)f(length)g Fo(p)p Fp(,)361 1520 y Fo(V)400 1514 y Fp(^)395 1520 y Fm(\012)p Fo(W)20 b Fp(is)13 b(a)h(simple)e(t)o(w)o(o-sided)i(ideal.)257 1619 y Fq(Pro)q(of.)36 b Fp(The)14 b(\014rst)g(three)h(statemen)o(ts)f (are)g(ob)o(vious.)e(W)m(e)h(no)o(w)g(pro)o(v)o(e)h(the)g(fourth.)f Fo(V)23 b Fp(is)257 1669 y(a)16 b(mo)q(dule)e(algebra)h(o)o(v)o(er)h Fo(H)s Fp(,)f(whereas)h Fo(W)22 b Fp(is)15 b(a)h(como)q(dule)e(algebra) h(o)o(v)o(er)h Fo(H)s Fp(.)f(F)m(or)g(eac)o(h)257 1719 y(suc)o(h)k(pair,)e(w)o(e)h(can)g(form)f(the)h(left)g(smash)f(pro)q (duct)i Fo(V)i Fm(\012)12 b Fo(W)24 b Fp(already)18 b(considered)h(in) 257 1769 y(P)o(aragraph)14 b(1.7,)e(where)j(the)g(m)o(ultiplicatio)o(n) c(is)j(de\014ned)h(as:)595 1860 y(\()p Fo(v)c Fm(\012)e Fo(w)q Fp(\)\()p Fo(v)767 1843 y Fl(0)789 1860 y Fm(\012)h Fo(w)862 1843 y Fl(0)873 1860 y Fp(\))i(:=)f Fo(v)q Fp(\()p Fo(w)1024 1843 y Fn(\(1\))1081 1860 y Fm(!)g Fo(v)1155 1843 y Fl(0)1167 1860 y Fp(\))e Fm(\012)h Fo(w)1265 1843 y Fn(\(2\))1309 1860 y Fo(w)1340 1843 y Fl(0)257 1952 y Fp(This)20 b(is)g(precisely)g(the)h(algebra)e(structure)j(of)d(the)h (ideal)f Fo(V)1268 1946 y Fp(^)1263 1952 y Fm(\012)p Fo(W)25 b Fp(in)20 b Fo(A)1451 1946 y Fp(^)1445 1952 y Fm(\012)p Fo(A)p Fp(.)g(W)m(e)f(no)o(w)257 2001 y(p)q(erform)c(a)g (discrete)i(F)m(ourier)e(transform)f(to)h(pass)h(to)f(a)g(new)h(basis)f Fo(c)1383 1986 y Fl(0)1383 2012 y Fn(0)1402 2001 y Fo(;)7 b(:)g(:)g(:)12 b(;)7 b(c)1520 1986 y Fl(0)1520 2012 y Fj(p)p Fl(\000)p Fn(1)1596 2001 y Fp(of)15 b Fo(W)257 2051 y Fp(de\014ned)g(as:)834 2132 y Fo(c)852 2115 y Fl(0)852 2142 y Fj(j)881 2132 y Fp(:=)937 2078 y Fj(p)p Fl(\000)p Fn(1)936 2093 y Fg(X)939 2181 y Fj(i)p Fn(=0)1003 2132 y Fo(\020)1024 2115 y Fl(\000)p Fj(ij)1080 2132 y Fo(e)1099 2115 y Fl(0)1099 2142 y Fj(i)257 2248 y Fp(It)e(is)g(easy)g (to)f(see)i(that)f(these)h(basis)f(elemen)o(ts)f(satisfy)h Fo(c)1157 2233 y Fl(0)1157 2259 y Fj(i)1171 2248 y Fo(c)1189 2233 y Fl(0)1189 2259 y Fj(j)1218 2248 y Fp(=)e Fo(c)1279 2233 y Fl(0)1279 2259 y Fj(i)p Fn(+)p Fj(j)1347 2248 y Fp(and)h Fo( )q Fp(\()p Fo(c)1488 2233 y Fl(0)1488 2259 y Fj(j)1506 2248 y Fp(\))g(=)g Fo(\020)1599 2233 y Fl(\000)p Fj(j)1643 2248 y Fo(c)1661 2233 y Fl(0)1661 2259 y Fj(j)1678 2248 y Fp(,)257 2298 y(i.)h(e.,)h(w)o(e)g(ha)o(v)o(e:) 828 2348 y Fo(\016)846 2354 y Fj(A)874 2348 y Fp(\()p Fo(c)908 2331 y Fl(0)908 2358 y Fj(j)925 2348 y Fp(\))e(=)g Fo(c)1015 2354 y Fj(j)1042 2348 y Fm(\012)d Fo(c)1101 2331 y Fl(0)1101 2358 y Fj(j)257 2423 y Fp(This)17 b(pro)o(v)o(es)g (that)g(the)g(mapping)e Fo(H)k Fm(!)c Fo(W)o(;)7 b(c)1013 2429 y Fj(j)1047 2423 y Fm(7!)15 b Fo(c)1122 2408 y Fl(0)1122 2434 y Fj(j)1156 2423 y Fp(is)i(an)f(isomorphism)e(of)i(como)q(d-)257 2473 y(ule)g(algebras,)f(where)h Fo(H)i Fp(is)d(considered)i(as)f(a)f (como)q(dule)f(algebra)h(via)f(the)i(left)f(regular)953 2628 y(91)p eop %%Page: 92 92 92 91 bop 257 262 a Fp(coaction,)12 b(i.)e(e.,)i(the)g(como)q(dule)f (structure)i(whose)g(coaction)e(is)h(equal)g(to)f(the)i(com)o(ultipli-) 257 311 y(cation.)h(It)h(is)f(ob)o(vious)g(that)h Fo(V)24 b Fp(is)14 b(isomorphic,)f(as)h(a)h(mo)q(dule)e(algebra,)h(to)g Fo(K)1518 296 y Fc(Z)1541 300 y Ff(p)1557 311 y Fp(,)g(where)257 361 y Fo(K)295 346 y Fc(Z)318 350 y Ff(p)349 361 y Fp(is)h(endo)o(w)o (ed)g(with)g(the)h(mo)q(dule)d(algebra)h(structure)j(considered)g(in)d (the)i(\014rst)f(step)257 411 y(of)g(the)g(pro)q(of)g(of)g(Prop)q (osition)f(2.3.1.)f(Therefore,)j Fo(V)1113 405 y Fp(^)1107 411 y Fm(\012)q Fo(W)21 b Fp(is)14 b(isomorphic)g(to)h(the)g(smash)257 461 y(pro)q(duct)e Fo(K)449 446 y Fc(Z)472 450 y Ff(p)493 461 y Fm(\012)5 b Fo(H)s Fp(,)11 b(and)g(th)o(us)i(is)e(simple)f(as)i (seen)h(in)e(the)h(pro)q(of)g(of)f(Prop)q(osition)g(2.3.1.)e Fi(2)257 580 y Fp(This)14 b(prop)q(osition)f(has)h(the)h(follo)o(wing)c (corollary)i(that)h(will)f(b)q(e)h(useful)g(later)g(on:)257 679 y Fq(Corollary)35 b Fp(Supp)q(ose)21 b(that)f Fo(V)29 b Fp(is)20 b(a)f(simple)f Fo(B)r Fp(-mo)q(dule)h(of)g(dimension)f Fo(p)i Fp(and)f(that)257 729 y Fo(e;)7 b(e)314 714 y Fl(0)344 729 y Fm(2)18 b Fo(E)i Fp(are)e(primitiv)o(e)e(idemp)q(oten)o (ts.)h(If)g Fo(e)1011 714 y Fl(0)1041 729 y Fp(=)i Fo( )q Fp(\()p Fo(e)p Fp(\),)f(the)h Fo(A)p Fp(-mo)q(dules)e Fo(V)1528 723 y Fp(^)1522 729 y Fm(\012)p Fo(Ae)i Fp(and)257 779 y Fo(V)297 773 y Fp(^)291 779 y Fm(\012)p Fo(Ae)373 764 y Fl(0)399 779 y Fp(are)14 b(isomorphic.)257 878 y Fq(Pro)q(of.)36 b Fp(According)15 b(to)e(P)o(aragraph)h(1.7,)e(the)j (mo)q(dules)e Fo(V)1226 872 y Fp(^)1220 878 y Fm(\012)q Fo(Ae)h Fp(and)g Fo(V)1437 872 y Fp(^)1431 878 y Fm(\012)p Fo(Ae)1513 863 y Fl(0)1539 878 y Fp(are)h(ev)o(en)257 928 y Fo(A)294 922 y Fp(^)288 928 y Fm(\012)q Fo(A)p Fp(-mo)q(dules,)c(and)h(w)o(e)h(shall)f(pro)o(v)o(e)g(the)i(stronger)f (statemen)o(t)g(that)f(they)h(are)g(isomor-)257 978 y(phic)k(as)f(suc)o (h.)h(By)f(Prop)q(osition)g(6.2,)f(w)o(e)i(can)f(assume)g(that)g Fo(V)25 b Fp(=)16 b(Span\()p Fo(O)q Fp(\))g(for)g(some)257 1028 y(purely)f(unstable)f Fo(\036)p Fp(-orbit)g Fo(O)q Fp(.)g(Denote)g(the)h Fo( )q Fp(-orbit)g(of)e Fo(e)i Fp(b)o(y)f Fo(O)1283 1013 y Fl(0)1294 1028 y Fp(;)g(it)g(is)g(also)g (the)g Fo( )q Fp(-orbit)257 1078 y(of)e Fo(e)322 1063 y Fl(0)346 1078 y Fp(and,)g(in)g(the)h(relev)n(an)o(t)g(case)g(where)g Fo(e)f Fm(6)p Fp(=)g Fo(e)1008 1063 y Fl(0)1020 1078 y Fp(,)g(it)g(has)g(length)h Fo(p)p Fp(.)f(Let)g Fo(W)18 b Fp(:=)11 b(Span\()p Fo(O)1650 1063 y Fl(0)1662 1078 y Fp(\).)257 1127 y(By)16 b(the)f(preceding)h(prop)q(osition,)e Fo(V)854 1121 y Fp(^)848 1127 y Fm(\012)q Fo(W)20 b Fp(is)15 b(a)g(simple)e(t)o(w)o(o-sided)i(ideal)f(of)g Fo(A)1520 1121 y Fp(^)1514 1127 y Fm(\012)q Fo(A)h Fp(of)f(di-)257 1177 y(mension)e Fo(p)439 1162 y Fn(2)458 1177 y Fp(;)g Fo(V)522 1171 y Fp(^)516 1177 y Fm(\012)p Fo(Ae)i Fp(and)f Fo(V)731 1171 y Fp(^)725 1177 y Fm(\012)p Fo(Ae)807 1162 y Fl(0)833 1177 y Fp(are)g Fo(p)p Fp(-dimensional)e(left)i(ideals)f(of) h Fo(A)1434 1171 y Fp(^)1428 1177 y Fm(\012)p Fo(A)p Fp(,)g(whic)o(h)g(are)257 1227 y(con)o(tained)k(in)g Fo(V)539 1221 y Fp(^)533 1227 y Fm(\012)p Fo(W)6 b Fp(.)17 b(Since)g(all)f(simple)f(mo)q(dules)h(of)g Fo(V)1199 1221 y Fp(^)1193 1227 y Fm(\012)p Fo(W)23 b Fp(are)17 b(isomorphic)e(and)i(of)257 1277 y(dimension)c Fo(p)p Fp(,)g(the)h(assertion)h(follo)o(ws.)d Fi(2)257 1412 y Fq(6.6)48 b Fp(W)m(e)17 b(ha)o(v)o(e)g(seen)i(in)e(the)h(preceding)g (paragraph)f(that)h(the)g(pro)q(duct)g(of)f(t)o(w)o(o)f(one-)257 1462 y(dimensional)h(c)o(haracters)22 b Fo(\021)e Fp(and)f Fo(\021)844 1447 y Fl(0)876 1462 y Fp(is)g(again)f(a)h(one-dimensional) f(c)o(haracter)i(if)f Fo(\021)i Fp(is)257 1512 y(linear)15 b(or)f Fo(\021)447 1497 y Fl(0)473 1512 y Fp(is)h(colinear)f(o)o(v)o (er)h Fo(H)s Fp(,)f(but)h(is)f(in)g(general)h(not)g(a)f(c)o(haracter)i (in)e(other)i(cases.)257 1562 y(W)m(e)e(no)o(w)f(in)o(v)o(estigate)h (what)g(can)g(b)q(e)g(said)g(instead.)257 1661 y Fq(Prop)q(osition)33 b Fp(Supp)q(ose)15 b(that)f Fo(\021)e Fp(:)f Fo(A)h Fm(!)f Fo(K)17 b Fp(is)d(a)f(c)o(haracter)i(that)f(is)g(not)g Fo(H)s Fp(-linear)f(and)257 1711 y(that)18 b Fo(\021)373 1696 y Fl(0)404 1711 y Fp(:)f Fo(A)i Fm(!)f Fo(K)j Fp(is)d(a)g(c)o (haracter)h(that)f(is)g(not)g(colinear.)f(Then)h(there)i(are)e (distinct)257 1761 y(c)o(haracters)e Fo(!)481 1767 y Fn(0)499 1761 y Fo(;)7 b(:)g(:)g(:)12 b(;)7 b(!)625 1767 y Fj(p)p Fl(\000)p Fn(1)700 1761 y Fp(with)14 b(the)g(follo)o(wing)d (prop)q(erties:)308 1880 y(1.)20 b(If)12 b Fo(\021)422 1886 y Fn(0)440 1880 y Fo(;)7 b(:)g(:)g(:)12 b(;)7 b(\021)561 1886 y Fj(p)p Fl(\000)p Fn(1)633 1880 y Fp(are)13 b(the)f(conjugates)h (of)e Fo(\021)i Fp(with)f(resp)q(ect)i(to)e Fo(\036)1359 1865 y Fl(\003)1390 1880 y Fp(and)g Fo(\021)1491 1865 y Fl(0)1490 1890 y Fn(0)1508 1880 y Fo(;)7 b(:)g(:)g(:)12 b(;)7 b(\021)1630 1865 y Fl(0)1629 1890 y Fj(p)p Fl(\000)p Fn(1)361 1930 y Fp(are)14 b(the)h(conjugates)f(of)g Fo(\021)776 1915 y Fl(0)801 1930 y Fp(with)g(resp)q(ect)i(to)e Fo( )1117 1915 y Fl(\003)1136 1930 y Fp(,)g(w)o(e)g(ha)o(v)o(e)766 2021 y Fo(\021)787 2027 y Fj(i)801 2021 y Fo(\021)823 2004 y Fl(0)822 2031 y Fj(j)850 2021 y Fm(2)e Fp(Span)o(\()p Fm(f)p Fo(!)1042 2027 y Fn(0)1061 2021 y Fo(;)7 b(:)g(:)g(:)12 b(;)7 b(!)1187 2027 y Fj(p)p Fl(\000)p Fn(1)1247 2021 y Fm(g)p Fp(\))361 2112 y(for)14 b(all)e Fo(i;)7 b(j)14 b Fm(2)d Fk(Z)616 2118 y Fj(p)632 2112 y Fp(.)j(In)f(particular,)h(w)o (e)g(ha)o(v)o(e)f Fo(\021)q(\021)1113 2097 y Fl(0)1137 2112 y Fm(2)e Fp(Span\()p Fm(f)p Fo(!)1329 2118 y Fn(0)1347 2112 y Fo(;)c(:)g(:)g(:)12 b(;)7 b(!)1473 2118 y Fj(p)p Fl(\000)p Fn(1)1534 2112 y Fm(g)p Fp(\).)308 2195 y(2.)20 b Fm(f)p Fo(!)408 2201 y Fn(0)426 2195 y Fo(;)7 b(:)g(:)g(:)12 b(;)7 b(!)552 2201 y Fj(p)p Fl(\000)p Fn(1)613 2195 y Fm(g)14 b Fp(is)f(in)o(v)n(arian)o(t)g(under)h Fo(\036)1006 2180 y Fl(\003)1039 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Fl(\000)p Fn(1)709 2503 y Fm(g)14 b Fp(is)f(in)o(v)n(arian)o(t)g(under)h Fo( )1105 2488 y Fl(\003)1125 2503 y Fp(.)953 2628 y(93)p eop %%Page: 94 94 94 93 bop 257 262 a Fp(\(4\))21 b(As)15 b(for)e Fo( )q Fp(,)h(w)o(e)g(ha)o(v)o(e)g(for)f Fo(\036)h Fp(that)620 353 y(\()p Fo(\036)c Fm(\012)f Fo(\036)p Fp(\)\()p Fo(a)p Fp(\()p Fo(v)i Fm(\012)e Fo(w)q Fp(\)\))j(=)g Fo(\036)p Fp(\()p Fo(a)p Fp(\)\()p Fo(\036)p Fp(\()p Fo(v)q Fp(\))e Fm(\012)f Fo(\036)p Fp(\()p Fo(w)q Fp(\)\))257 444 y(for)14 b Fo(a)f Fm(2)e Fo(A)p Fp(,)j Fo(v)g Fm(2)e Fo(V)e Fp(,)j(and)i Fo(w)e Fm(2)f Fo(W)6 b Fp(.)13 b(By)i(Prop)q(osition)f(6.4,)f(w)o(e)h (ha)o(v)o(e)h(that)f Fo(\036)p Fp(\()p Fo(e)1497 429 y Fl(0)1497 454 y Fn(0)1516 444 y Fp(\))e(=)h Fo(e)1608 429 y Fl(0)1608 455 y Fj(j)1640 444 y Fp(for)257 494 y(some)g Fo(j)h Fm(2)d Fk(Z)462 500 y Fj(p)478 494 y Fp(.)j(Applying)e Fo(\036)d Fm(\012)g Fo(\036)14 b Fp(to)g(the)g (equation)f Fo(a)p Fp(\()p Fo(v)1146 500 y Fj(i)1169 494 y Fm(\012)d Fo(e)1230 479 y Fl(0)1230 504 y Fn(0)1249 494 y Fp(\))h(=)h Fo(!)1346 500 y Fj(i)1360 494 y Fp(\()p Fo(a)p Fp(\))p Fo(v)1434 500 y Fj(i)1457 494 y Fm(\012)d Fo(e)1517 479 y Fl(0)1517 504 y Fn(0)1536 494 y Fp(,)14 b(w)o(e)g(get:)642 585 y Fo(a)p Fp(\()p Fo(\036)p Fp(\()p Fo(v)741 591 y Fj(i)755 585 y Fp(\))c Fm(\012)f Fo(e)841 568 y Fl(0)841 596 y Fj(j)859 585 y Fp(\))j(=)f Fo(!)956 591 y Fj(i)970 585 y Fp(\()p Fo(\036)1011 568 y Fl(\000)p Fn(1)1056 585 y Fp(\()p Fo(a)p Fp(\)\))p Fo(\036)p Fp(\()p Fo(v)1187 591 y Fj(i)1201 585 y Fp(\))e Fm(\012)h Fo(e)1287 568 y Fl(0)1287 596 y Fj(j)257 677 y Fp(Since)j Fo(V)i Fm(\012)6 b Fo(Ae)491 662 y Fl(0)491 687 y Fn(0)522 666 y Fm(\030)522 679 y Fp(=)565 677 y Fo(V)16 b Fm(\012)6 b Fo(Ae)693 662 y Fl(0)693 687 y Fj(j)723 677 y Fp(b)o(y)11 b(Corollary)g(6.5,)g(w)o(e)h(get)h(as)f(ab)q(o)o(v)o(e)g(that)g Fm(f)p Fo(!)1464 683 y Fn(0)1482 677 y Fo(;)7 b(:)g(:)g(:)12 b(;)7 b(!)1608 683 y Fj(p)p Fl(\000)p Fn(1)1669 677 y Fm(g)257 726 y Fp(is)14 b(in)o(v)n(arian)o(t)e(under)j 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998 y Fj(p)p Fl(\000)p Fn(1)1216 992 y Fp(\))p Fm(g)16 b Fp(=)h Fm(f)p Fo(\021)1360 998 y Fn(0)1378 992 y Fo(;)7 b(:)g(:)g(:)12 b(;)7 b(\021)1499 998 y Fj(p)p Fl(\000)p Fn(1)1559 992 y Fm(g)p Fp(.)17 b(This)257 1042 y(implies)12 b(the)j(\014rst)f(assertion)h(of)e(the)i(prop)q(osition.) 257 1125 y(\(6\))21 b(No)o(w)16 b(supp)q(ose)h(that)f Fo(\021)h Fp(is)f(colinear.)g(If)f Fm(f)p Fo(!)1019 1131 y Fn(0)1037 1125 y Fo(;)7 b(:)g(:)g(:)12 b(;)7 b(!)1163 1131 y Fj(p)p Fl(\000)p Fn(1)1224 1125 y Fm(g)16 b Fp(is)g(not)g(an)g (orbit)f(with)h(re-)257 1175 y(sp)q(ect)d(to)e Fo( )438 1160 y Fl(\003)457 1175 y Fp(,)g(it)g(consists)h(of)e(colinear)h(c)o (haracters.)h(Since)g Fo(\021)q(\021)1221 1160 y Fl(0)1244 1175 y Fm(2)f Fp(Span\()p Fm(f)p Fo(!)1436 1181 y Fn(0)1454 1175 y Fo(;)c(:)g(:)g(:)12 b(;)7 b(!)1580 1181 y Fj(p)p Fl(\000)p Fn(1)1641 1175 y Fm(g)p Fp(\),)257 1225 y(w)o(e)13 b(ha)o(v)o(e)f Fo(\021)433 1210 y Fl(0)456 1225 y Fm(2)g Fp(Span)o(\()p Fm(f)p Fo(\021)644 1210 y Fl(\000)p Fn(1)689 1225 y Fo(!)715 1231 y Fn(0)733 1225 y Fo(;)7 b(:)g(:)g(:)12 b(;)7 b(\021)855 1210 y Fl(\000)p Fn(1)899 1225 y Fo(!)925 1231 y Fj(p)p Fl(\000)p Fn(1)987 1225 y Fm(g)p Fp(\).)12 b(W)m(e)f(ha)o(v)o(e)i(seen)g(in)f(P)o(aragraph)g(1.5)f(that)257 1274 y(colinear)17 b(c)o(haracters)h(form)d(a)h(group,)g(therefore)i (the)f(linear)f(functions)g Fo(\021)1457 1259 y Fl(\000)p Fn(1)1502 1274 y Fo(!)1528 1280 y Fj(k)1565 1274 y Fp(are)g(co-)257 1324 y(linear)j(c)o(haracters.)h(Since)f(distinct)g(c)o(haracters)h (are)f(linearly)f(indep)q(enden)o(t,)i(w)o(e)f(ha)o(v)o(e)257 1374 y Fo(\021)279 1359 y Fl(0)303 1374 y Fp(=)11 b Fo(\021)368 1359 y Fl(\000)p Fn(1)413 1374 y Fo(!)439 1380 y Fj(k)473 1374 y Fp(for)i(some)f Fo(k)h Fm(2)e(f)p Fp(0)p Fo(;)c(:)g(:)g(:)k(;)c (p)h Fm(\000)g Fp(1)p Fm(g)p Fp(.)13 b(Therefore,)h Fo(\021)1211 1359 y Fl(0)1236 1374 y Fp(is)f(colinear,)g(whic)o(h)g(con)o(tra-)257 1424 y(dicts)i(our)f(assumptions.)257 1507 y(\(7\))21 b(No)o(w)e(supp)q(ose)i(that)e Fo(\021)710 1492 y Fl(0)741 1507 y Fp(is)h Fo(H)s Fp(-linear.)e(If)h Fm(f)p Fo(!)1067 1513 y Fn(0)1085 1507 y Fo(;)7 b(:)g(:)g(:)12 b(;)7 b(!)1211 1513 y Fj(p)p Fl(\000)p Fn(1)1272 1507 y Fm(g)19 b Fp(is)g(not)h(an)f (orbit)g(with)257 1557 y(resp)q(ect)i(to)d Fo(\036)484 1542 y Fl(\003)503 1557 y Fp(,)g(it)g(consists)i(of)d Fo(H)s Fp(-linear)h(c)o(haracters.)i(As)f(in)f(the)h(previous)f(step,)h (w)o(e)257 1606 y(get)c(that)g Fo(\021)e Fm(2)f Fp(Span\()p Fm(f)p Fo(!)645 1612 y Fn(0)664 1606 y Fo(\021)686 1591 y Fl(0)q(\000)p Fn(1)742 1606 y Fo(;)7 b(:)g(:)g(:)12 b(;)7 b(!)868 1612 y Fj(p)p Fl(\000)p Fn(1)928 1606 y Fo(\021)950 1591 y Fl(0)r(\000)p Fn(1)1007 1606 y Fm(g)p Fp(\))14 b(is)g(a)g(linear)g(com)o(bination)e(of)i Fo(H)s Fp(-linear)257 1656 y(c)o(haracters,)h(and)e(since)h(w)o(e)f(then)h(ha) o(v)o(e)f Fo(\021)g Fp(=)e Fo(!)999 1662 y Fj(k)1020 1656 y Fo(\021)1042 1641 y Fl(0)q(\000)p Fn(1)1111 1656 y Fp(for)i(some)f Fo(k)h Fm(2)e(f)p Fp(0)p Fo(;)c(:)g(:)g(:)k(;)c(p)g Fm(\000)h Fp(1)p Fm(g)p Fp(,)k Fo(\021)j Fp(is)257 1706 y Fo(H)s Fp(-linear)f(itself,)f(whic)o(h)g(con)o(tradicts)i(our)f (assumptions.)e Fi(2)257 1806 y Fq(Corollary)35 b Fp(If)17 b(all)f(primitiv)o(e)f(idemp)q(oten)o(ts)i(are)g(in)o(v)n(arian)o(t)f (or)h(coin)o(v)n(arian)o(t,)e(then)j(all)257 1856 y(primitiv)o(e)9 b(idemp)q(oten)o(ts)i(are)g(in)o(v)n(arian)o(t)e(or)i(all)f(primitiv)o (e)f(idemp)q(oten)o(ts)h(are)h(coin)o(v)n(arian)o(t.)257 1955 y Fq(Pro)q(of.)36 b Fp(If)18 b(this)f(is)h(not)f(the)i(case,)f (there)h(is)e(a)h(primitiv)o(e)d(idemp)q(oten)o(t)i Fo(e)h Fp(that)g(is)g(not)257 2005 y(in)o(v)n(arian)o(t)12 b(and)g(a)h (primitiv)o(e)d(idemp)q(oten)o(t)i Fo(e)957 1990 y Fl(0)982 2005 y Fp(that)h(is)g(not)f(coin)o(v)n(arian)o(t.)f(By)i(assumption,) 257 2055 y(w)o(e)21 b(then)h(ha)o(v)o(e)e(that)h Fo(e)g Fp(is)g(coin)o(v)n(arian)o(t)e(and)i(that)g Fo(e)1139 2040 y Fl(0)1171 2055 y Fp(is)g(in)o(v)n(arian)o(t.)e(W)m(e)h(denote)i (the)257 2105 y(c)o(haracters)12 b(corresp)q(onding)f(to)e Fo(e)i Fp(resp.)f Fo(e)904 2090 y Fl(0)926 2105 y Fp(b)o(y)g Fo(\021)g Fp(resp.)h Fo(\021)1129 2090 y Fl(0)1141 2105 y Fp(.)e(By)h(the)h(preceding)g(prop)q(osition,)257 2154 y(w)o(e)20 b(then)g(ha)o(v)o(e)g(c)o(haracters)h Fm(f)p Fo(!)776 2160 y Fn(0)794 2154 y Fo(;)7 b(:)g(:)g(:)12 b(;)7 b(!)920 2160 y Fj(p)p Fl(\000)p Fn(1)981 2154 y Fm(g)19 b Fp(satisfying)g Fo( )1240 2139 y Fl(\003)1259 2154 y Fp(\()p Fo(!)1301 2160 y Fj(j)1319 2154 y Fp(\))i(=)g Fo(!)1435 2160 y Fj(j)r Fn(+1)1514 2154 y Fp(suc)o(h)f(that)257 2204 y Fo(\021)q(\021)301 2189 y Fl(0)328 2204 y Fm(2)15 b Fp(Span\()p Fo(!)503 2210 y Fn(0)521 2204 y Fo(;)7 b(:)g(:)g(:)12 b(;)7 b(!)647 2210 y Fj(p)p Fl(\000)p Fn(1)708 2204 y Fp(\).)16 b(Since)g Fo(!)888 2210 y Fj(j)922 2204 y Fp(is)f(not)h(colinear,)g(it)f(m)o(ust)g(b)q(e)i(linear,)e(i.)g (e.,)g(w)o(e)257 2254 y(ha)o(v)o(e)f Fo(\036)378 2239 y Fl(\003)397 2254 y Fp(\()p Fo(!)439 2260 y Fj(j)456 2254 y Fp(\))e(=)g Fo(!)554 2260 y Fj(j)585 2254 y Fp(and)i(therefore)h Fo(\036)865 2239 y Fl(\003)884 2254 y Fp(\()p Fo(\021)q(\021)944 2239 y Fl(0)956 2254 y Fp(\))d(=)f Fo(\021)q(\021)1071 2239 y Fl(0)1083 2254 y Fp(.)j(This)f(implies:)627 2345 y Fo(\036)652 2328 y Fl(\003)670 2345 y Fp(\()p Fo(\021)q Fp(\))p Fo(\021)746 2328 y Fl(0)770 2345 y Fp(=)f Fo(\036)839 2328 y Fl(\003)858 2345 y Fp(\()p Fo(\021)q Fp(\))p Fo(\036)937 2328 y Fl(\003)956 2345 y Fp(\()p Fo(\021)994 2328 y Fl(0)1006 2345 y Fp(\))f(=)h Fo(\036)1102 2328 y Fl(\003)1121 2345 y Fp(\()p Fo(\021)q(\021)1181 2328 y Fl(0)1193 2345 y Fp(\))g(=)g Fo(\021)q(\021)1309 2328 y Fl(0)257 2437 y Fp(Since)i Fo(\021)387 2422 y Fl(0)411 2437 y Fp(is)e(an)h(in)o(v)o (ertible)f(elemen)o(t)g(b)o(y)h(Prop)q(osition)f(1.5.2,)e(w)o(e)j(ha)o (v)o(e)g Fo(\036)1410 2422 y Fl(\003)1429 2437 y Fp(\()p Fo(\021)q Fp(\))f(=)f Fo(\021)q Fp(,)i(whic)o(h)257 2487 y(is)h(a)g(con)o(tradiction.)f Fi(2)953 2628 y Fp(94)p eop %%Page: 95 95 95 94 bop 257 262 a Fq(6.7)48 b Fp(The)15 b(prop)q(osition)f(pro)o(v)o (ed)h(in)f(the)h(previous)g(paragraph)f(has)h(particularly)e(strik-)257 311 y(ing)i(consequences)j(if)c(applied)g(to)h(the)h(situation)e(where) j(the)e(t)o(w)o(o)g(c)o(haracters)i Fo(\021)f Fp(and)f Fo(\021)1678 296 y Fl(0)257 361 y Fp(corresp)q(ond)h(to)e(dual)f Fo(B)r Fp(-mo)q(dules:)257 451 y Fq(Prop)q(osition)33 b Fp(Supp)q(ose)20 b(that)g Fo(\021)i Fp(:)e Fo(A)h Fm(!)f Fo(K)i Fp(is)d(a)g(c)o(haracter)i(whic)o(h)e(is)h(neither)g Fo(H)s Fp(-)257 501 y(linear)13 b(nor)h(colinear.)e(Then)i(there)h (exists)f(a)f(c)o(haracter)i Fo(!)e Fp(:)e Fo(A)g Fm(!)g Fo(K)17 b Fp(with)c(the)h(follo)o(wing)257 551 y(prop)q(erties:)308 651 y(1.)20 b Fo(!)13 b Fp(:)e Fo(A)h Fm(!)f Fo(K)17 b Fp(is)d(an)f Fo(H)s Fp(-linear)g(and)h(colinear)g(c)o(haracter)h(of)e (order)i Fo(p)p Fp(.)308 728 y(2.)20 b(W)m(e)14 b(ha)o(v)o(e:)f Fo(\036)565 713 y Fl(\003)584 728 y Fp(\()p Fo(\021)q Fp(\))f(=)f Fo(!)q(\021)i Fp(=)f Fo(\021)q(!)847 713 y Fj(j)879 728 y Fp(for)h(some)h Fo(j)g Fm(2)d Fk(Z)1147 734 y Fj(p)1164 728 y Fp(.)308 804 y(3.)20 b(If)12 b Fo(e)g Fp(is)g(the)h(primitiv)o(e)d(idemp)q(oten)o(t)h(corresp)q (onding)i(to)f Fo(\021)q Fp(,)g Fo(O)g Fp(the)h(orbit)f(of)f Fo(e)i Fp(under)361 854 y(the)j(action)f(of)g Fo(C)s Fp(,)f(and)h Fo(V)24 b Fp(:=)14 b(Span\()p Fo(O)q Fp(\))h(the)h (corresp)q(onding)g(simple)e Fo(B)r Fp(-mo)q(dule,)361 904 y(the)d Fo(B)r Fp(-mo)q(dule)f Fo(V)i Fm(\012)s Fo(V)725 889 y Fl(\003)755 904 y Fp(decomp)q(oses)f(in)o(to)f Fo(p)1079 889 y Fn(2)1108 904 y Fp(one-dimensional)f(mo)q(dules,)g (whose)361 954 y(c)o(haracters)16 b(are)e Fo(!)655 939 y Fj(i)678 954 y Fm(\012)c Fo(\015)743 939 y Fj(j)761 954 y Fp(,)k(for)f Fo(i;)7 b(j)14 b Fp(=)e(0)p Fo(;)7 b(:)g(:)g(:)k(;)c(p)i Fm(\000)g Fp(1)14 b(and)g(a)f(generator)i Fo(\015)i Fp(of)1578 943 y(^)1569 954 y Fo(C)r Fp(.)257 1054 y(In)d(particular,)f Fo(p)h Fp(divides)g(dim)5 b Fo(A)p Fp(.)257 1144 y Fq(Pro)q(of.)36 b Fp(\(1\))21 b(Since)i Fo(V)647 1129 y Fl(\003)688 1144 y Fp(is)f(also)g(a)g(simple) e Fo(B)r Fp(-mo)q(dule,)h(there)i(is)f(a)g Fo(C)s Fp(-orbit)f Fo(O)1620 1129 y Fl(0)1657 1144 y Fp(=)257 1194 y Fm(f)p Fo(e)297 1179 y Fl(0)297 1204 y Fn(0)316 1194 y Fo(;)7 b(:)g(:)g(:)12 b(;)7 b(e)435 1179 y Fl(0)435 1204 y Fj(p)p Fl(\000)p Fn(1)496 1194 y Fm(g)17 b Fp(suc)o(h)i(that)e Fo(V)759 1179 y Fl(\003)795 1183 y Fm(\030)795 1196 y Fp(=)845 1194 y Fo(W)24 b Fp(:=)17 b(Span\()p Fo(O)1108 1179 y Fl(0)1120 1194 y Fp(\).)g(By)h(Prop)q(osition)f(6.4,)f(w)o(e)i (ha)o(v)o(e)257 1244 y(that)13 b Fo(O)g Fp(and)f Fo(O)503 1229 y Fl(0)527 1244 y Fp(are)h(in)o(v)n(arian)o(t)e(under)i Fo( )q Fp(.)g(The)g(fact)f(that)h Fo(\021)g Fp(is)f(neither)i(linear)e (nor)g(colin-)257 1293 y(ear)i(means)f(that)g Fo(O)h Fp(con)o(tains)g(no)f(\014xed)h(p)q(oin)o(ts)f(of)g Fo(\036)g Fp(and)g Fo( )q Fp(,)h(and)f(therefore,)h(according)257 1343 y(to)g(Prop)q(osition)f(6.3,)f Fo(O)642 1328 y Fl(0)667 1343 y Fp(con)o(tains)i(no)f(\014xed)h(p)q(oin)o(ts)g(of)f Fo(\036)g Fp(and)h Fo( )q Fp(,)f(and)h(th)o(us)g(is)f(an)h(orbit)257 1393 y(with)g(resp)q(ect)i(to)e Fo(\036)f Fp(and)h Fo( )q Fp(.)257 1456 y(As)j(in)f(the)h(pro)q(of)f(of)g(the)h(preceding)g(prop) q(osition,)f(there)h(exists)g(a)f(basis)h Fo(v)1490 1462 y Fn(0)1509 1456 y Fo(;)7 b(:)g(:)g(:)k(;)c(v)1628 1462 y Fj(p)p Fl(\000)p Fn(1)257 1506 y Fp(of)14 b Fo(V)23 b Fp(and)14 b(distinct)g(c)o(haracters)h Fo(!)806 1512 y Fn(0)825 1506 y Fo(;)7 b(:)g(:)g(:)12 b(;)7 b(!)951 1512 y Fj(p)p Fl(\000)p Fn(1)1025 1506 y Fp(suc)o(h)15 b(that:)749 1582 y Fo(a)p Fp(\()p Fo(v)807 1588 y Fj(i)830 1582 y Fm(\012)10 b Fo(e)891 1565 y Fl(0)891 1592 y Fn(0)910 1582 y Fp(\))h(=)h Fo(!)1007 1588 y Fj(i)1021 1582 y Fp(\()p Fo(a)p Fp(\))p Fo(v)1095 1588 y Fj(i)1118 1582 y Fm(\012)e Fo(e)1179 1565 y Fl(0)1179 1592 y Fn(0)257 1671 y Fp(\(2\))21 b(F)m(rom)12 b(the)j(decomp)q(osition)676 1789 y Fo(V)k Fm(\012)9 b Fo(V)794 1771 y Fl(\003)824 1778 y Fm(\030)824 1791 y Fp(=)868 1789 y Fo(V)19 b Fm(\012)9 b Fo(W)18 b Fp(=)1054 1735 y Fj(p)p 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Fl(\003)1678 2053 y Fp(.)257 2102 y(Therefore,)g(the)g(set)f Fm(f)p Fo(!)631 2108 y Fn(0)650 2102 y Fo(;)c(:)g(:)g(:)k(;)c(!)775 2108 y Fj(p)p Fl(\000)p Fn(1)836 2102 y Fm(g)p Fp(,)j(whic)o(h)h(is)g(in)o (v)n(arian)o(t)e(under)j Fo(\036)1345 2087 y Fl(\003)1375 2102 y Fp(and)e Fo( )1480 2087 y Fl(\003)1511 2102 y Fp(b)o(y)g(Prop)q(o-)257 2152 y(sition)i(6.6,)f(cannot)i(b)q(e)g(an)f (orbit)g(of)g(length)h Fo(p)f Fp(with)g(resp)q(ect)j(to)d(these)i (maps,)d(and)h(there-)257 2202 y(fore)i(m)o(ust)f(con)o(tain)h(only)f (\014xed)h(p)q(oin)o(ts.)f(This)h(implies:)719 2278 y Fo(\036)744 2261 y Fl(\003)763 2278 y Fp(\()p Fo(!)805 2284 y Fj(i)819 2278 y Fp(\))e(=)f Fo(!)916 2284 y Fj(i)1013 2278 y Fo( )1041 2261 y Fl(\003)1061 2278 y Fp(\()p Fo(!)1103 2284 y Fj(i)1116 2278 y Fp(\))h(=)g Fo(!)1214 2284 y Fj(i)257 2354 y Fp(for)f(all)g Fo(i)h Fp(=)f(0)p Fo(;)c(:)g(:)g(:)12 b(;)7 b(p)t Fm(\000)t Fp(1.)k(Therefore,)h(the)g Fo(B)r Fp(-mo)q(dule)f Fo(V)j Fm(\012)t Fo(V)1235 2339 y Fl(\003)1265 2354 y 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y(ing)e(on)f(the)i(left)e(and)h(on)g(the)g(righ)o (t)g(b)o(y)f Fo(\021)918 2408 y Fl(\000)p Fn(1)963 2423 y Fp(,)g(w)o(e)h(deduce)i(that:)578 2503 y(Span\()p Fo(!)711 2486 y Fl(0)r Fn(0)741 2503 y Fo(;)7 b(:)g(:)g(:)12 b(;)7 b(!)868 2486 y Fl(0)r Fj(p)p Fl(\000)p Fn(1)941 2503 y Fp(\))12 b(=)g(Span\()p Fo(!)1146 2486 y Fn(0)1165 2503 y Fo(;)7 b(:)g(:)g(:)12 b(;)7 b(!)1292 2486 y Fj(p)p Fl(\000)p Fn(1)1353 2503 y Fp(\))953 2628 y(96)p eop %%Page: 97 97 97 96 bop 257 262 a Fp(This)16 b(implies)d(that)i Fo(!)614 246 y Fl(0)641 262 y Fp(is)g(a)g(p)q(o)o(w)o(er)h(of)f Fo(!)q Fp(.)g(Replacing)f Fo(j)k Fp(b)o(y)d(a)g(di\013eren)o(t)h(one,)f (w)o(e)g(arriv)o(e)257 311 y(at)f Fo(\036)333 296 y Fl(\003)352 311 y Fp(\()p Fo(\021)q Fp(\))e(=)g Fo(\021)q(!)511 296 y Fj(j)529 311 y Fp(.)257 394 y(\(6\))21 b(F)m(rom)c(P)o(aragraph)g (1.2,)g(w)o(e)i(kno)o(w)e(that)i Fo(A)1038 379 y Fl(\003)1075 394 y Fp(is)f(a)g(\(righ)o(t\))g(Y)m(etter-Drinfel'd)g(Hopf)257 444 y(algebra)12 b(o)o(v)o(er)g Fo(H)s 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Fp(\()p Fo(\021)1136 1408 y Fl(0)1148 1425 y Fp(\))257 1506 y(and)i(therefore)h Fo(\036)537 1491 y Fl(\003)556 1506 y Fp(\()p Fo(\021)594 1491 y Fl(0)606 1506 y Fp(\))d(=)g Fo(!)705 1491 y Fn(\()p Fj(n)p Fl(\000)p Fn(1\))p Fj(m)825 1506 y Fo(\021)847 1491 y Fl(0)859 1506 y Fp(.)257 1571 y(Since)j Fo(\021)388 1556 y Fl(0)r(\000)p Fn(1)458 1571 y Fp(is)f(also)g(a)g(c)o(haracter)h (whic)o(h)f(is)g(colinear,)g(but)g(not)g(linear)g(o)o(v)o(er)g Fo(H)s Fp(,)f(w)o(e)i(ha)o(v)o(e)257 1621 y Fo(\036)282 1606 y Fl(\003)301 1621 y Fp(\()p Fo(\021)339 1606 y Fl(0)r(\000)p Fn(1)396 1621 y Fp(\))c(=)h Fo(!)494 1606 y Fj(k)515 1621 y Fo(\021)537 1606 y Fl(0)r(\000)p Fn(1)607 1621 y Fp(for)i(some)f Fo(k)f Fm(2)f Fk(Z)879 1627 y Fj(p)895 1621 y Fp(.)i(This)h(implies)e Fo(\036)1181 1606 y Fl(\003)1200 1621 y Fp(\()p Fo(\021)1238 1606 y Fl(0)1250 1621 y Fp(\))g(=)f Fo(\021)1343 1606 y Fl(0)1355 1621 y Fo(!)1382 1606 y Fl(\000)p Fj(k)1429 1621 y Fp(.)257 1700 y(\(4\))21 b(No)o(w)11 b(supp)q(ose)i(that)f Fo(\021)687 1685 y Fl(0)710 1700 y Fp(is)f(linear,)g(but)h(not)f(colinear)g(o)o(v)o (er)h Fo(H)s Fp(.)f(Consider)h(the)g(c)o(harac-)257 1750 y(ter)g Fo(\021)341 1735 y Fl(0)o(0)374 1750 y Fp(:=)f Fo(\021)451 1735 y Fl(0)463 1750 y Fo(\021)q Fp(.)f(Then)h Fo(\021)634 1735 y Fl(0)o(0)665 1750 y Fp(is)g(not)f Fo(H)s Fp(-linear,)g(b)q(ecause)i(in)e(this)g(case,)h(b)o(y)g(Prop)q (osition)f(1.5.2,)257 1799 y Fo(\021)j Fp(=)f Fo(\021)357 1784 y Fl(0)r(\000)p Fn(1)413 1799 y Fo(\021)435 1784 y Fl(00)467 1799 y Fp(w)o(ould)e(b)q(e)h Fo(H)s Fp(-linear,)f(to)q(o.)g (W)m(e)g(no)o(w)g(treat)i(separately)f(the)g(t)o(w)o(o)f(cases)i(where) 257 1849 y Fo(\021)279 1834 y Fl(00)316 1849 y Fp(is)j(colinear)h(and)f (where)h(it)g(is)f(not)g(colinear.)g(Supp)q(ose)h(\014rst)h(that)e Fo(\021)1414 1834 y Fl(00)1451 1849 y Fp(is)g(colinear.)g(If)257 1899 y Fo( )285 1884 y Fl(\003)305 1899 y Fp(\()p Fo(\021)q Fp(\))d(=)g Fo(!)442 1884 y Fj(s)460 1899 y Fo(\021)g Fp(=)g Fo(\021)q(!)586 1884 y Fj(sm)634 1899 y Fp(,)h(w)o(e)h(ha)o(v)o (e)553 1981 y Fo( )581 1964 y Fl(\003)601 1981 y Fp(\()p Fo(\021)639 1964 y Fl(0)651 1981 y Fp(\))p Fo(!)694 1964 y Fj(s)712 1981 y Fo(\021)f Fp(=)e Fo( )817 1964 y Fl(\003)837 1981 y Fp(\()p Fo(\021)875 1964 y Fl(0)887 1981 y Fp(\))p Fo( )931 1964 y Fl(\003)951 1981 y Fp(\()p Fo(\021)q Fp(\))h(=)f Fo( )1088 1964 y Fl(\003)1108 1981 y Fp(\()p Fo(\021)1146 1964 y Fl(00)1168 1981 y Fp(\))g(=)h Fo(\021)1261 1964 y Fl(00)1294 1981 y Fp(=)g Fo(\021)1360 1964 y Fl(0)1371 1981 y Fo(\021)257 2062 y Fp(and)i(therefore)h Fo( )540 2047 y Fl(\003)560 2062 y Fp(\()p Fo(\021)598 2047 y Fl(0)610 2062 y Fp(\))d(=)f Fo(\021)703 2047 y Fl(0)715 2062 y Fo(!)742 2047 y Fl(\000)p Fj(s)786 2062 y Fp(.)257 2127 y(No)o(w)18 b(supp)q(ose)g(that)g Fo(\021)631 2112 y Fl(00)670 2127 y Fp(is)f(not)h(colinear.)e(Then)j(w)o(e)e(can)h (apply)f(the)h(ab)q(o)o(v)o(e)f(result)i(to)257 2177 y(conclude)c(that)f Fo( )546 2162 y Fl(\003)566 2177 y Fp(\()p Fo(\021)604 2162 y Fl(0)o(0)625 2177 y Fp(\))e(=)g Fo(\021)719 2162 y Fl(0)o(0)740 2177 y Fo(!)767 2162 y Fj(k)801 2177 y Fp(for)i(some)f(other)h Fo(k)f Fm(2)e Fk(Z)1182 2183 y Fj(p)1198 2177 y Fp(.)i(This)h(implies:)579 2258 y Fo(\021)601 2241 y Fl(0)613 2258 y Fo(\021)q(!)662 2241 y Fj(k)694 2258 y Fp(=)e Fo( )766 2241 y Fl(\003)785 2258 y Fp(\()p Fo(\021)823 2241 y Fl(00)845 2258 y Fp(\))f(=)h Fo( )944 2241 y Fl(\003)964 2258 y Fp(\()p Fo(\021)1002 2241 y Fl(0)1014 2258 y Fp(\))p Fo( )1058 2241 y Fl(\003)1078 2258 y Fp(\()p Fo(\021)q Fp(\))f(=)h Fo( )1215 2241 y Fl(\003)1235 2258 y Fp(\()p Fo(\021)1273 2241 y Fl(0)1285 2258 y Fp(\))p Fo(!)1328 2241 y Fj(s)1346 2258 y Fo(\021)271 2340 y Fp(Since)j Fo(\021)q(!)e Fp(=)e Fo(!)511 2325 y Fj(t)526 2340 y Fo(\021)q Fp(,)j(where)h Fo(t)c Fm(2)h Fk(Z)790 2346 y Fj(p)820 2340 y Fp(is)i(the)g(m)o(ultiplicativ)o(e)d (in)o(v)o(erse)k(of)e Fo(m)p Fp(,)h(this)g(yields:)797 2421 y Fo(\021)819 2404 y Fl(0)831 2421 y Fo(!)858 2404 y Fj(k)q(t)891 2421 y Fo(\021)f Fp(=)f Fo( )997 2404 y Fl(\003)1016 2421 y Fp(\()p Fo(\021)1054 2404 y Fl(0)1066 2421 y Fp(\))p Fo(!)1109 2404 y Fj(s)1127 2421 y Fo(\021)257 2503 y Fp(and)i(therefore)h Fo( )540 2488 y Fl(\003)560 2503 y Fp(\()p Fo(\021)598 2488 y Fl(0)610 2503 y Fp(\))d(=)f Fo(\021)703 2488 y Fl(0)715 2503 y Fo(!)742 2488 y Fj(k)q(t)p Fl(\000)p Fj(s)817 2503 y Fp(.)953 2628 y(98)p eop %%Page: 99 99 99 98 bop 257 262 a Fp(Since)15 b Fo(\021)388 246 y Fl(0)r(\000)p Fn(1)458 262 y Fp(is)f(also)g(a)g(c)o(haracter)h(whic)o(h)f(is)g (linear,)f(but)i(not)f(colinear)g(o)o(v)o(er)g Fo(H)s Fp(,)f(w)o(e)i(ha)o(v)o(e)257 311 y Fo( )285 296 y Fl(\003)305 311 y Fp(\()p Fo(\021)343 296 y Fl(0)r(\000)p Fn(1)399 311 y Fp(\))d(=)g Fo(\021)493 296 y Fl(0)r(\000)p Fn(1)549 311 y Fo(!)576 296 y Fj(r)609 311 y Fp(for)h(some)g(other)i Fo(r)d Fm(2)f Fk(Z)986 317 y Fj(p)1002 311 y Fp(.)j(This)f(implies)f Fo( )1291 296 y Fl(\003)1311 311 y Fp(\()p Fo(\021)1349 296 y Fl(0)1361 311 y Fp(\))g(=)f Fo(!)1459 296 y Fl(\000)p Fj(r)1504 311 y Fo(\021)1526 296 y Fl(0)1538 311 y Fp(.)257 394 y(\(5\))21 b(The)c(remaining)e(case)i(is)f(that)h Fo(\021)865 379 y Fl(0)893 394 y Fp(is)g(linear)f(as)g(w)o(ell)g(as)h (colinear)f(o)o(v)o(er)g Fo(H)s Fp(.)g(This)h(is)257 444 y(the)e(trivial)d(case,)j(since)f(the)h(assertion)f(is)g (satis\014ed)h(with)e Fo(i;)7 b(j;)g(k)q(;)g(l)12 b Fp(=)g(0.)h Fi(2)953 2628 y Fp(99)p eop %%Page: 100 100 100 99 bop 257 262 a Fr(7)67 b(Co)r(comm)n(utativ)n(e)23 b(Y)-6 b(etter-Drinfel'd)24 b(Hopf)358 336 y(algebras)257 651 y Fq(7.1)48 b Fp(In)12 b(this)g(section,)g(w)o(e)g(assume)f(that)h Fo(p)f Fp(is)h(a)f(prime)g(and)g(that)h Fo(K)j Fp(is)d(an)f (algebraically)257 701 y(closed)16 b(\014eld)g(whose)g(c)o (haracteristic)g(is)g(di\013eren)o(t)g(from)e Fo(p)p Fp(.)g(W)m(e)h(can)h(therefore)h(c)o(ho)q(ose)f(a)257 750 y(primitiv)o(e)e Fo(p)p Fp(-th)h(ro)q(ot)h(of)f(unit)o(y)g(that)h (w)o(e)f(denote)i(b)o(y)e Fo(\020)s Fp(.)g(The)h(group)g(ring)f(of)g (the)h(cyclic)257 800 y(group)d Fk(Z)404 806 y Fj(p)433 800 y Fp(of)f(order)h Fo(p)f Fp(will)f(b)q(e)i(denoted)g(b)o(y)f Fo(H)j Fp(:=)c Fo(K)s Fp([)p Fk(Z)1147 806 y Fj(p)1164 800 y Fp(];)g(the)i(canonical)f(basis)g(elemen)o(t)257 850 y(of)18 b(the)h(group)f(ring)g(corresp)q(onding)h(to)f Fo(i)h Fm(2)g Fk(Z)1033 856 y Fj(p)1068 850 y Fp(will)d(b)q(e)j (denoted)g(b)o(y)f Fo(c)1452 856 y Fj(i)1466 850 y Fp(.)g(The)h(group) 257 900 y(of)g(grouplik)o(e)f(elemen)o(ts)i(of)e Fo(H)s Fp(,)h(whic)o(h)g(consists)h(precisely)g(of)f(these)i(canonical)d (basis)257 950 y(elemen)o(ts,)f(will)f(b)q(e)h(denoted)i(b)o(y)e Fo(C)s Fp(.)f Fo(A)h Fp(denotes)i(a)d(non)o(trivial)g(Y)m (etter-Drinfel'd)h(Hopf)257 999 y(algebra)j(o)o(v)o(er)g Fo(H)k Fp(that)c(is)g(co)q(comm)o(utativ)o(e)e(and)i(cosemisimple.)d Fo(A)k Fp(therefore)g(has)g(a)257 1049 y(unique)16 b(basis)g(that)f (consists)i(of)e(grouplik)o(e)f(elemen)o(ts,)h(whic)o(h)h(is)f(denoted) h(b)o(y)g Fo(G)p Fp(\()p Fo(A)p Fp(\).)f(If)257 1099 y Fo(\015)g Fp(:)c Fo(C)j Fm(!)d Fo(K)451 1084 y Fl(\002)493 1099 y Fp(is)i(the)h(group)f(homomorphism)c(that)14 b(maps)e Fo(c)1231 1105 y Fn(1)1263 1099 y Fp(to)h Fo(\020)s Fp(,)g(w)o(e)h(in)o (tro)q(duce)g(as)g(in)257 1149 y(P)o(aragraph)g(1.10)f(the)h(mappings) 470 1238 y Fo(\036)d Fp(:)g Fo(V)21 b Fm(!)11 b Fo(V)r(;)c(v)13 b Fm(7!)e Fp(\()p Fo(c)792 1244 y Fn(1)823 1238 y Fm(!)g Fo(v)q Fp(\))83 b Fo( )13 b Fp(:)e Fo(V)21 b Fm(!)11 b Fo(V)r(;)c(v)14 b Fm(7!)d Fo(\015)r Fp(\()p Fo(v)1349 1221 y Fn(\(1\))1395 1238 y Fp(\))p Fo(v)1432 1221 y Fn(\(2\))257 1326 y Fp(where)j(w)o(e)f(extend)g Fo(\015)i Fp(to)e Fo(H)i Fp(b)o(y)d(linearit)o(y)m(.)f(\(Note)i(that)f(the)h(sym) o(b)q(ol)e Fo(\015)r Fp(,)i(in)f(con)o(trast)h(to)f Fo( )q Fp(,)257 1376 y(will)g(also)g(b)q(e)h(used)h(for)f(other)g(c)o (haracters)i(b)q(elo)o(w.\))d(Since)h Fo(\036)g Fp(and)g Fo( )h Fp(are)f(coalgebra)g(auto-)257 1426 y(morphisms,)f(they)j (induce)g(p)q(erm)o(utations)f(of)g Fo(G)p Fp(\()p Fo(A)p Fp(\).)g(The)h(Radford)f(bipro)q(duct)h Fo(A)9 b Fm(\012)h Fo(H)257 1476 y Fp(will)j(b)q(e)h(denoted)h(b)o(y)f Fo(B)r Fp(.)257 1560 y(Throughout)f(the)g(section,)g(w)o(e)g(will)e(constan)o (tly)i(use)g(the)g(con)o(v)o(en)o(tion)g(that)f(indices)h(tak)o(e)257 1610 y(v)n(alues)h(b)q(et)o(w)o(een)i(0)e(and)g Fo(p)9 b Fm(\000)h Fp(1)k(and)g(are)g(reduced)i(mo)q(dulo)c Fo(p)i Fp(if)g(they)g(do)g(not)h(lie)e(within)257 1660 y(this)j(range.)f(In)h(notation,)e(w)o(e)i(shall)f(not)g(distinguish)g (b)q(et)o(w)o(een)i(an)f(in)o(teger)g Fo(i)e Fm(2)g Fk(Z)p Fp(and)257 1710 y(its)g(equiv)n(alence)g(class)h(in)e Fk(Z)711 1716 y Fj(p)738 1710 y Fp(:=)f Fk(Z)-13 b Fo(=)o(p)p Fk(Z)d Fp(.)257 1794 y(The)13 b(aim)d(of)i(the)g(section)h(is)f(to)g (pro)o(v)o(e)h(that)f Fo(A)g Fp(arises)h(from)e(the)h(construction)i (explained)257 1844 y(in)g(Section)g(3,)f(i.)g(e.,)g(that)h Fo(A)f Fp(is,)g(as)h(an)g(algebra,)f(a)g(crossed)i(pro)q(duct)g(of)e (the)i(dual)e(group)257 1894 y(ring)f Fo(K)379 1879 y Fc(Z)402 1883 y Ff(p)430 1894 y Fp(and)g(the)h(group)f(ring)g(of)g(a)g 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2414 y Fo(g)h Fp(=)e Fo(g)q(u)1315 2397 y Fj(l)257 2503 y Fp(In)j(particular,)f Fo(p)h Fp(divides)g(dim)5 b Fo(A)p Fp(.)942 2628 y(100)p eop %%Page: 101 101 101 100 bop 257 262 a Fq(Pro)q(of.)36 b Fp(F)m(rom)11 b(P)o(aragraph)i(1.2,)e(w)o(e)i(kno)o(w)f(that)h Fo(A)1097 246 y Fl(\003)1129 262 y Fp(is)g(a)f(righ)o(t)g(Y)m(etter-Drinfel'd)h (Hopf)257 311 y(algebra)j(o)o(v)o(er)g Fo(H)s Fp(,)f(and)h(therefore,)h (b)o(y)f(Lemma)d(1.2,)i(w)o(e)h(kno)o(w)f(that)h Fo(A)1419 296 y Fl(\003)p Fn(op)c(cop)1550 311 y Fp(is)k(a)g(left)257 361 y(Y)m(etter-Drinfel'd)e(Hopf)f(algebra)g(o)o(v)o(er)h Fo(H)s Fp(.)f(The)h(existence)h(of)e(grouplik)o(e)g(elemen)o(ts)g(that) 257 411 y(are)i(not)g(in)o(v)n(arian)o(t)e(and)i(not)f(coin)o(v)n (arian)o(t)f(no)o(w)i(follo)o(ws)e(b)o(y)h(applying)g(Corollary)f(6.6)h (to)257 461 y Fo(A)288 446 y Fl(\003)p Fn(op)e(cop)403 461 y Fp(,)d(and)g(the)h(existence)h(of)e Fo(u)g Fp(follo)o(ws)f(b)o(y) h(applying)e(Prop)q(osition)i(6.8)g(to)g Fo(A)1523 446 y Fl(\003)p Fn(op)i(cop)1638 461 y Fp(.)d Fi(2)257 580 y Fp(F)m(or)18 b(the)g(rest)h(of)e(this)h(section,)f(w)o(e)h(\014x)g(a) f(grouplik)o(e)g(elemen)o(t)g Fo(u)h Fp(of)f(order)h Fo(p)g Fp(that)f(has)257 629 y(the)i(prop)q(erties)g(stated)g(in)e(the) i(preceding)g(prop)q(osition.)d(The)j(subspace)g(spanned)g(b)o(y)257 679 y(the)f(p)q(o)o(w)o(ers)f(of)f Fo(u)h Fp(will)e(b)q(e)j(denoted)f (b)o(y)g Fo(U)5 b Fp(;)16 b(it)h(is)f(ob)o(viously)g(a)g(Y)m (etter-Drinfel'd)h(Hopf)257 729 y(subalgebra.)257 864 y Fq(7.3)48 b Fp(In)12 b(this)g(paragraph,)e(w)o(e)i(shall)f(construct) j(a)d(Y)m(etter-Drinfel'd)g(Hopf)h(algebra)f(quo-)257 914 y(tien)o(t)f(of)f Fo(A)h Fp(in)f(whic)o(h)h(the)g(equiv)n(alence)g (class)g(of)g Fo(u)f Fp(is)h(equal)f(to)h(the)g(unit.)f(Since)h(action) g(and)257 964 y(coaction)16 b(are)h(induced)f(b)o(y)g Fo(u)p Fp(,)g(they)g(b)q(ecome)g(trivial)f(in)g(this)i(quotien)o(t,)e (and)h(therefore)257 1014 y(this)e(quotien)o(t)f(is)g(an)g(ordinary)f (Hopf)h(algebra)g(whic)o(h)g(is,)f(in)h(addition,)f(co)q(comm)o(utativ) o(e.)257 1064 y(Therefore,)j(it)f(is)f(isomorphic)g(to)g(a)h(group)g (ring.)257 1149 y(W)m(e)k(shall)g(use)h(the)g(usual)g(notation)e Fo(U)904 1134 y Fn(+)951 1149 y Fp(:=)i(k)o(er\()p Fo(\017)1102 1155 y Fj(U)1130 1149 y Fp(\))g(=)h Fo(U)d Fm(\\)12 b Fp(k)o(er)7 b Fo(\017)1381 1155 y Fj(A)1408 1149 y Fp(.)18 b(Observ)o(e)i(that,)257 1199 y(since)c(the)g(pro)q(duct)g(of)e Fo(u)h Fp(and)g(a)g(grouplik)o(e)f(elemen)o(t)g Fo(g)j Fp(is)e(again)f(a)g(grouplik)o(e)g(elemen)o(t,)257 1249 y Fo(G)p Fp(\()p Fo(A)p Fp(\))g(can)g(b)q(e)h(decomp)q(osed)f(in)o(to)f (orbits)h(with)g(resp)q(ect)i(to)d(left)h(m)o(ultiplication)c(b)o(y)k Fo(u)p Fp(.)257 1342 y Fq(Prop)q(osition)308 1392 y Fp(1.)20 b Fo(u)9 b Fm(\000)h Fp(1)p Fo(;)d(u)500 1377 y Fn(2)526 1392 y Fm(\000)j Fp(1)p Fo(;)d(:)g(:)g(:)k(;)c(u)712 1377 y Fj(p)p Fl(\000)p Fn(1)783 1392 y Fm(\000)i Fp(1)14 b(is)f(a)h(basis)g(of)f Fo(U)1117 1377 y Fn(+)1145 1392 y Fp(.)308 1475 y(2.)20 b Fo(AU)425 1460 y Fn(+)470 1475 y Fp(is)d(a)h(t)o(w)o(o-sided)f(ideal,)f(a)h(t)o(w)o(o-sided)h (coideal,)e(an)h Fo(H)s Fp(-submo)q(dule)f(and)i(an)361 1525 y Fo(H)s Fp(-sub)q(como)q(dule)e(of)h Fo(A)g Fp(that)g(is)g(in)o (v)n(arian)o(t)e(with)i(resp)q(ect)i(to)e(the)h(an)o(tip)q(o)q(de)e Fo(S)1650 1531 y Fj(A)1678 1525 y Fp(.)361 1574 y Fo(A=)n(AU)475 1559 y Fn(+)516 1574 y Fp(is)e(therefore)h(a)f(Y)m(etter-Drinfel'd)f (Hopf)h(algebra.)308 1657 y(3.)20 b(If)i Fo(g)431 1663 y Fn(1)450 1657 y Fo(;)7 b(:)g(:)g(:)k(;)c(g)569 1663 y Fj(n)617 1657 y Fm(2)26 b Fo(G)p Fp(\()p Fo(A)p Fp(\))c(is)h(a)f (system)g(of)g(represen)o(tativ)o(es)j(for)d(the)h(orbits)g(of)361 1707 y(the)d(action)e(of)h Fo(u)g Fp(on)f Fo(G)p Fp(\()p Fo(A)p Fp(\),)h(the)g(equiv)n(alence)h(classes)h(\026)-22 b Fo(g)1310 1713 y Fn(1)1328 1707 y Fo(;)7 b(:)g(:)g(:)12 b(;)c Fp(\026)-22 b Fo(g)1448 1713 y Fj(n)1489 1707 y Fp(form)17 b(a)i(ba-)361 1757 y(sis)d(of)f Fo(A=)n(AU)584 1742 y Fn(+)626 1757 y Fp(consisting)h(of)e(grouplik)o(e)h(elemen)o (ts.)g(Therefore,)h(the)g(dimension)361 1807 y(of)d Fo(A=)n(AU)522 1792 y Fn(+)563 1807 y Fp(is)610 1791 y Fn(1)p 610 1798 18 2 v 610 1821 a Fj(p)639 1807 y Fp(dim)5 b Fo(A)p Fp(.)308 1897 y(4.)20 b(Action)c(and)f(coaction)g(of)g Fo(H)j Fp(on)d Fo(A=)n(AU)1020 1882 y Fn(+)1063 1897 y Fp(are)h(trivial.)d Fo(A=)n(AU)1384 1882 y Fn(+)1427 1897 y Fp(is)i(therefore)i(an)361 1947 y(ordinary)i(co)q(comm)o(utativ)o(e)f(Hopf)h(algebra,)g Fo(G)p Fp(\()p Fo(A=)n(AU)1262 1932 y Fn(+)1289 1947 y Fp(\))i(=)h Fm(f)q Fp(\026)-22 b Fo(g)1421 1953 y Fn(1)1439 1947 y Fo(;)7 b(:)g(:)g(:)12 b(;)c Fp(\026)-22 b Fo(g)1559 1953 y Fj(n)1581 1947 y Fm(g)19 b Fp(is)h(a)361 1997 y(group,)13 b(and)h(w)o(e)g(ha)o(v)o(e)g Fo(A=)n(AU)842 1982 y Fn(+)881 1986 y Fm(\030)881 1999 y Fp(=)924 1997 y Fo(K)s Fp([)p Fo(G)p Fp(\()p Fo(A=)n(AU)1137 1982 y Fn(+)1164 1997 y Fp(\)].)257 2097 y Fq(Pro)q(of.)36 b Fp(The)18 b(\014rst)h(statemen)o(t)e(holds)h(b)q(ecause)h(1)p Fo(;)7 b(u)k Fm(\000)h Fp(1)p Fo(;)7 b(u)1251 2081 y Fn(2)1280 2097 y Fm(\000)12 b Fp(1)p Fo(;)7 b(:)g(:)g(:)k(;)c(u)1468 2081 y Fj(p)p Fl(\000)p Fn(1)1541 2097 y Fm(\000)12 b Fp(1)17 b(is)h(a)257 2146 y(basis)i(of)f Fo(U)5 b Fp(.)19 b(It)g(is)g(ob)o(vious)g(that)h Fo(AU)895 2131 y Fn(+)941 2146 y Fp(is)g(a)f(left)g(ideal)g(of)g Fo(A)p Fp(.)g(W)m(e)g(no)o(w)g (pro)o(v)o(e)g(that)257 2196 y Fo(AU)321 2181 y Fn(+)360 2196 y Fp(=)12 b Fo(U)437 2181 y Fn(+)465 2196 y Fo(A)p Fp(,)h(whic)o(h)g(implies)f(that)i Fo(AU)934 2181 y Fn(+)975 2196 y Fp(is)f(also)g(a)g(righ)o(t)h(ideal.)e(Since)i(the)g(grouplik)o (e)257 2246 y(elemen)o(ts)f(form)e(a)i(basis)f(of)h Fo(A)p Fp(,)f(this)h(will)e(follo)o(w)g(if)h(w)o(e)h(can)g(pro)o(v)o(e)g(that) g Fo(U)5 b(g)13 b Fp(=)e Fo(g)q(U)18 b Fp(for)13 b(all)257 2296 y(grouplik)o(e)i(elemen)o(ts)h Fo(g)g Fm(2)f Fo(G)p Fp(\()p Fo(A)p Fp(\).)g(If)g Fo(g)i Fp(is)f(not)g(in)o(v)n(arian)o(t)e (or)i(not)g(coin)o(v)n(arian)o(t,)e(w)o(e)i(ha)o(v)o(e)257 2346 y(b)o(y)e(Prop)q(osition)f(7.2)g(that)601 2437 y Fo(\036)p Fp(\()p Fo(g)q Fp(\))f(=)g Fo(u)759 2420 y Fj(i)772 2437 y Fo(g)h Fp(=)f Fo(g)q(u)894 2420 y Fj(j)953 2437 y Fp(or)42 b Fo( )q Fp(\()p Fo(g)q Fp(\))12 b(=)g Fo(u)1193 2420 y Fj(i)1207 2437 y Fo(g)g Fp(=)g Fo(g)q(u)1328 2420 y Fj(j)942 2628 y Fp(101)p eop %%Page: 102 102 102 101 bop 257 262 a Fp(for)11 b(some)g(nonzero)h(elemen)o(ts)f Fo(i;)c(j)14 b Fm(2)d Fk(Z)873 268 y Fj(p)889 262 y Fp(,)g(whic)o(h)g (implies)f(the)i(assertion.)f(If)g Fo(g)i Fp(is)e(in)o(v)n(arian)o(t) 257 311 y(and)21 b(coin)o(v)n(arian)o(t,)d(c)o(ho)q(ose)k(a)e(grouplik) o(e)g(elemen)o(t)g Fo(g)1124 296 y Fl(0)1156 311 y Fp(that)h(is)f (neither)i(in)o(v)n(arian)o(t)c(nor)257 361 y(coin)o(v)n(arian)o(t,)13 b(whic)o(h)i(exists)h(b)o(y)f(Prop)q(osition)f(7.2.)g(Since)h Fo(g)q(g)1233 346 y Fl(0)1260 361 y Fp(is)g(also)f(neither)i(in)o(v)n (arian)o(t)257 411 y(nor)e(coin)o(v)n(arian)o(t,)e(w)o(e)i(then)h(ha)o (v)o(e)692 499 y Fo(g)713 482 y Fl(0)725 499 y Fo(u)c Fp(=)h Fo(u)828 482 y Fj(k)848 499 y Fo(g)869 482 y Fl(0)923 499 y Fp(and)41 b Fo(ug)q(g)1097 482 y Fl(0)1120 499 y Fp(=)12 b Fo(g)q(g)1206 482 y Fl(0)1218 499 y Fo(u)1242 482 y Fj(l)257 587 y Fp(for)k(some)f(nonzero)h(elemen)o(ts)g Fo(k)q(;)7 b(l)15 b Fm(2)f Fk(Z)899 593 y Fj(p)915 587 y Fp(.)i(This)f(implies)f Fo(ug)q(g)1248 572 y Fl(0)1275 587 y Fp(=)h Fo(g)q(u)1367 572 y Fj(k)q(l)1398 587 y Fo(g)1419 572 y Fl(0)1446 587 y Fp(and)h(therefore)257 636 y Fo(ug)d Fp(=)f Fo(g)q(u)403 621 y Fj(k)q(l)434 636 y Fp(.)257 721 y(W)m(e)i(ha)o(v)o(e)h(\001)461 727 y Fj(A)487 721 y Fp(\()p Fo(U)536 705 y Fn(+)564 721 y Fp(\))d Fm(\032)h Fp(k)o(er\()p Fo(\017)725 727 y Fj(U)763 721 y Fm(\012)c Fo(\017)821 727 y Fj(U)849 721 y Fp(\))k(=)f Fo(U)955 705 y Fn(+)992 721 y Fm(\012)e Fo(U)k Fp(+)c Fo(U)15 b Fm(\012)9 b Fo(U)1235 705 y Fn(+)1263 721 y Fp(.)14 b(If)g Fo(a)e Fm(2)g Fo(A)i Fp(and)h Fo(a)1554 705 y Fl(0)1578 721 y Fm(2)d Fo(U)1651 705 y Fn(+)1678 721 y Fp(,)257 770 y(w)o(e)i(ha)o(v)o(e)488 858 y(\001)523 864 y Fj(A)550 858 y Fp(\()p Fo(aa)610 841 y Fl(0)622 858 y Fp(\))d(=)h Fo(a)715 865 y Fn(\(1\))760 858 y Fp(\()p Fo(a)798 865 y Fn(\(2\))842 841 y(\(1\))899 858 y Fm(!)f Fo(a)974 841 y Fl(0)974 870 y Fn(\(1\))1018 858 y Fp(\))e Fm(\012)h Fo(a)1107 865 y Fn(\(2\))1151 841 y(\(2\))1196 858 y Fo(a)1218 841 y Fl(0)1218 870 y Fn(\(2\))649 929 y Fp(=)i Fo(a)715 936 y Fn(\(1\))760 929 y Fo(a)782 911 y Fl(0)782 940 y Fn(\(1\))836 929 y Fm(\012)d Fo(a)899 936 y Fn(\(2\))944 929 y Fo(a)966 911 y Fl(0)966 940 y Fn(\(2\))1022 929 y Fm(2)i Fo(AU)1125 911 y Fn(+)1162 929 y Fm(\012)e Fo(A)g Fp(+)h Fo(A)f Fm(\012)h Fo(AU)1431 911 y Fn(+)257 1022 y Fp(Therefore,)17 b Fo(AU)523 1007 y Fn(+)566 1022 y Fp(is)e(a)h(t)o(w)o(o-sided)f(coideal.)g(Since)h Fo(u)f Fp(is)h(in)o(v)n(arian)o(t)e(and)h(coin)o(v)n(arian)o(t,)f(it) 257 1072 y(is)g(an)g Fo(H)s Fp(-submo)q(dule)e(and)i(an)g Fo(H)s Fp(-sub)q(como)q(dule.)e(Since)j(w)o(e)f(ha)o(v)o(e)390 1160 y Fo(S)415 1166 y Fj(A)443 1160 y Fp(\()p Fo(aa)503 1143 y Fl(0)515 1160 y Fp(\))d(=)h Fo(S)611 1166 y Fj(A)639 1160 y Fp(\()p Fo(a)677 1143 y Fn(\(1\))733 1160 y Fm(!)f Fo(a)808 1143 y Fl(0)819 1160 y Fp(\))p Fo(S)860 1166 y Fj(A)888 1160 y Fp(\()p Fo(a)926 1143 y Fn(\(2\))971 1160 y Fp(\))g(=)h Fo(S)1067 1166 y Fj(A)1095 1160 y Fp(\()p Fo(a)1133 1143 y Fl(0)1144 1160 y Fp(\))p Fo(S)1185 1166 y Fj(A)1213 1160 y Fp(\()p Fo(a)p Fp(\))g Fm(2)f Fo(U)1351 1143 y Fn(+)1378 1160 y Fo(A)h Fp(=)g Fo(AU)1529 1143 y Fn(+)257 1248 y Fp(for)i Fo(a)343 1233 y Fl(0)366 1248 y Fm(2)d Fo(U)438 1233 y Fn(+)466 1248 y Fp(,)i(it)h(is)f(in)o(v)n (arian)o(t)g(with)g(resp)q(ect)j(to)e(the)g(an)o(tip)q(o)q(de.)257 1332 y(T)m(o)21 b(pro)o(v)o(e)g(the)h(third)f(statemen)o(t,)g(observ)o (e)h(that,)f(since)h Fo(A)i Fp(=)1331 1301 y Fg(L)1377 1311 y Fj(n)1377 1344 y(i)p Fn(=1)1440 1332 y Fo(g)1460 1338 y Fj(i)1474 1332 y Fo(U)5 b Fp(,)20 b(w)o(e)i(ha)o(v)o(e)257 1382 y Fo(AU)321 1367 y Fn(+)360 1382 y Fp(=)404 1351 y Fg(L)450 1361 y Fj(n)450 1394 y(i)p Fn(=1)513 1382 y Fo(g)533 1388 y Fj(i)547 1382 y Fo(U)580 1367 y Fn(+)607 1382 y Fp(,)14 b(and)f(therefore:)652 1505 y Fo(A=)n(AU)766 1488 y Fn(+)805 1505 y Fp(=)870 1453 y Fj(n)849 1466 y Fg(M)853 1554 y Fj(i)p Fn(=1)919 1505 y Fo(g)939 1511 y Fj(i)952 1505 y Fo(U)r(=g)1023 1511 y Fj(i)1037 1505 y Fo(U)1070 1488 y Fn(+)1109 1505 y Fp(=)1174 1453 y Fj(n)1153 1466 y Fg(M)1157 1554 y Fj(i)p Fn(=1)1223 1505 y Fo(K)t Fp(\026)-22 b Fo(g)1281 1511 y Fj(i)257 1629 y Fp(Since)15 b(the)f Fo(n)g Fp(orbits)g(all)f(consist)h(of)f Fo(p)h Fp(elemen)o(ts,)f(w)o(e)i(ha)o(v)o(e)e(dim)5 b Fo(A)12 b Fp(=)g Fo(pn)p Fp(.)257 1713 y(T)m(o)h(pro)o(v)o(e)g(the)h (last)f(statemen)o(t,)g(observ)o(e)h(that,)f(if)g Fo(g)f Fm(2)g Fo(G)p Fp(\()p Fo(A)p Fp(\))h(is)g(an)g(arbitrary)g(grouplik)o (e)257 1763 y(elemen)o(t,)g(w)o(e)h(ha)o(v)o(e)g(b)o(y)g(Prop)q (osition)f(7.2)g(that)736 1851 y Fo(\036)p Fp(\()p Fo(g)q Fp(\))e(=)h Fo(g)q(u)914 1834 y Fj(i)1011 1851 y Fo( )q Fp(\()p Fo(g)q Fp(\))h(=)f Fo(g)q(u)1194 1834 y Fj(j)257 1939 y Fp(for)i(some)e Fo(i;)7 b(j)14 b Fm(2)d Fk(Z)558 1945 y Fj(p)574 1939 y Fp(.)i(This)h(implies)d(that)j Fo(\036)p Fp(\()p Fo(g)q Fp(\))9 b Fm(\000)g Fo(g)k Fp(=)f Fo(g)q Fp(\()p Fo(u)1190 1924 y Fj(i)1212 1939 y Fm(\000)d Fp(1\))j Fm(2)f Fo(AU)1405 1924 y Fn(+)1433 1939 y Fp(,)i(whic)o(h)g (means)257 1989 y(that)h Fo(\036)p Fp(,)f(and)h(similarly)d Fo( )q Fp(,)j(induces)h(the)f(iden)o(tit)o(y)f(on)h Fo(A=)n(AU)1246 1973 y Fn(+)1273 1989 y Fp(.)f(Therefore,)i(action)f(and)257 2038 y(coaction)g(on)g Fo(A=)n(AU)593 2023 y Fn(+)634 2038 y Fp(are)g(trivial.)e(The)i(remaining)e(assertions)j(are)f(ob)o (vious.)f Fi(2)257 2172 y Fq(7.4)48 b Fp(W)m(e)17 b(in)o(tro)q(duce)h (some)e(more)h(notation.)f(W)m(e)h(denote)h(b)o(y)f Fo(G)g Fp(:=)g Fo(G)p Fp(\()p Fo(A=)n(AU)1572 2157 y Fn(+)1598 2172 y Fp(\))h(the)257 2222 y(group)13 b(of)e(grouplik)o(e)h(elemen)o (ts)g(of)g Fo(A=)n(AU)926 2207 y Fn(+)953 2222 y Fp(.)f(In)i(the)g(pro) q(of)f(of)f(Prop)q(osition)h(7.3,)f(w)o(e)i(ha)o(v)o(e)257 2272 y(seen)22 b(that,)d(for)h(all)f Fo(g)k Fm(2)f Fo(G)p Fp(\()p Fo(A)p Fp(\),)e(there)h(exists)g Fo(j)j Fm(2)e Fk(Z)1170 2278 y Fj(p)1207 2272 y Fp(suc)o(h)f(that)f Fo(g)q(u)i Fp(=)g Fo(u)1548 2257 y Fj(j)1565 2272 y Fo(g)q Fp(.)e(The)257 2322 y(n)o(um)o(b)q(er)13 b Fo(j)j Fp(ob)o(viously)c (only)h(dep)q(ends)i(on)f(the)g Fo(u)p Fp(-orbit)f(of)g Fo(g)q Fp(.)g(W)m(e)g(therefore)i(ha)o(v)o(e)f(a)f(map)257 2371 y Fo(\027)h Fp(:)d Fo(G)g Fm(!)g Fk(Z)443 2356 y Fl(\002)443 2382 y Fj(p)482 2371 y Fp(suc)o(h)j(that)869 2430 y Fo(g)q(u)d Fp(=)h Fo(u)993 2413 y Fj(\027)r Fn(\()q(\026)-18 b Fj(g)q Fn(\))1057 2430 y Fo(g)257 2503 y Fp(where)15 b Fk(Z)408 2488 y Fl(\002)408 2513 y Fj(p)445 2503 y Fp(=)c Fk(Z)519 2509 y Fj(p)538 2503 y Fm(n)r(f)p Fp(0)p Fm(g)i Fp(denotes)i(the)f(m)o(ultiplicativ)o(e)d(group)j(of)f(the)i (\014nite)f(\014eld)g Fk(Z)1583 2509 y Fj(p)1600 2503 y Fp(.)942 2628 y(102)p eop %%Page: 103 103 103 102 bop 257 262 a Fp(By)22 b(Prop)q(osition)f(7.2,)g(w)o(e)h(ha)o (v)o(e,)f(for)g(ev)o(ery)h(grouplik)o(e)f(elemen)o(t)g Fo(g)26 b Fm(2)e Fo(G)p Fp(\()p Fo(A)p Fp(\),)d(n)o(um-)257 311 y(b)q(ers)16 b Fo(i;)7 b(j)13 b Fm(2)e Fk(Z)480 317 y Fj(p)510 311 y Fp(suc)o(h)k(that:)736 392 y Fo(\036)p Fp(\()p Fo(g)q Fp(\))c(=)h Fo(u)893 375 y Fj(i)907 392 y Fo(g)84 b( )q Fp(\()p Fo(g)q Fp(\))13 b(=)f Fo(u)1173 375 y Fj(j)1190 392 y Fo(g)257 472 y Fp(Again)k(the)g(n)o(um)o(b)q(ers) g Fo(i)g Fp(and)g Fo(j)j Fp(only)c(dep)q(end)i(on)f(the)h(orbit)f(of)f Fo(g)j Fp(with)d(resp)q(ect)k(to)d(left)257 522 y(m)o(ultiplication)h (b)o(y)j Fo(u)p Fp(,)g(since)h Fo(u)f Fp(is)g(in)o(v)n(arian)o(t)f(and) h(coin)o(v)n(arian)o(t.)e(W)m(e)i(therefore)i(ha)o(v)o(e)257 572 y(mappings)12 b Fo(\013)g Fp(:)f Fo(G)g Fm(!)g Fk(Z)634 578 y Fj(p)664 572 y Fp(and)i Fo(\014)i Fp(:)c Fo(G)g Fm(!)g Fk(Z)932 578 y Fj(p)962 572 y Fp(suc)o(h)k(that:)685 658 y Fo(\036)p Fp(\()p Fo(g)q Fp(\))d(=)g Fo(u)843 641 y Fj(\013)p Fn(\()q(\026)-18 b Fj(g)p Fn(\))909 658 y Fo(g)85 b( )q Fp(\()p Fo(g)q Fp(\))12 b(=)g Fo(u)1175 641 y Fj(\014)q Fn(\()q(\026)-18 b Fj(g)q Fn(\))1240 658 y Fo(g)257 744 y Fq(Prop)q(osition)308 794 y Fp(1.)20 b Fo(\027)c Fp(:)d Fo(G)g Fm(!)h Fk(Z)555 779 y Fl(\002)555 805 y Fj(p)595 794 y Fp(is)h(a)g(group)g(homomorphism)o(.)c(Therefore,) 16 b Fk(Z)1331 800 y Fj(p)1362 794 y Fp(is)f(a)g Fo(G)p Fp(-mo)q(dule)e(via)361 844 y Fo(g)q(:i)f Fp(:=)f Fo(\027)s Fp(\()p Fo(g)q Fp(\))p Fo(i)p Fp(.)308 923 y(2.)20 b Fo(\013)e Fp(and)h Fo(\014)i Fp(are)e(1-co)q(cycles)g(with)g(resp)q (ect)h(to)f(this)f(mo)q(dule)f(structure,)k(i.)c(e.,)h(w)o(e)361 973 y(ha)o(v)o(e:)579 1022 y Fo(\013)p Fp(\()p Fo(st)p Fp(\))12 b(=)g Fo(\013)p Fp(\()p Fo(s)p Fp(\))e(+)f Fo(\027)s Fp(\()p Fo(s)p Fp(\))p Fo(\013)p Fp(\()p Fo(t)p Fp(\))42 b Fo(\014)r Fp(\()p Fo(st)p Fp(\))13 b(=)e Fo(\014)r Fp(\()p Fo(s)p Fp(\))g(+)f Fo(\027)s Fp(\()p Fo(s)p Fp(\))p Fo(\014)r Fp(\()p Fo(t)p Fp(\))361 1091 y(for)k(all)e Fo(s;)7 b(t)12 b Fm(2)f Fo(G)p Fp(.)257 1184 y Fq(Pro)q(of.)36 b Fp(F)m(or)14 b Fo(g)q(;)7 b(g)559 1169 y Fl(0)582 1184 y Fm(2)k Fo(G)p Fp(\()p Fo(A)p Fp(\),)j(w)o(e)g(ha)o(v)o(e:)582 1270 y Fo(u)606 1253 y Fj(\027)r Fn(\()q(\026)-18 b Fj(g)r Fn(\026)g Fj(g)672 1241 y Fh(0)683 1253 y Fn(\))698 1270 y Fo(g)q(g)740 1253 y Fl(0)764 1270 y Fp(=)12 b Fo(g)q(g)850 1253 y Fl(0)862 1270 y Fo(u)f Fp(=)h Fo(g)q(u)986 1253 y Fj(\027)r Fn(\()q(\026)-18 b Fj(g)1035 1241 y Fh(0)1046 1253 y Fn(\))1061 1270 y Fo(g)1082 1253 y Fl(0)1106 1270 y Fp(=)12 b Fo(u)1174 1253 y Fj(\027)r Fn(\()q(\026)-18 b Fj(g)p Fn(\))p Fj(\027)r Fn(\()q(\026)g Fj(g)1284 1241 y Fh(0)1295 1253 y Fn(\))1310 1270 y Fo(g)q(g)1352 1253 y Fl(0)257 1351 y Fp(This)14 b(implies)e(that)i Fo(\027)s Fp(\()q(\026)-22 b Fo(g)r Fp(\026)g Fo(g)665 1336 y Fl(0)677 1351 y Fp(\))11 b(=)h Fo(\027)s Fp(\()q(\026)-22 b Fo(g)q Fp(\))p Fo(\027)s Fp(\()q(\026)g Fo(g)886 1336 y Fl(0)897 1351 y Fp(\).)13 b(Similarly)m(,)d(w)o(e)k(ha)o(v)o(e:)328 1437 y Fo(u)352 1420 y Fj(\013)p Fn(\()q(\026)-18 b Fj(g)r Fn(\026)g Fj(g)421 1408 y Fh(0)432 1420 y Fn(\))447 1437 y Fo(g)q(g)489 1420 y Fl(0)513 1437 y Fp(=)12 b Fo(\036)p Fp(\()p Fo(g)q(g)640 1420 y Fl(0)652 1437 y Fp(\))g(=)f Fo(\036)p Fp(\()p Fo(g)q Fp(\))p Fo(\036)p Fp(\()p Fo(g)863 1420 y Fl(0)875 1437 y Fp(\))h(=)g Fo(u)971 1420 y Fj(\013)p Fn(\()q(\026)-18 b Fj(g)p Fn(\))1038 1437 y Fo(g)q(u)1083 1420 y Fj(\013)p Fn(\()q(\026)g Fj(g)1135 1408 y Fh(0)1146 1420 y Fn(\))1161 1437 y Fo(g)1182 1420 y Fl(0)903 1510 y Fp(=)12 b Fo(u)971 1493 y Fj(\013)p Fn(\()q(\026)-18 b Fj(g)p Fn(\))1038 1510 y Fo(u)1062 1493 y Fj(\027)r Fn(\()q(\026)g Fj(g)p Fn(\))p Fj(\013)p Fn(\()q(\026)g Fj(g)1175 1480 y Fh(0)1186 1493 y Fn(\))1201 1510 y Fo(g)q(g)1243 1493 y Fl(0)1267 1510 y Fp(=)12 b Fo(u)1335 1493 y Fj(\013)p Fn(\()q(\026)-18 b Fj(g)p Fn(\)+)p Fj(\027)r Fn(\()q(\026)g Fj(g)r Fn(\))p Fj(\013)p Fn(\()q(\026)g Fj(g)1539 1480 y Fh(0)1549 1493 y Fn(\))1565 1510 y Fo(g)q(g)1607 1493 y Fl(0)257 1591 y Fp(whic)o(h)14 b(implies)e Fo(\013)p Fp(\()q(\026)-22 b Fo(g)s Fp(\026)f Fo(g)602 1576 y Fl(0)614 1591 y Fp(\))12 b(=)g Fo(\013)p Fp(\()q(\026)-22 b Fo(g)q Fp(\))9 b(+)h Fo(\027)s Fp(\()q(\026)-22 b Fo(g)p Fp(\))p Fo(\013)p Fp(\()q(\026)g Fo(g)957 1576 y Fl(0)969 1591 y Fp(\).)14 b(The)g(pro)q(of)f(for)h Fo(\014)j Fp(is)c(similar.)e Fi(2)257 1701 y Fp(As)i(in)f(P)o(aragraph)f(3.3,)g(w)o(e)h(will)f (denote)i Fk(Z)940 1707 y Fj(p)968 1701 y Fp(b)o(y)1024 1707 y Fj(G)1052 1701 y Fk(Z)1083 1707 y Fj(p)1111 1701 y Fp(if)e(it)h(is)g(considered)h(as)g(a)e Fo(G)p Fp(-mo)q(dule)257 1751 y(via)i Fo(\027)j Fp(as)e(in)g(the)g(preceding)h(prop)q(osition.) 257 1882 y Fq(7.5)48 b Fp(The)17 b(fact)f(that)h Fo(A=)n(AU)745 1867 y Fn(+)788 1882 y Fp(is)g(a)f(quotien)o(t)g(coalgebra)g(of)g Fo(A)h Fp(leads)f(to)h(a)f(como)q(dule)257 1932 y(structure)c(of)d Fo(A)g Fp(o)o(v)o(er)h Fo(A=)n(AU)713 1917 y Fn(+)740 1932 y Fp(.)f(If)g Fo(\031)k Fp(:)e Fo(A)g Fm(!)g Fo(A=)n(AU)1067 1917 y Fn(+)1104 1932 y Fp(denotes)g(the)f(canonical)f(pro)r(jection,) 257 1982 y(w)o(e)14 b(in)o(tro)q(duce)h(the)g(righ)o(t)e(coaction)629 2062 y Fo(\016)647 2068 y Fj(G)687 2062 y Fp(:)e Fo(A)h Fm(!)f Fo(A)e Fm(\012)h Fo(A=)n(AU)1002 2045 y Fn(+)1029 2062 y Fo(;)d(a)k Fm(7!)g Fo(a)1156 2069 y Fn(\(1\))1210 2062 y Fm(\012)f Fp(\026)-22 b Fo(a)1273 2069 y Fn(\(2\))257 2143 y Fp(whic)o(h)10 b(should)f(b)q(e)i(distinguished)e(from)f(the)i (left)g(coaction)f Fo(\016)1206 2149 y Fj(A)1243 2143 y Fp(that)h(is)f(part)h(of)f(the)i(Y)m(etter-)257 2193 y(Drinfel'd)18 b(structure.)j(Since)f Fo(A=)n(AU)859 2178 y Fn(+)906 2181 y Fm(\030)906 2195 y Fp(=)958 2193 y Fo(K)s Fp([)p Fo(G)p Fp(])e(is)h(a)g(group)g(ring,)f Fo(A)h Fp(b)q(ecomes)g(a)g Fo(G)p Fp(-)257 2242 y(graded)13 b(v)o(ector)f(space,)h(where,)g(for)e(a)h(grouplik)o(e)f(elemen)o(t)g Fo(g)i Fm(2)e Fo(G)p Fp(\()p Fo(A)p Fp(\),)h(the)g(homogeneous)257 2292 y(comp)q(onen)o(t)i(corresp)q(onding)g(to)i(\026)-23 b Fo(g)13 b Fm(2)e Fo(G)j Fp(is:)700 2373 y Fo(A)732 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Fo(A=)n(AU)1315 284 y Fn(+)1343 299 y Fp(.)308 381 y(2.)20 b(F)m(or)14 b Fo(s)e Fm(2)f Fo(G)p Fp(,)i(the)h (homogeneous)f(comp)q(onen)o(t)g Fo(A)1132 387 y Fj(s)1164 381 y Fp(is:)693 469 y Fo(A)724 475 y Fj(s)754 469 y Fp(=)e(Span\()p Fo(g)923 475 y Fj(s)941 469 y Fo(;)c(ug)1004 475 y Fj(s)1021 469 y Fo(;)g(u)1064 452 y Fn(2)1082 469 y Fo(g)1102 475 y Fj(s)1119 469 y Fo(;)g(:)g(:)g(:)12 b(;)7 b(u)1243 452 y Fj(p)p Fl(\000)p Fn(1)1304 469 y Fo(g)1324 475 y Fj(s)1341 469 y Fp(\))361 557 y(In)14 b(particular,)f(w)o(e)h(ha)o(v)o(e)g Fo(A)804 563 y Fn(1)834 557 y Fp(=)e Fo(U)5 b Fp(.)308 639 y(3.)20 b(The)14 b(extension)h Fo(U)h Fm(\032)c Fo(A)i Fp(is)g(cleft)g(o)o(v)o(er)g Fo(K)s Fp([)p Fo(G)p Fp(])f(with)g(resp)q(ect)j(to)e(the)h(map)860 728 y Fo(K)s Fp([)p Fo(G)p Fp(])c Fm(!)g Fo(A;)c(s)12 b Fm(7!)f Fo(g)1173 734 y Fj(s)308 832 y Fp(4.)20 b Fo(A)14 b Fp(is)g(a)f(crossed)j(pro)q(duct)f(of)e Fo(U)19 b Fp(and)13 b Fo(K)s Fp([)p Fo(G)p Fp(],)g(with)g(co)q(cycle)738 920 y Fo(\033)g Fp(:)e Fo(G)e Fm(\002)g Fo(G)i Fm(!)g Fo(U;)c Fp(\()p Fo(s;)g(t)p Fp(\))12 b Fm(7!)f Fo(g)1195 926 y Fj(s)1212 920 y Fo(g)1232 926 y Fj(t)1247 920 y Fo(g)1268 903 y Fl(\000)p Fn(1)1267 931 y Fj(st)361 1009 y Fp(with)j(resp)q(ect)i(to)e(the)g Fo(G)p Fp(-mo)q(dule)e(structure)k (on)e Fo(U)k Fp(determined)c(b)o(y)g Fo(s:u)d Fp(=)h Fo(u)1604 994 y Fj(\027)r Fn(\()p Fj(s)p Fn(\))1666 1009 y Fp(.)257 1107 y Fq(Pro)q(of.)36 b Fp(The)16 b(\014rst)g(statemen)o(t) g(follo)o(ws)d(from)h(the)i(fact)f(that)h Fo(A=)n(AU)1391 1092 y Fn(+)1433 1107 y Fp(is)g(a)f(trivial)f Fo(H)s Fp(-)257 1157 y(como)q(dule:)f(F)m(or)g Fo(a;)7 b(a)592 1141 y Fl(0)615 1157 y Fm(2)k Fo(A)p Fp(,)i(w)o(e)h(ha)o(v)o(e:)274 1245 y Fo(\016)292 1251 y Fj(G)320 1245 y Fp(\()p Fo(aa)380 1228 y Fl(0)392 1245 y Fp(\))e(=)f Fo(a)485 1252 y Fn(\(1\))530 1245 y Fp(\()p Fo(a)568 1252 y Fn(\(2\))613 1228 y(\(1\))669 1245 y Fm(!)g Fo(a)744 1228 y Fl(0)744 1256 y Fn(\(1\))788 1245 y Fp(\))f Fm(\012)g Fp(\026)-22 b Fo(a)877 1252 y Fn(\(2\))922 1228 y(\(2\))967 1245 y Fp(\026)g Fo(a)988 1228 y Fl(0)988 1256 y Fn(\(2\))1044 1245 y Fp(=)12 b Fo(a)1110 1252 y Fn(\(1\))1155 1245 y Fo(a)1177 1228 y Fl(0)1177 1256 y Fn(\(1\))1230 1245 y Fm(\012)e Fp(\026)-21 b Fo(a)1294 1252 y Fn(\(2\))1339 1245 y Fp(\026)f Fo(a)1360 1228 y Fl(0)1360 1256 y Fn(\(2\))1416 1245 y Fp(=)12 b Fo(\016)1478 1251 y Fj(G)1507 1245 y Fp(\()p Fo(a)p Fp(\))p Fo(\016)1579 1251 y Fj(G)1607 1245 y Fp(\()p Fo(a)1645 1228 y Fl(0)1657 1245 y Fp(\))257 1368 y(T)m(o)e(pro)o(v)o(e) h(the)g(second)g(statemen)o(t,)f(observ)o(e)i(that,)e(since)h Fo(u)g Fp(is)f(homogeneous)f(of)h(degree)i(1,)257 1418 y(w)o(e)i(ha)o(v)o(e)g Fo(u)438 1403 y Fj(i)452 1418 y Fo(g)472 1424 y Fj(s)501 1418 y Fm(2)d Fo(A)571 1424 y Fj(s)589 1418 y Fp(.)i(Since)i(w)o(e)f(already)f(ha)o(v)o(e:)664 1512 y Fo(A)f Fp(=)751 1473 y Fg(M)750 1562 y Fj(s)p Fl(2)p Fj(G)821 1512 y Fp(Span\()p Fo(g)947 1518 y Fj(s)965 1512 y Fo(;)7 b(ug)1028 1518 y Fj(s)1045 1512 y Fo(;)g(:)g(:)g(:)k(;)c (u)1168 1495 y 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2471 y Fj(s)942 2628 y Fp(104)p eop %%Page: 105 105 105 104 bop 257 262 a Fp(where)14 b Fo(\020)j Fp(is)12 b(the)i(primitiv)o(e)d Fo(p)p Fp(-th)i(ro)q(ot)f(of)h(unit)o(y)f (\014xed)i(at)e(the)i(b)q(eginning)e(of)h(the)g(section.)257 311 y(Then)i Fo(e)385 317 y Fn(0)404 311 y Fp(\()p Fo(s)p Fp(\))p Fo(;)7 b(:)g(:)g(:)12 b(;)7 b(e)574 317 y Fj(p)p Fl(\000)p Fn(1)636 311 y Fp(\()p Fo(s)p Fp(\))14 b(is)g(a)g(basis)f(of) h Fo(A)958 317 y Fj(s)976 311 y Fp(.)f(These)i(basis)f(elemen)o(ts)g (satisfy:)831 403 y Fo(ue)874 409 y Fj(i)888 403 y Fp(\()p Fo(s)p Fp(\))f(=)e Fo(\020)1016 386 y Fj(i)1031 403 y Fo(e)1050 409 y Fj(i)1064 403 y Fp(\()p Fo(s)p Fp(\))257 494 y(W)m(e)j(therefore)i(see)f(that)f(these)i(elemen)o(ts)e (explicitly)f(dep)q(end)j(on)d(the)i(c)o(hosen)g(represen-)257 544 y(tativ)o(e,)g(but)g(only)g(up)g(to)g(a)g(ro)q(ot)g(of)g(unit)o(y:) f(T)m(o)g(replace)i Fo(g)1191 550 y Fj(s)1224 544 y Fp(b)o(y)f(another) h(represen)o(tativ)o(e)257 594 y Fo(u)281 579 y Fj(k)302 594 y Fo(g)322 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1763 y Fj(m)1134 1757 y Fp(\()p Fo(st)p Fp(\))j Fm(\012)f Fo(e)1270 1763 y Fj(i)p Fl(\000)p Fj(m)1340 1757 y Fp(\()p Fo(st)p Fp(\))257 1920 y(On)15 b(the)f(other)h(hand,)e(w)o(e)h(ha)o(v)o (e:)338 2050 y(\001)373 2056 y Fj(A)399 2050 y Fp(\()p Fo(e)434 2056 y Fj(i)449 2050 y Fp(\()p Fo(s)p Fp(\))p Fo(e)519 2057 y Fj(i\027)r Fn(\()p Fj(s)p Fn(\))594 2050 y Fp(\()p Fo(t)p Fp(\)\))e(=)g(\001)748 2056 y Fj(A)774 2050 y Fp(\()p Fo(\033)814 2056 y Fj(i)828 2050 y Fp(\()p Fo(s;)7 b(t)p Fp(\))p Fo(e)932 2056 y Fj(i)946 2050 y Fp(\()p Fo(st)p Fp(\)\))13 b(=)1090 1997 y Fj(p)p Fl(\000)p Fn(1)1090 2011 y Fg(X)1084 2099 y Fj(m)p Fn(=0)1163 2050 y Fo(\033)1187 2056 y Fj(i)1200 2050 y Fp(\()p Fo(s;)7 b(t)p Fp(\))p Fo(e)1304 2056 y Fj(m)1336 2050 y Fp(\()p Fo(st)p Fp(\))j Fm(\012)g Fo(e)1473 2056 y Fj(i)p Fl(\000)p Fj(m)1542 2050 y Fp(\()p Fo(st)p Fp(\))257 2177 y(By)15 b(comparing)d(co)q(e\016cien)o(ts,)i(w)o(e)g(get:)561 2269 y Fo(\033)585 2275 y Fj(i)599 2269 y Fp(\()p Fo(s;)7 b(t)p Fp(\))k(=)h Fo(\020)760 2252 y Fj(m)p Fn(\()p Fj(i)p Fl(\000)p Fj(m)p Fn(\))p Fj(\027)r Fn(\()p Fj(s)p Fn(\))p Fj(\014)q Fn(\()p Fj(s)p Fn(\))p Fj(\013)p Fn(\()p Fj(t)p Fn(\))1067 2269 y Fo(\033)1091 2275 y Fj(m)1122 2269 y Fp(\()p Fo(s;)7 b(t)p Fp(\))p Fo(\033)1231 2275 y Fj(i)p Fl(\000)p Fj(m)1301 2269 y Fp(\()p Fo(s;)g(t)p Fp(\))257 2360 y(By)15 b(replacing)e Fo(i)h Fp(b)o(y)g Fo(i)c Fp(+)f Fo(m)p Fp(,)14 b(w)o(e)g(arriv)o(e)g(at)g(the)g(assertion.)g Fi(2)942 2628 y Fp(106)p eop %%Page: 107 107 107 106 bop 257 262 a Fq(7.7)48 b Fp(W)m(e)15 b(shall)f(no)o(w,)g(in)g (the)i(case)g(where)g Fo(p)e Fp(is)h(o)q(dd,)g(pro)o(v)o(e)g(the)g (main)e(result,)i(namely)257 311 y(that)20 b Fo(A)g Fp(is)f(isomorphic) f(to)i(a)f(Y)m(etter-Drinfel'd)g(Hopf)g(algebra)g(of)g(the)i(form)c (consid-)257 361 y(ered)f(in)e(P)o(aragraph)g(3.4.)f(The)i(remaining)e (task)i(is)f(to)g(determine)h(the)g(solutions)f(of)g(the)257 411 y(ab)q(o)o(v)o(e)19 b(compatibilit)o(y)d(condition.)i(Note)h(that,) 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Fd(2)557 1157 y Fj(\027)r Fn(\()p Fj(s)p Fn(\))p Fj(\014)q Fn(\()p Fj(s)p Fn(\))p Fj(\013)p Fn(\()p Fj(t)p Fn(\))p Fj(=)p Fn(2)774 1174 y Fo(\033)798 1180 y Fj(i)p Fn(+)p Fj(j)853 1174 y Fp(\()p Fo(s;)7 b(t)p Fp(\))292 1249 y(=)12 b Fo(\020)357 1231 y Fl(\000)p Fj(i)395 1219 y Fd(2)411 1231 y Fj(\027)r Fn(\()p Fj(s)p Fn(\))p Fj(\014)q Fn(\()p Fj(s)p Fn(\))p Fj(\013)p Fn(\()p Fj(t)p Fn(\))p Fj(=)p Fn(2)629 1249 y Fo(\020)650 1231 y Fl(\000)p Fj(ij)r(\027)r Fn(\()p Fj(s)p Fn(\))p Fj(\014)q Fn(\()p Fj(s)p Fn(\))p Fj(\013)p Fn(\()p Fj(t)p Fn(\))888 1249 y Fo(\020)909 1231 y Fl(\000)p Fj(j)951 1219 y Fd(2)966 1231 y Fj(\027)r Fn(\()p Fj(s)p Fn(\))p Fj(\014)q Fn(\()p Fj(s)p Fn(\))p Fj(\013)p Fn(\()p Fj(t)p Fn(\))p Fj(=)p Fn(2)1184 1249 y Fo(\020)1205 1231 y Fj(ij)r(\027)r Fn(\()p Fj(s)p Fn(\))p Fj(\014)q Fn(\()p Fj(s)p Fn(\))p Fj(\013)p Fn(\()p Fj(t)p Fn(\))1417 1249 y Fo(\033)1441 1255 y Fj(i)1454 1249 y Fp(\()p Fo(s;)7 b(t)p Fp(\))p Fo(\033)1563 1255 y Fj(j)1581 1249 y Fp(\()p Fo(s;)g(t)p Fp(\))292 1311 y(=)14 b(~)-23 b Fo(\033)360 1317 y Fj(i)376 1311 y Fp(~)g Fo(\033)398 1317 y Fj(j)257 1401 y Fp(This)15 b(means)e(that)k(~)-23 b Fo(\033)15 b Fp(de\014nes)h(a)e(c)o(haracter)i(to)e(the)h(base)g (\014eld.)f(Therefore,)h(there)h(exists)257 1450 y(an)f(elemen)o(t)g Fo(q)q Fp(\()p Fo(s;)7 b(t)p Fp(\))14 b Fm(2)f Fk(Z)662 1456 y Fj(p)678 1450 y Fp(,)i(dep)q(ending)g(on)g Fo(s)h Fp(and)f Fo(t)p Fp(,)f(suc)o(h)i(that)i(~)-24 b Fo(\033)1332 1456 y Fj(i)1360 1450 y Fp(=)14 b Fo(\020)1427 1435 y Fj(iq)q Fn(\()p Fj(s;t)p Fn(\))1521 1450 y Fp(,)g(i.)h(e.,)f(w)o(e)257 1507 y(ha)o(v)o(e)g Fo(\033)377 1513 y Fj(i)391 1507 y Fp(\()p Fo(s;)7 b(t)p Fp(\))k(=)h Fo(\020)552 1492 y Fj(iq)q Fn(\()p Fj(s;t)p Fn(\))646 1507 y Fo(\020)667 1492 y Fj(i)679 1480 y Fd(2)696 1492 y Fj(\027)r Fn(\()p Fj(s)p Fn(\))p Fj(\014)q Fn(\()p Fj(s)p Fn(\))p Fj(\013)p Fn(\()p Fj(t)p Fn(\))p Fj(=)p Fn(2)913 1507 y Fp(.)257 1592 y(W)m(e)21 b(no)o(w)f(pro)o(v)o(e)g(that)h(the)g(so-de\014ned)h Fo(q)g Fp(is)e(a)g(normalized)f(2-co)q(cycle.)i(The)g(co)q(cycle)257 1642 y(condition)16 b(for)g Fo(\033)q Fp(,)g(i.)g(e.,)g(the)h(equalit)o (y)f Fo(r)o(:\033)q Fp(\()p Fo(s;)7 b(t)p Fp(\))p Fo(\033)q Fp(\()p Fo(r)o(;)g(st)p Fp(\))15 b(=)h Fo(\033)q Fp(\()p Fo(r)q(s;)7 b(t)p Fp(\))p Fo(\033)q Fp(\()p Fo(r)o(;)g(s)p Fp(\),)16 b(implies)f(for)257 1692 y(the)g(comp)q(onen)o(ts)e(that:)650 1782 y Fo(\033)674 1789 y Fj(i\027)r Fn(\()p Fj(r)q Fn(\))748 1782 y Fp(\()p Fo(s;)7 b(t)p Fp(\))p Fo(\033)857 1788 y Fj(i)871 1782 y Fp(\()p Fo(r)o(;)g(st)p Fp(\))k(=)h Fo(\033)1053 1788 y Fj(i)1067 1782 y Fp(\()p Fo(r)q(s;)7 b(t)p Fp(\))p Fo(\033)1196 1788 y Fj(i)1209 1782 y Fp(\()p Fo(r)o(;)g(s)p Fp(\))257 1871 y(As)16 b(explained)e(in)g(P)o(aragraph)h (3.3,)e(w)o(e)i(can)g(use)h(the)f(isomorphism)d Fk(Z)1396 1877 y Fj(p)1422 1871 y Fm(\012)1454 1877 y Fc(Z)1486 1871 y Fk(Z)1517 1877 y Fj(p)1546 1860 y Fm(\030)1546 1873 y Fp(=)1591 1871 y Fk(Z)1622 1877 y Fj(p)1653 1871 y Fp(to)257 1921 y(regard)j(the)f(cup)g(pro)q(duct)g Fo(\014)e Fm([)c Fo(\013)k Fm(2)f Fo(Z)871 1906 y Fn(2)890 1921 y Fp(\()p Fo(G;)c Fk(Z)988 1927 y Fj(p)1013 1921 y Fm(\012)1045 1927 y Fc(Z)1075 1921 y Fk(Z)1106 1927 y Fj(p)1122 1921 y Fp(\))14 b(as)f(an)h(elemen)o(t)f(of)g Fo(Z)1491 1906 y Fn(2)1510 1921 y Fp(\()p Fo(G;)7 b Fk(Z)1608 1927 y Fj(p)1624 1921 y Fp(\),)13 b(if)257 1971 y(the)i Fo(G)p Fp(-mo)q(dule)d(structure)k(on)d Fk(Z)788 1977 y Fj(p)818 1971 y Fp(is)h(c)o(hosen)h(correctly)m(.)f(W)m(e)f(then)i (ha)o(v)o(e:)707 2070 y Fo(\033)731 2076 y Fj(i)745 2070 y Fp(\()p Fo(s;)7 b(t)p Fp(\))k(=)h Fo(\020)906 2053 y Fj(iq)q Fn(\()p Fj(s;t)p Fn(\))1001 2070 y Fo(\020)1022 2053 y Fj(i)1034 2040 y Fd(2)1050 2053 y Fn(\()p Fj(\014)q Fl([)p Fj(\013)p Fn(\)\()p Fj(s;t)p Fn(\))p Fj(=)p Fn(2)257 2159 y Fp(The)j(ab)q(o)o(v)o(e)e(condition)g(then)i(reads:)327 2249 y Fo(\020)348 2232 y Fj(i\027)r Fn(\()p Fj(r)q Fn(\))p Fj(q)q Fn(\()p Fj(s;t)p Fn(\))503 2249 y Fo(\020)524 2232 y Fj(i)536 2219 y Fd(2)552 2232 y Fj(\027)r Fn(\()p Fj(r)q Fn(\))613 2219 y Fd(2)629 2232 y Fn(\()p Fj(\014)q Fl([)p Fj(\013)p Fn(\)\()p Fj(s;t)p Fn(\))p Fj(=)p Fn(2)819 2249 y Fo(\020)840 2232 y Fj(iq)q Fn(\()p Fj(r)o(;st)p Fn(\))949 2249 y Fo(\020)970 2232 y Fj(i)982 2219 y Fd(2)998 2232 y Fn(\()p Fj(\014)q Fl([)p Fj(\013)p Fn(\)\()p Fj(r)o(;st)p Fn(\))p Fj(=)p Fn(2)831 2324 y Fp(=)d Fo(\020)896 2307 y Fj(iq)q Fn(\()p Fj(r)q(s;t)p Fn(\))1006 2324 y Fo(\020)1027 2307 y Fj(i)1039 2294 y Fd(2)1055 2307 y Fn(\()p Fj(\014)q Fl([)p Fj(\013)p Fn(\)\()p Fj(r)q(s;t)p Fn(\))p Fj(=)p Fn(2)1262 2324 y Fo(\020)1283 2307 y Fj(iq)q Fn(\()p Fj(r)o(;s)p Fn(\))1379 2324 y Fo(\020)1400 2307 y Fj(i)1412 2294 y Fd(2)1428 2307 y Fn(\()p Fj(\014)q Fl([)p Fj(\013)p Fn(\)\()p Fj(r)o(;s)p Fn(\))p Fj(=)p Fn(2)257 2413 y Fp(Since)j(w)o(e)f(already)f(kno)o(w)h(that)g Fo(\014)e Fm([)d Fo(\013)k Fp(is)h(a)f(2-co)q(cycle,)i(this)e(implies:)627 2503 y Fo(\027)s Fp(\()p Fo(r)q Fp(\))p Fo(q)q Fp(\()p Fo(s;)7 b(t)p Fp(\))i(+)g Fo(q)q Fp(\()p Fo(r)o(;)e(st)p Fp(\))12 b(=)f Fo(q)q Fp(\()p Fo(r)o(;)c(s)p Fp(\))i(+)h Fo(q)q Fp(\()p Fo(r)q(s;)d(t)p Fp(\))942 2628 y(107)p eop %%Page: 108 108 108 107 bop 257 262 a Fp(and)19 b(therefore)g Fo(q)h Fp(is)e(a)g(co)q(cycle.)h(Since)g Fo(\033)g Fp(is)g(normalized,)d(w)o (e)i(ha)o(v)o(e)h Fo(\033)1425 268 y Fj(i)1438 262 y Fp(\(1)p Fo(;)7 b Fp(1\))18 b(=)i(1)e(for)257 311 y(all)h Fo(i)i Fm(2)f Fk(Z)435 317 y Fj(p)451 311 y Fp(.)f(F)m(rom)f(Lemma)e (1.13,)i(w)o(e)i(ha)o(v)o(e)f Fo(\014)r Fp(\(1\))i(=)h(0.)c(Therefore,) j(w)o(e)e(m)o(ust)g(ha)o(v)o(e)257 361 y Fo(q)q Fp(\(1)p Fo(;)7 b Fp(1\))k(=)h(0,)h(whic)o(h)h(means)f(that)h Fo(q)h Fp(is)e(normalized.)257 447 y(It)19 b(is)e(ob)o(vious)h(from)e (the)j(description)f(of)g(the)h(structure)h(elemen)o(ts)e(of)f Fo(A)1467 453 y Fj(G)1495 447 y Fp(\()p Fo(\013;)7 b(\014)r(;)g(q)q Fp(\))18 b(in)257 497 y(P)o(aragraph)k(3.4)e(and)i(the)g(description)g (of)f(the)h(structure)i(elemen)o(ts)d(of)g Fo(A)h Fp(in)f(P)o(ara-)257 546 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b(structure)k(on)d Fk(Z)719 1100 y Fj(p)750 1094 y Fp(is)g(trivial.)f(By)i(Theorem)f(7.7,) f Fo(A)h Fp(is)h(isomorphic)e(to)h(some)257 1144 y(algebra)h(of)g(the)g (t)o(yp)q(e)h Fo(A)651 1150 y Fj(p)670 1144 y Fp(\()p Fo(\013;)7 b(\014)r(;)g(q)q Fp(\),)14 b(for)h(group)g(homomorphi)o(sm)o (s)e Fo(\013;)7 b(\014)15 b Fm(2)f Fp(Hom)n(\()p Fk(Z)1592 1150 y Fj(p)1608 1144 y Fo(;)7 b Fk(Z)1657 1150 y Fj(p)1673 1144 y Fp(\))257 1194 y(and)19 b(a)f(normalized)f(2-co)q(cycle)i Fo(q)i Fm(2)e Fo(Z)899 1179 y Fn(2)918 1194 y Fp(\()p Fk(Z)965 1200 y Fj(p)981 1194 y Fo(;)7 b Fk(Z)1030 1200 y Fj(p)1046 1194 y Fp(\).)18 b(F)m(rom)f(Prop)q(osition)h(1.13.2,)e(w)o (e)j(see)257 1244 y(that)d Fo(A)g Fp(is)f(comm)o(utativ)o(e.)d(As)k (explained)f(in)h(P)o(aragraph)f(4.6,)f(there)j(are)f Fo(p)p Fp(\()p Fo(p)10 b Fm(\000)h Fp(1\))k(iso-)257 1294 y(morphism)c(classes)16 b(of)d(these)i(Y)m(etter-Drinfel'd)f(Hopf) f(algebras.)257 1379 y(In)k(the)h(case)g Fo(p)f Fp(=)g(2,)f(w)o(e)i(ha) o(v)o(e)f(card\()p Fo(G)p Fp(\))g(=)g(2,)f(and)h(therefore)i Fo(G)1347 1368 y Fm(\030)1347 1381 y Fp(=)1396 1379 y Fk(Z)1427 1385 y Fn(2)1443 1379 y Fp(.)d(F)m(rom)f(Theo-)257 1429 y(rem)d(7.8)g(and)g(the)i(discussion)f(in)f(P)o(aragraph)g(3.6,)g (w)o(e)g(see)i(that)f Fo(A)g Fp(is)f(isomorphic)f(to)i Fo(A)1662 1435 y Fn(+)257 1479 y Fp(or)g Fo(A)338 1485 y Fl(\000)367 1479 y Fp(.)f(Therefore,)i Fo(A)f Fp(is)g(comm)o(utativ)o (e.)d(Since,)j(b)o(y)g(Prop)q(osition)g(4.9,)e Fo(A)1441 1485 y Fn(+)1482 1479 y Fp(and)i Fo(A)1593 1485 y Fl(\000)1634 1479 y Fp(are)257 1529 y(not)h(isomorphic,)e(the)i(assertions)h(follo)o (w.)d Fi(2)942 2628 y Fp(110)p eop %%Page: 111 111 111 110 bop 257 268 a Fr(8)67 b(Semisimple)24 b(Hopf)e(algebras)g(of)g (dimension)h Fb(p)1564 247 y Fp(3)257 584 y Fq(8.1)48 b Fp(In)14 b(this)g(section,)h(w)o(e)f(assume)g(that)g Fo(K)j Fp(is)d(an)g(algebraically)e(closed)j(\014eld)f(of)f(c)o(har-) 257 634 y(acteristic)20 b(zero,)f(and)g(that)g Fo(p)f Fp(is)h(a)f(prime)g(n)o(um)o(b)q(er.)f Fo(B)k Fp(denotes)f(a)f (semisimple)d(Hopf)257 684 y(algebra)h(of)f(dimension)f Fo(p)676 669 y Fn(3)712 684 y Fp(o)o(v)o(er)i Fo(K)j Fp(that)d(is)f(neither)i(comm)o(utativ)o(e)c(nor)j(co)q(comm)o(uta-)257 734 y(tiv)o(e.)12 b(W)m(e)f(w)o(an)o(t)h(to)g(pro)o(v)o(e)g(that)g Fo(B)j Fp(is)c(a)h(Radford)f(bipro)q(duct)i(of)e(a)h(group)g(ring)f(of) h(a)g(group)257 783 y(of)e(order)h Fo(p)g Fp(and)f(a)g(Y)m (etter-Drinfel'd)g(Hopf)g(algebra)g(of)g(the)h(form)e(describ)q(ed)j (in)e(Section)h(3.)257 833 y(W)m(e)17 b(then)h(apply)e(this)h(result)h (to)f(giv)o(e)f(a)h(new)h(pro)q(of)e(of)h(the)g(theorem)g(of)f(A.)h (Masuok)n(a)257 883 y(that,)e(if)g Fo(p)g Fp(is)g(o)q(dd,)g(there)i (are)f Fo(p)10 b Fp(+)h(1)k(isomorphism)d(classes)17 b(of)d(semisimple)f(Hopf)i(alge-)257 933 y(bras)h(of)e(dimension)f Fo(p)615 918 y Fn(3)648 933 y Fp(that)i(are)g(neither)h(comm)o(utativ)o (e)c(nor)j(comm)o(utativ)n(e)e(\(cf.)h([50)o(]\).)257 983 y(In)i(Masuok)n(a's)g(approac)o(h,)f(these)j(Hopf)e(algebras)g(are) 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Fp(and)j(its)g(equiv)n(alence) g(class)g(in)g(a)f(quotien)o(t)h(group)g Fk(Z)1351 1423 y Fj(n)1371 1417 y Fp(.)257 1552 y Fq(8.2)48 b Fp(First,)14 b(w)o(e)g(determine)g(the)g(group)g Fo(G)p Fp(\()p Fo(B)r Fp(\))g(of)g(grouplik)o(e)f(elemen)o(ts)g(of)h Fo(B)r Fp(.)257 1652 y Fq(Prop)q(osition)33 b Fp(The)20 b(group)g Fo(G)p Fp(\()p Fo(B)r Fp(\))g(of)f(grouplik)o(e)g(elemen)o(ts)g(of)g Fo(B)j Fp(is)e(isomorphic)e(to)257 1702 y Fk(Z)288 1708 y Fj(p)313 1702 y Fm(\002)10 b Fk(Z)386 1708 y Fj(p)402 1702 y Fp(.)257 1801 y Fq(Pro)q(of.)36 b Fp(\(1\))21 b(By)13 b(a)f(result)h(of)f(G.)g(I.)g(Kac)h(and)f(A.)g(Masuok)n(a)g (\([33)o(],)f(Cor.)h(2,)g(p.)g(159,)f([49)o(],)257 1851 y(Thm.)16 b(1,)h(p.)g(736\),)f Fo(B)k Fp(con)o(tains)d(a)h(non)o (trivial)d(cen)o(tral)j(grouplik)o(e)f(elemen)o(t)g Fo(g)q Fp(.)g(By)h(the)257 1901 y(Nic)o(hols-Zo)q(eller)13 b(theorem,)f(the)h (order)g(of)f Fo(g)i Fp(is)e(a)h(p)q(o)o(w)o(er)f(of)g Fo(p)p Fp(;)g(w)o(e)h(can)g(therefore)h(assume)257 1951 y(b)o(y)j(Cauc)o(h)o(y's)g(theorem)g(\(cf.)f([3],)g(Chap.)g(2,)h (Exerc.)g(3,)g(p.)f(20,)g([25)o(],)g(Kap.)h(I,)g(Satz)g(7.4,)257 2001 y(p.)d(34,)g([34)o(],)f(Satz)i(3.9,)e(p.)h(41,)g([82)o(],)f(Chap.) h(2,)g Fm(x)g Fp(2,)g(p.)g(97\))g(that)h(its)f(order)h(is)g(exactly)f Fo(p)p Fp(.)257 2050 y(Denote)k(b)o(y)e Fo(R)h Fp(the)g(group)g(ring)f (spanned)i(b)o(y)e(the)h(p)q(o)o(w)o(ers)h(of)e Fo(g)q Fp(;)h Fo(R)f Fp(is)h(then)g(a)g(normal)257 2100 y(Hopf)10 b(subalgebra)g(of)g(dimension)e Fo(p)p Fp(.)i(By)g(the)h(normal)d (basis)j(theorem)e(\(cf.)h([72)o(],)g(Thm.)e(2.4,)257 2150 y(p.)i(300,)f([57)o(],)g(Cor.)h(8.4.7,)e(p.)h(142\),)g(the)i (corresp)q(onding)g(Hopf)f(algebra)f(quotien)o(t)h Fo(B)r(=B)r(R)1661 2135 y Fn(+)257 2200 y Fp(has)k(dimension)e Fo(p)548 2185 y Fn(2)567 2200 y Fp(.)h(By)h(a)f(theorem)h(of)f(A.)g(Masuok)n(a)g (\(cf.)h([49)o(],)e(Thm.)g(2,)h(p.)g(736\),)g(Hopf)257 2250 y(algebras)h(of)f(dimension)g Fo(p)684 2235 y Fn(2)716 2250 y Fp(are)h(group)g(rings.)257 2333 y(\(2\))21 b(Recall)c(that)h (groups)g(of)f(order)i Fo(p)877 2318 y Fn(2)913 2333 y Fp(are)f(isomorphic)e(to)i Fk(Z)1285 2339 y Fj(p)1313 2333 y Fm(\002)13 b Fk(Z)1388 2339 y Fj(p)1422 2333 y Fp(or)18 b Fk(Z)1508 2340 y Fj(p)1525 2332 y Fd(2)1558 2333 y Fp(\(cf.)f([3)o(],)257 2382 y(Chap.)c(2,)f(Exerc.)i(4,)e(p.)g (20,)g([25)o(],)g(Kap.)h(I,)f(Satz)h(6.10,)f(p.)g(31,)g([34)o(],)g (Satz)h(4.3,)f(p.)g(57,)g([82)o(],)257 2432 y(Chap.)17 b(1,)g Fm(x)h Fp(3,)f(p.)h(27\).)f(Assume)g(that)h(the)g(quotien)o(t)g Fo(B)r(=B)r(R)1272 2417 y Fn(+)1318 2432 y Fp(is)f(isomorphic)f(to)i (the)257 2482 y(group)f(ring)g(of)f Fk(Z)548 2489 y Fj(p)565 2481 y Fd(2)580 2482 y Fp(.)h(If)g Fo(\031)g Fp(:)g Fo(B)i Fm(!)d Fo(B)r(=B)r(R)951 2467 y Fn(+)997 2482 y Fp(denotes)i(the)g (corresp)q(onding)g(pro)r(jection,)942 2628 y(111)p eop %%Page: 112 112 112 111 bop 257 262 a Fp(the)15 b(set)g(of)e(coin)o(v)n(arian)o(t)f (elemen)o(ts)i(with)g(resp)q(ect)i(to)e(this)g(pro)r(jection)g(is)g (precisely)g Fo(R)p 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y(b)q(ecause)c(in)d(this)g(case)i Fo(G)p Fp(\()p Fo(B)r Fp(\))f(w)o(ould)e(b)q(e)i(con)o(tained)g(in)f (the)h(set)h(of)e(coin)o(v)n(arian)o(t)f(elemen)o(ts)257 539 y Fm(f)p Fo(b)16 b Fm(2)h Fo(B)i Fm(j)d Fp(\(id)6 b Fm(\012)p Fo(\031)q Fp(\)\001)600 545 y Fj(B)629 539 y Fp(\()p Fo(b)p Fp(\))17 b(=)f Fo(b)11 b Fm(\012)h Fp(1)p Fm(g)p Fp(,)k(whic)o(h)h(has)g(dimension)e Fo(p)i Fp(b)o(y)f(the)i (normal)c(basis)257 589 y(theorem.)257 674 y(The)j(group)e Fo(G)p Fp(\()p Fo(B)r Fp(\))578 663 y Fm(\030)578 676 y Fp(=)625 674 y Fk(Z)656 680 y Fj(p)682 674 y Fm(\002)c Fk(Z)756 680 y Fj(p)788 674 y Fp(is)16 b(elemen)o(tary)f(ab)q(elian,)g (and)h(therefore)h(ma)o(y)d(b)q(e)i(con-)257 724 y(sidered)e(as)f(a)g (v)o(ector)h(space)g(o)o(v)o(er)f(the)g(\014eld)g(with)g Fo(p)f Fp(elemen)o(ts.)h(Considered)h(in)e(this)h(w)o(a)o(y)m(,)257 774 y Fo(f)21 b Fp(is)15 b(a)g(linear)g(mapping)e(of)i(v)o(ector)h (spaces.)h(W)m(e)e(ha)o(v)o(e)g(just)h(seen)g(that)g(the)g(k)o(ernel)g (of)f Fo(f)257 823 y 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y(p.)c(1368,)e([65)o(],)h(Prop.)g(3,)g(p.)g(333,)g([20)o(],)g (Cor.)g(5.8,)f(p.)h(4885,)f([81)o(],)h(Prop.)h(2.14,)e(p.)h(22\).)g(W)m (e)257 1774 y(no)o(w)i(pro)o(v)o(e:)257 1872 y Fq(Prop)q(osition)33 b Fo(A)14 b Fp(is)g(comm)o(utativ)o(e.)257 1970 y Fq(Pro)q(of.)36 b Fp(Supp)q(ose)20 b(that)f Fo(W)24 b Fp(is)19 b(a)f(simple)g Fo(A)p Fp(-mo)q(dule,)e(and)j(denote)h(the)f(corresp)q(ond-)257 2020 y(ing)f(cen)o(trally)g(primitiv)o(e)f(idemp)q(oten)o(t)g(b)o(y)h Fo(e)p Fp(.)g(W)m(e)g(ha)o(v)o(e)h(to)f(pro)o(v)o(e)g(that)h(dim)5 b Fo(W)25 b Fp(=)19 b(1.)257 2070 y(By)c(Corollary)d(2.3,)h(there)i(is) f(a)f(simple)g Fo(B)r Fp(-mo)q(dule)g Fo(V)23 b Fp(suc)o(h)15 b(that)e Fo(e)f Fm(2)g Fo(\024)p Fp(\()p Fo(V)d Fp(\))14 b(\(cf.)g(De\014ni-)257 2120 y(tion)d(2.2\).)e(By)i(a)g(result)g(of)g (S.)f(Mon)o(tgomery)f(and)i(S.)f(J.)h(Withersp)q(o)q(on)g(\(cf.)g([58)o (],)f(Cor.)g(3.6,)257 2169 y(p.)h(325\),)g Fo(V)21 b Fp(has)11 b(dimension)f(1,)h Fo(p)p Fp(,)g Fo(p)824 2154 y Fn(2)843 2169 y 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(p.)h(335,)e([81)o(],)257 1062 y(Cor.)15 b(2.14,)e(p.)i(23\).)f(If)h (the)h(action)e(of)h Fo(H)j Fp(on)d Fo(A)g Fp(w)o(ere)h(trivial,)d(it)i (w)o(ould)f(follo)o(w)f(directly)257 1111 y(from)h(the)i(form)o(ula)d (for)i(the)h(m)o(ultiplication)c(of)j(the)h(Radford)f(bipro)q(duct)h (that)f Fo(B)k Fp(w)o(ere)257 1161 y(comm)o(utativ)o(e.)c(Similarly)l (,)g(if)i(the)i(coaction)e(w)o(ere)i(trivial,)e(it)g(w)o(ould)g(follo)o (w)g(from)f(the)257 1211 y(com)o(ultiplication)11 b(of)j(the)h(Radford) e(bipro)q(duct)i(that)f Fo(B)j Fp(w)o(ere)e(co)q(comm)o(utativ)o(e.)c (There-)257 1261 y(fore,)i(action)f(and)h(coaction)f(are)i(non)o (trivial,)c(and)j(therefore)h Fo(A)f Fp(is)g(non)o(trivial)e(b)o(y)h (Prop)q(o-)257 1311 y(sition)i(1.11.)257 1439 y Fq(8.5)48 b Fp(No)o(w)16 b(supp)q(ose)h(that)g Fo(p)f Fp(is)g(o)q(dd)g(and)g (that)g Fo(\020)k Fp(is)c(a)g(primitiv)o(e)e Fo(p)p Fp(-th)i(ro)q(ot)g (of)f(unit)o(y.)257 1489 y(Non)o(trivial,)10 b(co)q(comm)o(utativ)o(e,) 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b(is)g(a)g(normalized)e(2-co)q(cycle.)i (Denote)h(b)o(y)f Fo(\013)p Fp(,)f(resp.)h Fo(\014)r Fp(,)h(the)f(group)g(auto-)257 1877 y(morphism)e(of)h Fk(Z)528 1883 y Fj(p)557 1877 y Fp(giv)o(en)g(b)o(y)h(m)o (ultiplication)d(b)o(y)j Fo(a)p Fp(,)f(resp.)i Fo(b)p Fp(.)e(In)h(the)h(Radford)e(bipro)q(duct)257 1927 y Fo(B)288 1933 y Fj(p)308 1927 y Fp(\()p Fo(a;)7 b(b;)g(q)q Fp(\))k(:=)g Fo(A)535 1933 y Fj(p)554 1927 y Fp(\()p Fo(\013;)c(\014)r(;)g(q)q Fp(\))i Fm(\012)h Fo(H)s Fp(,)j(w)o(e)h(in)o(tro)q(duce)g(the)h(basis) 801 2001 y Fo(b)819 2007 y Fj(ij)r(k)878 2001 y Fp(:=)d Fo(e)953 2007 y Fj(i)976 2001 y Fm(\012)d Fo(c)1035 2007 y Fj(j)1062 2001 y Fm(\012)h Fo(d)1126 2007 y Fj(k)257 2075 y Fp(where)22 b Fo(i;)7 b(j;)g(k)23 b Fm(2)f Fk(Z)579 2081 y Fj(p)595 2075 y Fp(.)e(Here)h Fo(e)750 2081 y Fj(i)785 2075 y Fp(denotes,)g(as)g(in)f(P)o(aragraph)g(2.3,)f(the)i Fo(i)p Fp(-th)g(primitiv)o(e)257 2125 y(idemp)q(oten)o(t)c(in)g Fo(K)570 2110 y Fc(Z)592 2114 y Ff(p)609 2125 y Fp(,)g Fo(c)656 2131 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Fj(p)p Fl(\000)p Fn(1)257 2503 y Fp(w)o(ould)13 b(corresp)q(ond)j(to)e Fo(e)660 2509 y Fn(0)679 2503 y Fo(;)7 b(:)g(:)g(:)k(;)c(e)797 2509 y Fj(p)p Fl(\000)p Fn(1)873 2503 y Fp(.)942 2628 y(114)p eop %%Page: 115 115 115 114 bop 257 262 a Fp(With)14 b(resp)q(ect)i(to)e(this)f(basis,)h (the)g(structure)i(maps)d(of)g Fo(B)1188 268 y Fj(p)1208 262 y Fp(\()p Fo(a;)7 b(b;)g(q)q Fp(\))13 b(tak)o(e)h(the)g(form:)308 387 y(1.)20 b(Multiplication:)12 b Fo(b)662 393 y Fj(ij)r(k)709 387 y Fo(b)727 393 y Fj(lmn)801 387 y Fp(=)g Fo(\016)863 393 y Fj(k)q Fl(\000)p Fj(n;alm)998 387 y Fo(\016)1016 393 y Fj(il)1041 387 y Fo(\020)1062 372 y Fj(iq)q Fn(\()p Fj(j;m)p Fn(\))1171 387 y Fo(\020)1192 372 y Fj(abj)r(mi)1282 360 y Fd(2)1298 372 y Fj(=)p Fn(2)1334 387 y Fo(b)1352 393 y Fj(i;j)r Fn(+)p Fj(m;n)308 476 y Fp(2.)20 b(Unit:)13 b(1)f(=)544 445 y Fg(P)588 455 y Fj(p)p Fl(\000)p Fn(1)588 489 y Fj(i;k)q Fn(=0)679 476 y Fo(b)697 482 y Fj(i;)p Fn(0)p Fj(;k)308 571 y Fp(3.)20 b(Com)o(ultiplicatio)o(n:)11 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Fp(is)g(a)f(grouplik)o(e)g(elemen)o(t)h(whic)o(h)f(is)h(not)g (cen)o(tral.)308 938 y(3.)20 b(The)14 b(elemen)o(t)g Fo(\037)g Fp(is)f(a)h(cen)o(tral)g(grouplik)o(e)f(elemen)o(t)h(of)f Fo(B)1265 944 y Fj(p)1285 938 y Fp(\()p Fo(a;)7 b(b;)g(q)q Fp(\))1415 923 y Fl(\003)1433 938 y Fp(.)308 1018 y(4.)20 b(The)14 b(basis)g(elemen)o(t)g Fo(b)720 1024 y Fj(ij)r(k)781 1018 y Fp(satis\014es)525 1101 y Fo(g)545 1107 y Fj(Z)572 1101 y Fo(b)590 1107 y Fj(ij)r(k)649 1101 y Fp(=)e Fo(\020)714 1084 y Fj(i)728 1101 y Fo(b)746 1107 y Fj(ij)r(k)876 1101 y Fo(b)894 1107 y Fj(ij)r(k)941 1101 y Fo(g)961 1107 y Fj(N)1004 1101 y Fp(=)g Fo(\020)1069 1084 y Fj(k)1090 1101 y Fo(b)1108 1107 y Fj(ij)r(k)1238 1101 y Fp(\012\()p Fo(b)1302 1107 y Fj(ij)r(k)1350 1101 y Fp(\))f(=)h Fo(\020)1442 1084 y Fj(j)1460 1101 y Fo(b)1478 1107 y Fj(ij)r(k)361 1185 y Fp(Ev)o(ery)i(other)f(elemen)o(t)f(satisfying)g(these)j (equations)e(is)f(a)h(scalar)g(m)o(ultiple)d(of)j Fo(b)1631 1191 y Fj(ij)r(k)1678 1185 y Fp(.)257 1279 y Fq(Pro)q(of.)36 b Fo(g)443 1285 y Fj(Z)487 1279 y Fp(is)17 b(grouplik)o(e)f(since)j Fo(u)e Fp(is)g(coin)o(v)n(arian)o(t,)e(and)i(it)g(is)g(cen)o(tral)h (since)g Fo(u)f Fp(is)g(in-)257 1329 y(v)n(arian)o(t.)e(Similarly)l(,)e Fo(\037)j Fp(is)g(grouplik)o(e,)e(i.)h(e.,)h(a)g(c)o(haracter,)h(since) g Fo(!)g Fp(is)f Fo(H)s Fp(-linear,)f(and)h(it)257 1379 y(is)e(cen)o(tral)h(since)f Fo(!)i Fp(is)d(colinear.)h Fo(g)808 1385 y Fj(N)853 1379 y Fp(is)g(not)f(cen)o(tral)i(since)g (conjugation)e(b)o(y)g Fo(g)1510 1385 y Fj(N)1556 1379 y Fp(induces)257 1429 y(the)k(action)f(of)f Fo(c)525 1435 y Fn(1)560 1429 y Fp(on)h Fo(A)651 1435 y Fj(p)670 1429 y Fp(\()p Fo(\013;)7 b(\014)r(;)g(q)q Fp(\),)16 b(whic)o(h)g(is)g(non)o(trivial.)e(This)i(pro)o(v)o(es)g(the)h(\014rst) g(three)257 1479 y(assertions.)257 1561 y(It)f(is)f(easy)h(to)f(v)o (erify)g(that)g Fo(b)718 1567 y Fj(ij)r(k)781 1561 y Fp(in)g(fact)g(satis\014es)h(the)g(equations)g(stated)g(in)f(the)h (fourth)257 1611 y(assertion.)h(In)f(particular,)g(w)o(e)g(see)i(that)e (the)h(op)q(erators)g(`Left)f(m)o(ultiplication)d(b)o(y)j Fo(g)1640 1617 y Fj(Z)1666 1611 y Fp(',)257 1660 y(`Righ)o(t)f(m)o (ultiplicatio)o(n)e(b)o(y)j Fo(g)734 1666 y Fj(N)766 1660 y Fp(',)f(and)h(\012)g(comm)o(ute.)d(This)j(also)g(follo)o(ws)e (from)h(the)h(fact)257 1710 y(that)j(\012)f(is)g(an)g(algebra)g (homomorphi)o(sm)d(that,)j(since)h Fo(\037)p Fp(\()p Fo(g)1230 1716 y Fj(Z)1256 1710 y Fp(\))g(=)h Fo(!)q Fp(\()p Fo(u)p Fp(\))f(=)g(1)f(and)g(also)257 1760 y Fo(\037)p Fp(\()p Fo(g)319 1766 y Fj(N)351 1760 y Fp(\))11 b(=)h Fo(!)q Fp(\(1\))g(=)g(1,)h(satis\014es)i(\012\()p Fo(g)823 1766 y Fj(Z)850 1760 y Fp(\))5 b(=)g Fo(g)928 1766 y Fj(Z)954 1760 y Fo(\037)p Fp(\()p Fo(g)1016 1766 y Fj(Z)1043 1760 y Fp(\))g(=)g Fo(g)1121 1766 y Fj(Z)1161 1760 y Fp(and)14 b(\012\()p Fo(g)1308 1766 y Fj(N)1339 1760 y Fp(\))e(=)g Fo(g)1431 1766 y Fj(N)1462 1760 y Fo(\037)p Fp(\()p Fo(g)1524 1766 y Fj(N)1555 1760 y Fp(\))g(=)g Fo(g)1647 1766 y Fj(N)1678 1760 y Fp(.)257 1810 y(Since)k(the)g(basis)f Fo(b)561 1816 y Fj(ij)r(k)623 1810 y Fp(is)g(a)g(basis)g(of)f(sim)o (ultaneous)g(eigen)o(v)o(ectors)i(for)f(these)h(op)q(erators,)257 1860 y(the)f(sim)o(ultaneous)d(eigenspaces)k(cannot)e(ha)o(v)o(e)g(a)f (dimension)f(greater)j(than)f(one.)g Fi(2)257 1973 y Fp(Since)j(w)o(e)f(kno)o(w)g(from)e(Prop)q(osition)i(8.2)f(that)h Fo(B)1060 1979 y Fj(p)1080 1973 y Fp(\()p Fo(a;)7 b(b;)g(q)q Fp(\))15 b(con)o(tains)h Fo(p)1411 1958 y Fn(2)1445 1973 y Fp(grouplik)o(e)f(ele-)257 2022 y(men)o(ts,)e(w)o(e)g(see)i(that)e Fo(g)626 2028 y Fj(Z)666 2022 y Fp(and)g Fo(g)766 2028 y Fj(N)810 2022 y Fp(generate)i Fo(G)p Fp(\()p Fo(B)1056 2028 y Fj(p)1075 2022 y Fp(\()p Fo(a;)7 b(b;)g(q)q Fp(\)\).)12 b(In)i(addition,)d(since)j(not)g(all)257 2072 y(grouplik)o(e)e(elemen)o (ts)h(are)g(cen)o(tral,)g(the)g(subgroup)g(of)g(cen)o(tral)g(grouplik)o (e)f(elemen)o(ts)g(m)o(ust)257 2122 y(ha)o(v)o(e)i(order)h Fo(p)p Fp(,)e(and)h(is)f(therefore)j(generated)f(b)o(y)f Fo(g)1072 2128 y Fj(Z)1098 2122 y Fp(.)257 2254 y Fq(8.7)48 b Fp(W)m(e)12 b(ha)o(v)o(e)f(seen)i(in)f(Theorem)f(7.10)g(that,)g(up)h (to)g(isomorphism)o(,)d(there)k(are)f Fo(p)p Fp(\()p Fo(p)5 b Fm(\000)g Fp(1\))257 2304 y(isomorphism)i(classes)j(of)f(non)o (trivial,)e(co)q(comm)o(utativ)o(e,)g(cosemisimple)g(Y)m (etter-Drinfel'd)257 2354 y(Hopf)13 b(algebras)f(of)g(dimension)f Fo(p)780 2339 y Fn(2)812 2354 y Fp(o)o(v)o(er)h Fo(K)s Fp([)p Fk(Z)980 2360 y Fj(p)996 2354 y Fp(].)g(Ho)o(w)o(ev)o(er,)g(t)o (w)o(o)h(nonisomorphic)e(Y)m(etter-)257 2403 y(Drinfel'd)e(Hopf)h (algebras)g(ma)o(y)f(ha)o(v)o(e)h(isomorphic)e(Radford)i(bipro)q (ducts,)g(and)g(this)h(turns)257 2453 y(out)h(to)g(b)q(e)g(the)g(case)h (in)e(our)h(situation.)f(The)h(follo)o(wing)d(lemma)f(giv)o(es)k(a)f (\014rst)i(description)257 2503 y(of)h(the)g(form)e(of)i(suc)o(h)g(an)g (isomorphism:)942 2628 y(116)p eop %%Page: 117 117 117 116 bop 257 262 a Fq(Lemma)36 b Fp(Supp)q(ose)15 b(that)f Fo(a;)7 b(b;)g(a)800 246 y Fl(0)811 262 y Fo(;)g(b)848 246 y Fl(0)870 262 y Fm(2)k Fk(Z)940 246 y Fl(\002)940 272 y Fj(p)979 262 y Fp(and)j(that)g Fo(q)q(;)7 b(q)1209 246 y Fl(0)1232 262 y Fm(2)k Fo(Z)1302 246 y Fn(2)1321 262 y Fp(\()p Fk(Z)1368 268 y Fj(p)1384 262 y Fo(;)c Fk(Z)1433 268 y Fj(p)1449 262 y Fp(\))14 b(are)h(normal-)257 311 y(ized)g(2-co)q(cycles.)f(Supp)q(ose)h(that)714 403 y Fo(f)i Fp(:)11 b Fo(B)805 409 y Fj(p)824 403 y Fp(\()p Fo(a;)c(b;)g(q)q Fp(\))k Fm(!)g Fo(B)1049 409 y Fj(p)1069 403 y Fp(\()p Fo(a)1107 386 y Fl(0)1118 403 y Fo(;)c(b)1155 386 y Fl(0)1166 403 y Fo(;)g(q)1205 386 y Fl(0)1216 403 y Fp(\))257 494 y(is)15 b(a)g(Hopf)g(algebra)f(isomorphism.)e(Then)j (there)h(are)g Fo(r)o(;)7 b(s;)g(t)12 b Fm(2)h Fk(Z)1302 479 y Fl(\002)1302 504 y Fj(p)1327 494 y Fp(,)h Fo(u)g Fm(2)f Fk(Z)1462 500 y Fj(p)1494 494 y Fp(and)h(a)h(map)257 544 y Fo(\034)i Fp(:)11 b Fk(Z)345 550 y Fj(p)371 544 y Fm(\002)e Fk(Z)443 550 y Fj(p)468 544 y Fm(\002)h Fk(Z)541 550 y Fj(p)568 544 y Fm(!)h Fo(K)659 529 y Fl(\002)702 544 y Fp(suc)o(h)j(that)719 635 y Fo(f)t Fp(\()p Fo(b)777 641 y Fj(ij)r(k)825 635 y Fp(\))e(=)g Fo(\034)5 b Fp(\()p Fo(i;)i(j;)g(k)q Fp(\))p Fo(b)1062 618 y Fl(0)1062 645 y Fj(r)q(i;sj;tk)q Fn(+)p Fj(ui)257 726 y Fp(for)14 b(all)f Fo(i;)7 b(j;)g(k)k Fm(2)h Fk(Z)551 732 y Fj(p)567 726 y Fp(.)i(Here)h(the)f(basis)g(elemen)o(ts)g(in)g Fo(B)1114 732 y Fj(p)1133 726 y Fp(\()p Fo(a)1171 711 y Fl(0)1183 726 y Fo(;)7 b(b)1220 711 y Fl(0)1231 726 y Fo(;)g(q)1270 711 y Fl(0)1281 726 y Fp(\))14 b(are)g(denoted)h(b)o(y)f Fo(b)1613 711 y Fl(0)1613 738 y Fj(ij)r(k)1660 726 y Fp(.)257 826 y Fq(Pro)q(of.)36 b Fp(Denote)12 b(the)g(corresp)q(onding) h(ob)r(jects)f(of)f Fo(B)1112 832 y Fj(p)1132 826 y Fp(\()p Fo(a)1170 811 y Fl(0)1182 826 y Fo(;)c(b)1219 811 y Fl(0)1230 826 y Fo(;)g(q)1269 811 y Fl(0)1280 826 y Fp(\))k(b)o(y)h Fo(g)1384 811 y Fl(0)1383 838 y Fj(Z)1409 826 y Fo(;)7 b(g)1449 811 y Fl(0)1448 838 y Fj(N)1479 826 y Fo(;)g(\037)1524 811 y Fl(0)1535 826 y Fp(,)k(and)g(\012)1666 811 y Fl(0)1678 826 y Fp(.)257 876 y Fo(g)278 861 y Fl(0)277 887 y Fj(Z)320 876 y Fp(m)o(ust)k(b)q(e)i(the)g(image)e(of)g(a)h(cen)o(tral)h (grouplik)o(e)f(elemen)o(t)f(in)h Fo(B)1327 882 y Fj(p)1347 876 y Fp(\()p Fo(a;)7 b(b;)g(q)q Fp(\).)15 b(Therefore,)257 926 y(there)h(exists)e Fo(r)f Fm(2)e Fk(Z)579 911 y Fl(\002)579 936 y Fj(p)618 926 y Fp(suc)o(h)k(that:)871 1017 y Fo(f)t Fp(\()p Fo(g)932 1000 y Fj(r)931 1027 y(Z)958 1017 y Fp(\))d(=)g Fo(g)1051 1000 y Fl(0)1050 1027 y Fj(Z)257 1108 y Fp(Similarly)m(,)d Fo(g)462 1093 y Fl(0)461 1120 y Fj(N)505 1108 y Fp(m)o(ust)i(b)q(e)j(the)f(image)e(of)h(a)g(grouplik) o(e)g(elemen)o(t)g(in)g Fo(B)1339 1114 y Fj(p)1359 1108 y Fp(\()p Fo(a;)7 b(b;)g(q)q Fp(\))k(that)i(is)g(not)257 1158 y(cen)o(tral.)h(Therefore,)h(there)g(exist)f Fo(t)e Fm(2)f Fk(Z)907 1143 y Fl(\002)907 1168 y Fj(p)946 1158 y Fp(and)i Fo(u)f Fm(2)f Fk(Z)1132 1164 y Fj(p)1162 1158 y Fp(suc)o(h)j(that:)843 1255 y Fo(f)t Fp(\()p Fo(g)904 1238 y Fj(u)903 1265 y(Z)930 1255 y Fo(g)951 1238 y Fj(t)950 1265 y(N)981 1255 y Fp(\))e(=)g Fo(g)1074 1238 y Fl(0)1073 1265 y Fj(N)257 1346 y Fp(Of)i(course,)g(this)g Fo(u)g Fp(is)f(di\013eren)o(t)i(from)d(the)i(grouplik)o(e)f(elemen)o(t)g(used) h(in)g(P)o(aragraph)f(8.6.)257 1396 y(Since)18 b(the)f(transp)q(ose)h Fo(f)656 1381 y Fl(\003)692 1396 y Fp(of)e(maps)g(cen)o(tral)h (grouplik)o(es)f(to)g(cen)o(tral)h(grouplik)o(es,)f(there)257 1446 y(exists)f Fo(s)d Fm(2)f Fk(Z)473 1431 y Fl(\002)473 1456 y Fj(p)512 1446 y Fp(suc)o(h)j(that:)867 1496 y Fo(f)891 1478 y Fl(\003)911 1496 y Fp(\()p Fo(\037)953 1478 y Fl(0)965 1496 y Fp(\))d(=)h Fo(\037)1062 1478 y Fj(s)257 1570 y Fp(W)m(e)i(therefore)h(get)f(that:)275 1662 y(\012)305 1644 y Fl(0)326 1662 y Fm(\016)9 b Fo(f)16 b Fp(=)c(\(id)6 b Fm(\012)p Fo(\037)551 1644 y Fl(0)563 1662 y Fp(\))k Fm(\016)f Fp(\001)g Fm(\016)g Fo(f)16 b Fp(=)c(\(id)6 b Fm(\012)p Fo(\037)888 1644 y Fl(0)900 1662 y Fp(\))k Fm(\016)f Fp(\()p Fo(f)14 b Fm(\012)9 b Fo(f)t Fp(\))h Fm(\016)f Fp(\001)j(=)f Fo(f)j Fm(\016)9 b Fp(\(id)e Fm(\012)p Fo(\037)1397 1644 y Fj(s)1415 1662 y Fp(\))i Fm(\016)g Fp(\001)i(=)h Fo(f)i Fm(\016)9 b Fp(\012)1654 1644 y Fj(s)257 1789 y Fp(W)m(e)14 b(no)o(w)f(ha)o(v)o(e)h (the)h(follo)o(wing)c(equations:)604 1880 y Fo(g)625 1863 y Fl(0)624 1890 y Fj(Z)651 1880 y Fo(f)t Fp(\()p Fo(b)709 1886 y Fj(ij)r(k)757 1880 y Fp(\))h(=)g Fo(f)t Fp(\()p Fo(g)890 1863 y Fj(r)889 1890 y(Z)916 1880 y Fo(b)934 1886 y Fj(ij)r(k)982 1880 y Fp(\))f(=)h Fo(\020)1074 1863 y Fj(r)q(i)1104 1880 y Fo(f)t Fp(\()p Fo(b)1162 1886 y Fj(ij)r(k)1211 1880 y Fp(\))604 1949 y Fo(f)t Fp(\()p Fo(b)662 1955 y Fj(ij)r(k)711 1949 y Fp(\))p Fo(g)748 1932 y Fl(0)747 1959 y Fj(N)790 1949 y Fp(=)f Fo(f)t Fp(\()p Fo(b)891 1955 y Fj(ij)r(k)940 1949 y Fo(g)961 1932 y Fj(u)960 1959 y(Z)986 1949 y Fo(g)1007 1932 y Fj(t)1006 1959 y(N)1038 1949 y Fp(\))g(=)h Fo(\020)1130 1932 y Fj(tk)q Fn(+)p Fj(ui)1220 1949 y Fo(f)t Fp(\()p Fo(b)1278 1955 y Fj(ij)r(k)1326 1949 y Fp(\))604 2017 y(\012)634 2000 y Fl(0)646 2017 y Fp(\()p Fo(f)t Fp(\()p Fo(b)720 2023 y Fj(ij)r(k)768 2017 y Fp(\)\))g(=)g Fo(f)t Fp(\(\012)926 2000 y Fj(s)945 2017 y Fp(\()p Fo(b)979 2023 y Fj(ij)r(k)1026 2017 y Fp(\)\))g(=)g Fo(\020)1135 2000 y Fj(sj)1168 2017 y Fo(f)t Fp(\()p Fo(b)1226 2023 y Fj(ij)r(k)1275 2017 y Fp(\))257 2108 y(By)j(Prop)q(osition)e(8.6,)f Fo(f)t Fp(\()p Fo(b)682 2114 y Fj(ij)r(k)731 2108 y Fp(\))i(m)o(ust)f (b)q(e)h(prop)q(ortional)f(to)h Fo(b)1228 2093 y Fl(0)1228 2120 y Fj(r)q(i;sj;tk)q Fn(+)p Fj(ui)1394 2108 y Fp(.)g Fi(2)257 2213 y Fq(Prop)q(osition)33 b Fp(Supp)q(ose)20 b(that)g Fo(a;)7 b(b;)g(a)900 2198 y Fl(0)910 2213 y Fo(;)g(b)947 2198 y Fl(0)978 2213 y Fm(2)20 b Fk(Z)1057 2198 y Fl(\002)1057 2223 y Fj(p)1101 2213 y Fp(and)f(that)g Fo(q)q(;)7 b(q)1341 2198 y Fl(0)1373 2213 y Fm(2)20 b Fo(Z)1452 2198 y Fn(2)1470 2213 y Fp(\()p Fk(Z)1517 2219 y Fj(p)1533 2213 y Fo(;)7 b Fk(Z)1583 2219 y Fj(p)1599 2213 y Fp(\))19 b(are)257 2263 y(normalized)10 b(2-co)q(cycles.)j(Supp) q(ose)f(that)g Fo(B)950 2269 y Fj(p)969 2263 y Fp(\()p Fo(a;)7 b(b;)g(q)q Fp(\))k(and)g Fo(B)1219 2269 y Fj(p)1239 2263 y Fp(\()p Fo(a)1277 2248 y Fl(0)1289 2263 y Fo(;)c(b)1326 2248 y Fl(0)1337 2263 y Fo(;)g(q)1376 2248 y Fl(0)1387 2263 y Fp(\))12 b(are)g(isomorphic.)257 2313 y(Then)i(there)h(are)f Fo(r)o(;)7 b(t)k Fm(2)g Fk(Z)673 2298 y Fl(\002)673 2323 y Fj(p)711 2313 y Fp(suc)o(h)k(that)e Fo(a)916 2298 y Fl(0)939 2313 y Fp(=)988 2296 y Fj(ta)p 988 2303 31 2 v 995 2327 a(r)1024 2313 y Fp(,)g Fo(b)1067 2298 y Fl(0)1090 2313 y Fp(=)1146 2296 y Fj(b)p 1138 2303 29 2 v 1138 2327 a(r)q(t)1172 2313 y Fp(,)g(and)h Fo(q)g Fp(and)g Fo(r)q(q)1432 2298 y Fl(0)1457 2313 y Fp(are)g(cohomol-)257 2363 y(ogous.)942 2628 y(117)p eop %%Page: 118 118 118 117 bop 257 262 a Fq(Pro)q(of.)36 b Fp(\(1\))21 b(Supp)q(ose)16 b(that)g Fo(f)j Fp(:)14 b Fo(B)849 268 y Fj(p)868 262 y Fp(\()p Fo(a;)7 b(b;)g(q)q Fp(\))14 b Fm(!)f Fo(B)1098 268 y Fj(p)1118 262 y Fp(\()p Fo(a)1156 246 y Fl(0)1168 262 y Fo(;)7 b(b)1205 246 y Fl(0)1216 262 y Fo(;)g(q)1255 246 y Fl(0)1266 262 y Fp(\))15 b(is)h(a)f(Hopf)g(algebra)g(iso-)257 311 y(morphism.)d(By)j(the)h(preceding)g(lemma,)c(there)k(are)g Fo(r)o(;)7 b(s;)g(t)12 b Fm(2)h Fk(Z)1290 296 y Fl(\002)1290 322 y Fj(p)1315 311 y Fp(,)i Fo(u)e Fm(2)g Fk(Z)1451 317 y Fj(p)1467 311 y Fp(,)h(and)h(a)g(map)257 361 y Fo(\034)i Fp(:)11 b Fk(Z)345 367 y Fj(p)371 361 y Fm(\002)e Fk(Z)443 367 y Fj(p)468 361 y Fm(\002)h Fk(Z)541 367 y Fj(p)568 361 y Fm(!)h Fo(K)659 346 y Fl(\002)702 361 y Fp(suc)o(h)j(that)g Fo(f)t Fp(\()p Fo(b)943 367 y Fj(ij)r(k)991 361 y Fp(\))e(=)g Fo(\034)5 b Fp(\()p Fo(i;)i(j;)g(k)q Fp(\))p Fo(b)1228 346 y Fl(0)1228 373 y Fj(r)q(i;sj;tk)q Fn(+)p Fj(ui)1394 361 y Fp(.)13 b(W)m(e)h(ha)o(v)o(e:)260 497 y(\001\()p Fo(f)t Fp(\()p Fo(b)369 503 y Fj(ij)r(k)417 497 y Fp(\)\))e(=)g Fo(\034)5 b Fp(\()p Fo(i;)i(j;)g(k)q Fp(\))674 444 y Fj(p)p Fl(\000)p Fn(1)674 458 y Fg(X)659 547 y Fj(l;m)p Fn(=0)757 497 y Fo(\020)778 480 y Fj(b)793 468 y Fh(0)804 480 y Fj(r)q Fn(\()p Fj(i)p Fl(\000)p Fj(l)p Fn(\))p Fj(sj)r(m)957 497 y Fo(b)975 480 y Fl(0)975 508 y Fj(r)q(l;sj;m)1092 497 y Fm(\012)i Fo(b)1151 480 y Fl(0)1151 509 y Fj(r)q Fn(\()p Fj(i)p Fl(\000)p Fj(l)p Fn(\))p Fj(;sj;tk)q Fn(+)p Fj(ui)p Fl(\000)p Fj(m)461 650 y Fp(=)j Fo(\034)5 b Fp(\()p Fo(i;)i(j;)g(k)q Fp(\))674 597 y Fj(p)p Fl(\000)p Fn(1)674 611 y Fg(X)659 700 y Fj(l;m)p Fn(=0)757 650 y Fo(\020)778 633 y Fj(b)793 621 y Fh(0)804 633 y Fj(r)q Fn(\()p Fj(i)p Fl(\000)p Fj(l)p Fn(\))p Fj(sj)r Fn(\()p Fj(tm)p Fn(+)p Fj(ul)p Fn(\))1051 650 y Fo(b)1069 633 y Fl(0)1069 661 y Fj(r)q(l;sj;tm)p Fn(+)p Fj(ul)1254 650 y Fm(\012)j Fo(b)1314 633 y Fl(0)1314 662 y Fj(r)q Fn(\()p Fj(i)p Fl(\000)p Fj(l)p Fn(\))p Fj(;sj;t)p Fn(\()p Fj(k)q Fl(\000)p Fj(m)p Fn(\)+)p Fj(u)p Fn(\()p Fj(i)p Fl(\000)p Fj(l)p Fn(\))257 797 y Fp(On)15 b(the)f(other)h(hand,)e(w)o(e)h(ha)o(v)o(e:)296 885 y(\()p Fo(f)g Fm(\012)c Fo(f)t Fp(\)\001\()p Fo(b)497 891 y Fj(ij)r(k)545 885 y Fp(\))h(=)319 939 y Fj(p)p Fl(\000)p Fn(1)319 954 y Fg(X)303 1043 y Fj(l;m)p Fn(=0)402 993 y Fo(\020)423 976 y Fj(b)p Fn(\()p Fj(i)p Fl(\000)p Fj(l)p Fn(\))p Fj(j)r(m)559 993 y Fo(\034)5 b Fp(\()p Fo(l)q(;)i(j;)g(m)p Fp(\))p Fo(\034)e Fp(\()p Fo(i)k Fm(\000)h Fo(l)q(;)d(j;)g(k)i Fm(\000)h Fo(m)p Fp(\))p Fo(b)1033 976 y Fl(0)1033 1003 y Fj(r)q(l;sj;tm)p Fn(+)p Fj(ul)1218 993 y Fm(\012)g Fo(b)1278 976 y Fl(0)1278 1004 y Fj(r)q Fn(\()p Fj(i)p Fl(\000)p Fj(l)p Fn(\))p Fj(;sj;t)p Fn(\()p Fj(k)q Fl(\000)p Fj(m)p Fn(\)+)p Fj(u)p Fn(\()p Fj(i)p Fl(\000)p Fj(l)p Fn(\))257 1140 y Fp(By)15 b(comparing)d(co)q(e\016cien)o(ts,)i(w)o(e)g (get:)419 1227 y Fo(\034)5 b Fp(\()p Fo(i;)i(j;)g(k)q Fp(\))p Fo(\020)587 1210 y Fj(b)602 1198 y Fh(0)612 1210 y Fj(r)q Fn(\()p Fj(i)p Fl(\000)p Fj(l)p Fn(\))p Fj(sj)r Fn(\()p Fj(tm)p Fn(+)p Fj(ul)p Fn(\))871 1227 y Fp(=)12 b Fo(\034)5 b Fp(\()p Fo(i)k Fm(\000)h Fo(l)q(;)d(j;)g(k)i Fm(\000)h Fo(m)p Fp(\))p Fo(\034)5 b Fp(\()p Fo(l)q(;)i(j;)g(m)p Fp(\))p Fo(\020)1392 1210 y Fj(b)p Fn(\()p Fj(i)p Fl(\000)p Fj(l)p Fn(\))p Fj(j)r(m)257 1315 y Fp(or,)14 b(b)o(y)f(replacing)h Fo(i)g Fp(with)g Fo(i)9 b Fp(+)h Fo(l)15 b Fp(and)f Fo(k)g Fp(with)g Fo(k)c Fp(+)g Fo(m)p Fp(:)481 1403 y Fo(\034)5 b Fp(\()p Fo(i)10 b Fp(+)f Fo(l)q(;)e(j;)g(k)j Fp(+)f Fo(m)p Fp(\))p Fo(\020)799 1386 y Fj(b)814 1373 y Fh(0)826 1386 y Fj(r)q(isj)r Fn(\()p Fj(tm)p Fn(+)p Fj(ul)p Fn(\))1022 1403 y Fp(=)j Fo(\034)5 b Fp(\()p Fo(i;)i(j;)g(k)q Fp(\))p Fo(\034)e Fp(\()p Fo(l)q(;)i(j;)g(m)p Fp(\))p Fo(\020)1393 1386 y Fj(bij)r(m)257 1507 y Fp(\(2\))21 b(The)14 b(last)g(equation)g (can)g(b)q(e)g(rewritten)h(in)f(the)g(form)615 1590 y Fo(\034)5 b Fp(\()p Fo(i;)i(j;)g(k)q Fp(\))p Fo(\034)e Fp(\()p Fo(l)q(;)i(j;)g(m)p Fp(\))p 615 1609 306 2 v 619 1647 a Fo(\034)e Fp(\()p Fo(i)10 b Fp(+)f Fo(l)q(;)e(j;)g(k)j Fp(+)f Fo(m)p Fp(\))937 1618 y(=)j Fo(\020)1002 1601 y Fn(\()p Fj(b)1030 1589 y Fh(0)1041 1601 y Fj(r)q(st)p Fl(\000)p Fj(b)p Fn(\))p Fj(ij)r(m)1198 1618 y Fo(\020)1219 1601 y Fj(b)1234 1589 y Fh(0)1245 1601 y Fj(r)q(suij)r(l)257 1728 y Fp(The)i(left)g(hand)f(side)h(of)f(this)g(equation)g(remains)g (in)o(v)n(arian)o(t)f(if)g(w)o(e)i(exc)o(hange)g(sim)o(ultane-)257 1778 y(ously)e Fo(i)h Fp(and)f Fo(l)h Fp(as)f(w)o(ell)g(as)g Fo(k)h Fp(and)g Fo(m)p Fp(.)f(Therefore,)h(w)o(e)f(ha)o(v)o(e)g Fo(\020)1222 1763 y Fn(\()p Fj(b)1250 1750 y Fh(0)1261 1763 y Fj(r)q(st)p Fl(\000)p Fj(b)p Fn(\))p Fj(ij)r(m)1430 1778 y Fp(=)f Fo(\020)1494 1763 y Fn(\()p Fj(b)1522 1750 y Fh(0)1533 1763 y Fj(r)q(st)p Fl(\000)p Fj(b)p Fn(\))p Fj(lj)r(k)1678 1778 y Fp(.)257 1828 y(This)16 b(yields)g(for)f Fo(i)g Fp(=)h Fo(j)h Fp(=)e Fo(m)g Fp(=)g(1)h(and)g Fo(k)f Fp(=)g Fo(l)h Fp(=)g(0)f(that)h Fo(\020)1223 1813 y Fj(b)1238 1800 y Fh(0)1249 1813 y Fj(r)q(st)p Fl(\000)p Fj(b)1351 1828 y Fp(=)f(1,)g(and)h(therefore)257 1878 y Fo(b)c Fp(=)g Fo(b)349 1862 y Fl(0)360 1878 y Fo(r)q(st)p Fp(.)257 1959 y(\(3\))21 b(W)m(e)13 b(no)o(w)g(argue)g(as)g(in)g(the)h(pro)q(of) e(of)h(Theorem)g(7.7.)e(Fix)i Fo(j)h Fm(2)d Fk(Z)1350 1965 y Fj(p)1366 1959 y Fp(.)i(F)m(or)f Fo(i;)7 b(k)13 b Fm(2)e Fk(Z)1601 1965 y Fj(p)1618 1959 y Fp(,)h(w)o(e)257 2009 y(de\014ne:)725 2059 y(~)-23 b Fo(\034)t Fp(\()p Fo(i;)7 b(k)q Fp(\))12 b(:=)f Fo(\034)5 b Fp(\()p Fo(i;)i(j;)g(k)q Fp(\))p Fo(\020)1068 2042 y Fj(b)1083 2029 y Fh(0)1094 2042 y Fj(r)q(suj)r(i)1174 2029 y Fd(2)1189 2042 y Fj(=)p Fn(2)257 2131 y Fp(Note)18 b(that,)g(since)g Fo(p)f Fp(is)h(o)q(dd,)f (2)g(is)h(an)f(in)o(v)o(ertible)g(elemen)o(t)g(of)g Fk(Z)1324 2137 y Fj(p)1340 2131 y Fp(,)g(and)h(therefore)h(the)257 2181 y(expression)c Fo(i=)p Fp(2)f(in)g(the)g(ab)q(o)o(v)o(e)g(form)o (ula)d(mak)o(es)i(sense.)i(W)m(e)e(then)i(ha)o(v)o(e:)422 2277 y(~)-23 b Fo(\034)t Fp(\()p Fo(i)10 b Fp(+)g Fo(l)q(;)d(k)i Fp(+)h Fo(m)p Fp(\))i(=)g Fo(\034)5 b Fp(\()p Fo(i)k Fp(+)h Fo(l)q(;)d(j;)g(k)i Fp(+)h Fo(m)p Fp(\))p Fo(\020)1055 2260 y Fj(b)1070 2247 y Fh(0)1082 2260 y Fj(r)q(suj)r Fn(\()p Fj(i)p Fn(+)p Fj(l)p Fn(\))1223 2247 y Fd(2)1239 2260 y Fj(=)p Fn(2)693 2351 y Fp(=)i Fo(\034)5 b Fp(\()p Fo(i)k Fp(+)h Fo(l)q(;)d(j;)g(k)i Fp(+)h Fo(m)p Fp(\))p Fo(\020)1055 2334 y Fj(b)1070 2322 y Fh(0)1082 2334 y Fj(r)q(suj)r(i)1162 2322 y Fd(2)1176 2334 y Fj(=)p Fn(2)1212 2351 y Fo(\020)1233 2334 y Fj(b)1248 2322 y Fh(0)1259 2334 y Fj(r)q(suj)r(il)1351 2351 y Fo(\020)1372 2334 y Fj(b)1387 2322 y Fh(0)1398 2334 y Fj(r)q(suj)r(l)1476 2322 y Fd(2)1492 2334 y Fj(=)p Fn(2)693 2426 y Fp(=)i Fo(\034)5 b Fp(\()p Fo(i;)i(j;)g(k)q Fp(\))p Fo(\034)e Fp(\()p Fo(l)q(;)i(j;)g(m)p Fp(\))p Fo(\020)1064 2409 y Fj(b)1079 2396 y Fh(0)1089 2409 y Fj(r)q(suj)r(i)1169 2396 y Fd(2)1184 2409 y Fj(=)p Fn(2)1220 2426 y Fo(\020)1241 2409 y Fj(b)1256 2396 y Fh(0)1267 2409 y Fj(r)q(suj)r(l)1345 2396 y Fd(2)1361 2409 y Fj(=)p Fn(2)693 2488 y Fp(=)14 b(~)-23 b Fo(\034)5 b Fp(\()p Fo(i;)i(k)q Fp(\))r(~)-23 b Fo(\034)5 b Fp(\()p Fo(l)q(;)i(m)p Fp(\))942 2628 y(118)p eop %%Page: 119 119 119 118 bop 257 262 a Fp(This)13 b(sho)o(ws)h(that)h(~)-23 b Fo(\034)17 b Fp(:)11 b Fk(Z)647 268 y Fj(p)671 262 y Fm(\002)d Fk(Z)742 268 y Fj(p)770 262 y Fm(!)j Fo(K)861 246 y Fl(\002)902 262 y Fp(is)i(a)g(group)g(homomorphism)n(.)d (Therefore,)k(there)257 311 y(exist,)e(for)g(ev)o(ery)h Fo(j)h Fm(2)d Fk(Z)636 317 y Fj(p)652 311 y Fp(,)h(elemen)o(ts)g Fo(v)q Fp(\()p Fo(j)r Fp(\))p Fo(;)7 b(w)q Fp(\()p Fo(j)r Fp(\))13 b Fm(2)e Fk(Z)1099 317 y Fj(p)1128 311 y Fp(suc)o(h)i(that)h (~)-23 b Fo(\034)5 b Fp(\()p Fo(i;)i(k)q Fp(\))k(=)h Fo(\020)1495 296 y Fj(v)q Fn(\()p Fj(j)r Fn(\))p Fj(i)p Fn(+)p Fj(w)q Fn(\()p Fj(j)r Fn(\))p Fj(k)1678 311 y Fp(,)257 361 y(i.)h(e.:)669 411 y Fo(\034)5 b Fp(\()p Fo(i;)i(j;)g(k)q Fp(\))k(=)h Fo(\020)892 394 y Fj(v)q Fn(\()p Fj(j)r Fn(\))p Fj(i)p Fn(+)p Fj(w)q Fn(\()p Fj(j)r Fn(\))p Fj(k)1075 411 y Fo(\020)1096 394 y Fl(\000)p Fj(b)1137 381 y Fh(0)1148 394 y Fj(r)q(suj)r(i)1228 381 y Fd(2)1242 394 y Fj(=)p Fn(2)257 488 y Fp(\(4\))21 b(W)m(e)14 b(ha)o(v)o(e:)293 569 y Fo(f)t Fp(\()p Fo(b)351 575 y Fj(ij)r(k)399 569 y Fp(\))p Fo(f)t Fp(\()p Fo(b)473 575 y Fj(ilm)528 569 y Fp(\))d(=)h Fo(\020)620 551 y Fj(v)q Fn(\()p Fj(j)r Fn(\))p Fj(i)p Fn(+)p Fj(w)q Fn(\()p Fj(j)r Fn(\))p Fj(k)q Fn(+)p Fj(v)q Fn(\()p Fj(l)p Fn(\))p Fj(i)p Fn(+)p Fj(w)q Fn(\()p Fj(l)p Fn(\))p Fj(m)1011 569 y Fo(\020)1032 551 y Fl(\000)p Fj(b)1073 539 y Fh(0)1084 551 y Fj(r)q(su)p Fn(\()p Fj(j)r Fn(+)p Fj(l)p Fn(\))p Fj(i)1225 539 y Fd(2)1241 551 y Fj(=)p Fn(2)1277 569 y Fo(b)1295 551 y Fl(0)1295 579 y Fj(r)q(i;sj;tk)q Fn(+)p Fj(ui)1461 569 y Fo(b)1479 551 y Fl(0)1479 579 y Fj(r)q(i;sl;tm)p Fn(+)p Fj(ui)555 644 y Fp(=)g Fo(\016)617 651 y Fj(t)p Fn(\()p Fj(k)q Fl(\000)p Fj(m)p Fn(\))p Fj(;a)758 643 y Fh(0)769 651 y Fj(r)q(isl)825 644 y Fo(\020)846 627 y Fj(i)p Fn(\()p Fj(v)q Fn(\()p Fj(j)r Fn(\)+)p Fj(v)q Fn(\()p Fj(l)p Fn(\)\)+)p Fj(w)q Fn(\()p Fj(j)r Fn(\))p Fj(k)q Fn(+)p Fj(w)q Fn(\()p Fj(l)p Fn(\))p Fj(m)1252 644 y Fo(\020)1273 627 y Fl(\000)p Fj(b)1314 614 y Fh(0)1325 627 y Fj(r)q(su)p Fn(\()p Fj(j)r Fn(+)p Fj(l)p Fn(\))p Fj(i)1466 614 y Fd(2)1482 627 y Fj(=)p Fn(2)599 719 y Fo(\020)620 702 y Fj(r)q(iq)664 689 y Fh(0)676 702 y Fn(\()p Fj(sj;sl)p Fn(\))769 719 y Fo(\020)790 702 y Fj(a)808 689 y Fh(0)820 702 y Fj(b)835 689 y Fh(0)846 702 y Fj(sj)r(sl)p Fn(\()p Fj(r)q(i)p Fn(\))958 689 y Fd(2)973 702 y Fj(=)p Fn(2)1009 719 y Fo(b)1027 702 y Fl(0)1027 730 y Fj(r)q(i;s)p Fn(\()p Fj(j)r Fn(+)p Fj(m)p Fn(\))p Fj(;tm)p Fn(+)p Fj(ui)257 805 y Fp(On)j(the)f(other)h(hand,)e (w)o(e)h(ha)o(v)o(e:)330 883 y Fo(f)t Fp(\()p Fo(b)388 889 y Fj(ij)r(k)437 883 y Fo(b)455 889 y Fj(ilm)508 883 y Fp(\))e(=)g Fo(\016)598 889 y Fj(k)q Fl(\000)p Fj(m;ail)724 883 y Fo(\020)745 866 y Fj(iq)q Fn(\()p Fj(j;l)p Fn(\))836 883 y Fo(\020)857 866 y Fj(abj)r(li)928 854 y Fd(2)944 866 y Fj(=)p Fn(2)979 883 y Fo(f)t Fp(\()p Fo(b)1037 889 y Fj(i;j)r Fn(+)p Fj(l;m)1153 883 y Fp(\))342 958 y(=)g Fo(\016)404 964 y Fj(k)q Fl(\000)p Fj(m;ail)530 958 y Fo(\020)551 941 y Fj(iq)q Fn(\()p Fj(j;l)p Fn(\))642 958 y Fo(\020)663 941 y Fj(abj)r(li)734 928 y Fd(2)749 941 y Fj(=)p Fn(2)785 958 y Fo(\020)806 941 y Fj(iv)q Fn(\()p Fj(j)r Fn(+)p Fj(l)p Fn(\)+)p Fj(mw)q Fn(\()p Fj(j)r Fn(+)p Fj(l)p Fn(\))1072 958 y Fo(\020)1093 941 y Fl(\000)p Fj(b)1134 928 y Fh(0)1145 941 y Fj(r)q(su)p Fn(\()p Fj(j)r Fn(+)p Fj(l)p Fn(\))p Fj(i)1286 928 y Fd(2)1303 941 y Fj(=)p Fn(2)1338 958 y Fo(b)1356 941 y Fl(0)1356 969 y Fj(r)q(i;s)p Fn(\()p Fj(j)r Fn(+)p Fj(m)p Fn(\))p Fj(;tm)p Fn(+)p Fj(ui)257 1044 y Fp(No)o(w)17 b(supp)q(ose)h(that)f Fo(k)12 b Fm(\000)f Fo(m)17 b Fp(=)g Fo(ail)q Fp(.)f(Then)i(w)o(e)f(ha)o(v)o(e)f Fo(f)t Fp(\()p Fo(b)1195 1050 y Fj(ij)r(k)1243 1044 y Fo(b)1261 1050 y Fj(ilm)1315 1044 y Fp(\))h Fm(6)p Fp(=)f(0,)g(and)h(therefore)257 1094 y(also)f Fo(f)t Fp(\()p Fo(b)401 1100 y Fj(ij)r(k)450 1094 y Fp(\))p Fo(f)t Fp(\()p Fo(b)524 1100 y Fj(ilm)578 1094 y Fp(\))h Fm(6)p Fp(=)f(0.)g(This)g(implies)e(that)j Fo(t)p Fp(\()p Fo(k)12 b Fm(\000)f Fo(m)p Fp(\))17 b(=)f Fo(a)1288 1079 y Fl(0)1300 1094 y Fo(r)q(isl)q Fp(.)h(In)f(particular,) g(in)257 1144 y(the)f(case)g Fo(k)d Fp(=)g Fo(a;)7 b(m)k Fp(=)h(0)p Fo(;)7 b(i)k Fp(=)h Fo(l)h Fp(=)f(1,)h(w)o(e)h(ha)o(v)o(e)g (that)f Fo(at)f Fp(=)g Fo(a)1212 1129 y Fl(0)1223 1144 y Fo(r)q(s)p Fp(.)257 1219 y(\(5\))21 b(Inserting)15 b(this)f(result)g(in)o(to)f(the)i(\014rst)g(calculation)d(ab)q(o)o(v)o (e,)i(w)o(e)g(get:)281 1292 y Fo(f)t Fp(\()p Fo(b)339 1298 y Fj(ij)r(k)387 1292 y Fp(\))p Fo(f)t Fp(\()p Fo(b)461 1298 y Fj(ilm)516 1292 y Fp(\))398 1367 y(=)e Fo(\016)460 1373 y Fj(k)q Fl(\000)p Fj(m;ail)587 1367 y Fo(\020)608 1350 y Fj(i)p Fn(\()p Fj(v)q Fn(\()p Fj(j)r Fn(\)+)p Fj(v)q Fn(\()p Fj(l)p Fn(\)\)+)p Fj(w)q Fn(\()p Fj(j)r Fn(\))p Fj(k)q Fn(+)p Fj(w)q Fn(\()p Fj(l)p Fn(\))p Fj(m)1013 1367 y Fo(\020)1034 1350 y Fl(\000)p Fj(b)1075 1337 y Fh(0)1086 1350 y Fj(r)q(su)p Fn(\()p Fj(j)r Fn(+)p Fj(l)p Fn(\))p Fj(i)1227 1337 y Fd(2)1243 1350 y Fj(=)p Fn(2)1279 1367 y Fo(\020)1300 1350 y Fj(r)q(iq)1344 1337 y Fh(0)1355 1350 y Fn(\()p Fj(sj;sl)p Fn(\))1449 1367 y Fo(\020)1470 1350 y Fj(abj)r(li)1541 1337 y Fd(2)1557 1350 y Fj(=)p Fn(2)1388 1429 y Fo(b)1406 1412 y Fl(0)1406 1441 y Fj(r)q(i;s)p Fn(\()p Fj(j)r Fn(+)p Fj(m)p Fn(\))p Fj(;tm)p Fn(+)p Fj(ui)257 1515 y Fp(In)h(the)f(case)i Fo(k)6 b Fm(\000)g Fo(m)12 b Fp(=)g Fo(ail)q Fp(,)g(a)g(comparison)f(with)h(the)h (expression)g(for)f Fo(f)t Fp(\()p Fo(b)1427 1521 y Fj(ij)r(k)1475 1515 y Fo(b)1493 1521 y Fj(ilm)1547 1515 y Fp(\))g(yields:)257 1571 y Fo(\020)278 1555 y Fj(iq)q Fn(\()p Fj(j;l)p Fn(\))369 1571 y Fo(\020)390 1555 y Fj(iv)q Fn(\()p Fj(j)r Fn(+)p Fj(l)p Fn(\)+)p Fj(mw)q Fn(\()p Fj(j)r Fn(+)p Fj(l)p Fn(\))672 1571 y Fp(=)k Fo(\020)741 1555 y Fj(i)p Fn(\()p Fj(v)q Fn(\()p Fj(j)r Fn(\)+)p Fj(v)q Fn(\()p Fj(l)p Fn(\)\)+)p Fj(k)q(w)q Fn(\()p Fj(j)r Fn(\)+)p Fj(mw)q Fn(\()p Fj(l)p Fn(\))1146 1571 y Fo(\020)1167 1555 y Fj(ir)q(q)1211 1543 y Fh(0)1223 1555 y Fn(\()p Fj(sj;sl)p Fn(\))1332 1571 y Fp(Therefore,)h(if)f(w)o(e)h(set)257 1620 y Fo(k)c Fp(=)f Fo(m)d Fp(+)h Fo(ail)q Fp(,)k(w)o(e)g(get:)257 1693 y Fo(iq)q Fp(\()p Fo(j;)7 b(l)q Fp(\))s Fm(\000)s Fo(ir)q(q)464 1676 y Fl(0)477 1693 y Fp(\()p Fo(sj;)g(sl)q Fp(\))13 b(=)e Fo(i)p Fp(\()p Fo(v)q Fp(\()p Fo(j)r Fp(\))s(+)s Fo(v)q Fp(\()q Fo(l)q Fp(\))s Fm(\000)s Fo(v)r Fp(\()p Fo(j)6 b Fp(+)s Fo(l)q Fp(\)\))t(+)s Fo(ail)r(w)q Fp(\()p Fo(j)r Fp(\))s(+)t Fo(m)p Fp(\()p Fo(w)q Fp(\()p Fo(j)s Fp(\))s(+)s Fo(w)q Fp(\()p Fo(l)r Fp(\))s Fm(\000)s Fo(w)q Fp(\()p Fo(j)g Fp(+)s Fo(l)q Fp(\)\))257 1767 y(for)k(all)e Fo(i;)f(j;)g(l)q(;)g(m)k Fm(2)g Fk(Z)588 1773 y Fj(p)604 1767 y Fp(.)e(F)m(or)h Fo(i)h Fp(=)h(0)e(and)f Fo(m)j Fp(=)g(1,)d(this)g(yields)h Fo(w)q Fp(\()p Fo(j)r Fp(\))g(+)f Fo(w)q Fp(\()p Fo(l)q Fp(\))h Fm(\000)f Fo(w)q Fp(\()p Fo(j)j Fp(+)e Fo(l)q Fp(\))h(=)h(0.)257 1816 y(Therefore,)19 b(w)o(e)f(ha)o(v)o(e)g Fo(w)q Fp(\()p Fo(j)r Fp(\))h(=)f Fo(w)q(j)j Fp(for)c(some)g Fo(w)i Fm(2)f Fk(Z)1147 1822 y Fj(p)1163 1816 y Fp(,)g(for)f(whic)o(h)h(w)o(e)g(use)h(the)f(same)257 1866 y(notation)c(as)g(for)g(the)g(function)g Fo(w)q Fp(.)g(On)g(the)h(other)g(hand,)e(w)o(e)i(get)f(for)g Fo(i)e Fp(=)g(1)i(and)g Fo(m)f Fp(=)f(0)257 1916 y(that:)526 1966 y Fo(q)q Fp(\()p Fo(j;)7 b(l)q Fp(\))j Fm(\000)f Fo(r)q(q)718 1949 y Fl(0)730 1966 y Fp(\()p Fo(sj;)e(sl)q Fp(\))12 b(=)g Fo(v)q Fp(\()p Fo(j)r Fp(\))f(+)e Fo(v)q Fp(\()p Fo(l)q Fp(\))i Fm(\000)e Fo(v)q Fp(\()p Fo(j)j Fp(+)e Fo(l)q Fp(\))g(+)f Fo(aw)q(j)r(l)257 2043 y Fp(\(6\))21 b(No)o(w)14 b(de\014ne)i(~)-22 b Fo(v)q Fp(\()p Fo(j)r Fp(\))13 b(:=)e Fo(v)q Fp(\()p Fo(j)r Fp(\))f Fm(\000)815 2026 y Fj(aw)p 815 2033 43 2 v 828 2057 a Fn(2)863 2043 y Fo(j)882 2028 y Fn(2)901 2043 y Fp(.)j(Then)i(w)o(e)f(ha)o(v)o(e:)296 2130 y(~)-23 b Fo(v)r Fp(\()p Fo(j)r Fp(\))10 b(+)h(~)-23 b Fo(v)r Fp(\()p Fo(l)q Fp(\))10 b Fm(\000)h Fp(~)-23 b Fo(v)r Fp(\()p Fo(j)12 b Fp(+)d Fo(l)q Fp(\))j(=)g Fo(v)q Fp(\()p Fo(j)r Fp(\))e(+)g Fo(v)q Fp(\()p Fo(l)q Fp(\))g Fm(\000)g Fo(v)q Fp(\()p Fo(j)i Fp(+)e Fo(l)q Fp(\))f(+)1164 2102 y Fo(aw)p 1164 2121 53 2 v 1180 2159 a Fp(2)1221 2130 y(\()p Fo(j)j Fp(+)e Fo(l)q Fp(\))1337 2113 y Fn(2)1365 2130 y Fm(\000)1412 2102 y Fo(aw)p 1412 2121 V 1428 2159 a Fp(2)1469 2130 y Fo(j)1488 2113 y Fn(2)1517 2130 y Fm(\000)1563 2102 y Fo(aw)p 1563 2121 V 1579 2159 a Fp(2)1621 2130 y Fo(l)1634 2113 y Fn(2)685 2210 y Fp(=)i Fo(v)q Fp(\()p Fo(j)r Fp(\))e(+)g Fo(v)q Fp(\()p Fo(l)q Fp(\))g Fm(\000)g Fo(v)q Fp(\()p Fo(j)i Fp(+)e Fo(l)q Fp(\))f(+)h Fo(aw)q(j)r(l)685 2272 y Fp(=)i Fo(q)q Fp(\()p Fo(j;)7 b(l)q Fp(\))i Fm(\000)h Fo(r)q(q)921 2255 y Fl(0)932 2272 y Fp(\()p Fo(sj;)d(sl)q Fp(\))257 2346 y(Therefore,)20 b Fo(q)h Fp(is)e(cohomologous)e(to)i(the)h(co)q (cycle)g(\()p Fo(j;)7 b(l)q Fp(\))21 b Fm(7!)f Fo(r)q(q)1305 2331 y Fl(0)1316 2346 y Fp(\()p Fo(sj;)7 b(sl)q Fp(\).)20 b(By)g(Prop)q(osi-)257 2395 y(tion)e(1.13.2,)e(w)o(e)i(no)o(w)g(ha)o(v) o(e)g(that)h(this)f(co)q(cycle,)h(and)f(therefore)i Fo(q)q Fp(,)e(is)g(cohomologous)257 2445 y(to)e Fo(r)q(sq)369 2430 y Fl(0)381 2445 y Fp(.)f(No)o(w)g(the)h(assertion)h(follo)o(ws)d (if)h(w)o(e)g(c)o(hange)h(notation)f(and)h(denote)g(the)g(pro)q(d-)257 2495 y(uct)f Fo(r)q(s)f Fp(b)o(y)g Fo(r)q Fp(.)f Fi(2)942 2628 y Fp(119)p eop %%Page: 120 120 120 119 bop 257 262 a Fq(8.8)48 b Fp(In)17 b(the)g(preceding)g (paragraph,)f(w)o(e)g(ha)o(v)o(e)g(obtained)h(a)f(necessary)i (condition)e(for)257 311 y Fo(B)288 317 y Fj(p)308 311 y Fp(\()p Fo(a;)7 b(b;)g(q)q Fp(\))18 b(and)g Fo(B)572 317 y Fj(p)592 311 y Fp(\()p Fo(a)630 296 y Fl(0)641 311 y Fo(;)7 b(b)678 296 y Fl(0)689 311 y Fo(;)g(q)728 296 y Fl(0)739 311 y Fp(\))19 b(to)g(b)q(e)g(isomorphic.)d(Ho)o(w)o(ev) o(er,)j(this)g(condition)f(is)g(also)257 361 y(su\016cien)o(t.)h(F)m (or,)g(if)f(there)i(are)g Fo(r)o(;)7 b(t)19 b Fm(2)h Fk(Z)918 346 y Fl(\002)918 371 y Fj(p)962 361 y Fp(suc)o(h)g(that)f Fo(q)h Fp(and)f Fo(r)q(q)1321 346 y Fl(0)1352 361 y Fp(are)g (cohomologous,)257 418 y Fo(a)279 403 y Fl(0)302 418 y Fp(=)351 401 y Fj(ta)p 351 408 31 2 v 358 432 a(r)387 418 y Fp(,)13 b(and)h Fo(b)511 403 y Fl(0)534 418 y Fp(=)590 401 y Fj(b)p 583 408 29 2 v 583 432 a(r)q(t)617 418 y Fp(,)f(it)g(can)h(b)q(e)h(v)o(eri\014ed)f(directly)h(that)537 511 y Fo(f)h Fp(:)11 b Fo(B)627 517 y Fj(p)647 511 y Fp(\()p Fo(a;)c(b;)g(q)q Fp(\))k Fm(!)g Fo(B)872 517 y Fj(p)891 511 y Fp(\()p Fo(a)929 494 y Fl(0)941 511 y Fo(;)c(b)978 494 y Fl(0)989 511 y Fo(;)g(q)1028 494 y Fl(0)1039 511 y Fp(\))p Fo(;)g(b)1092 517 y Fj(ij)r(k)1151 511 y Fm(!)k Fo(\020)1225 494 y Fj(iv)q Fn(\()p Fj(j)r Fn(\))1298 511 y Fo(b)1316 494 y Fl(0)1316 521 y Fj(r)q(i;j;tk)257 597 y Fp(is)f(a)g(Hopf)f(algebra)h(isomorphism)o(,)d(where)k Fo(v)i Fp(:)e Fk(Z)1018 603 y Fj(p)1045 597 y Fm(!)g Fk(Z)1129 603 y Fj(p)1155 597 y Fp(is)f(a)f(1-co)q(c)o(hain)h(whose)g (cob)q(ound-)257 646 y(ary)k(is)g Fo(r)q(q)412 631 y Fl(0)433 646 y Fm(\000)9 b Fo(q)q Fp(,)14 b(i.)e(e.,)i(w)o(e)g(ha)o(v)o (e:)614 732 y Fo(r)q(q)654 715 y Fl(0)665 732 y Fp(\()p Fo(j;)7 b(l)q Fp(\))j Fm(\000)f Fo(q)q Fp(\()p Fo(j;)e(l)q Fp(\))12 b(=)g Fo(v)q Fp(\()p Fo(j)g Fp(+)e Fo(l)q Fp(\))f Fm(\000)h Fo(v)q Fp(\()p Fo(j)r Fp(\))g Fm(\000)g Fo(v)q Fp(\()p Fo(l)q Fp(\))257 851 y(The)16 b(remaining)d(task)i(is)g(to)g (determine)g(the)h(n)o(um)o(b)q(er)e(of)g(isomorphism)e(classes)17 b(among)257 901 y(the)e Fo(B)360 907 y Fj(p)380 901 y Fp(\()p Fo(a;)7 b(b;)g(q)q Fp(\).)12 b(F)m(or)h(this,)h(the)g(follo)o (wing)e(lemma)e(is)k(helpful:)257 997 y Fq(Lemma)36 b Fp(Supp)q(ose)20 b(that)f Fo(p)g Fp(is)f(o)q(dd.)h(Consider)g(the)g (action)g(of)f Fk(Z)1362 982 y Fl(\002)1362 1007 y Fj(p)1399 997 y Fm(\002)13 b Fk(Z)1475 982 y Fl(\002)1475 1007 y Fj(p)1519 997 y Fp(on)18 b Fo(M)25 b Fp(:=)257 1047 y Fk(Z)288 1032 y Fl(\002)288 1057 y Fj(p)322 1047 y Fm(\002)10 b Fk(Z)394 1032 y Fl(\002)394 1057 y Fj(p)429 1047 y Fm(\002)f Fk(Z)501 1053 y Fj(p)531 1047 y Fp(de\014ned)15 b(b)o(y)733 1160 y(\()p Fo(r)o(;)7 b(t)p Fp(\))p Fo(:)p Fp(\()p Fo(a;)g(b;)g(q)q Fp(\))i(:=)i(\()1044 1132 y Fo(ta)p 1044 1150 37 2 v 1053 1188 a(r)1086 1160 y(;)1118 1132 y(b)p 1110 1150 35 2 v 1110 1188 a(r)q(t)1150 1160 y(;)1173 1132 y(q)p 1173 1150 21 2 v 1173 1188 a(r)1198 1160 y Fp(\))257 1257 y(Then)k Fo(M)j Fp(can)c(b)q(e)h(decomp)q(osed)f (in)o(to)f Fo(p)c Fp(+)h(1)j(orbits)h(with)g(resp)q(ect)i(to)e(this)g (action.)257 1353 y Fq(Pro)q(of.)36 b Fp(W)m(e)11 b(denote)h(the)f (same)g(set)h Fk(Z)885 1338 y Fl(\002)885 1363 y Fj(p)914 1353 y Fm(\002)t Fk(Z)980 1338 y Fl(\002)980 1363 y Fj(p)1009 1353 y Fm(\002)t Fk(Z)1075 1359 y Fj(p)1103 1353 y Fp(b)o(y)e Fo(N)16 b Fp(if)10 b(endo)o(w)o(ed)i(with)f(the)g(action:)724 1455 y(\()p Fo(r)o(;)c(t)p Fp(\))p Fo(:)p Fp(\()p Fo(a;)g(b;)g(q)q Fp(\))j(:=)h(\()p Fo(t)1046 1438 y Fn(2)1065 1455 y Fo(a;)c(r)q(tb;) 1181 1427 y(q)p 1181 1445 V 1181 1483 a(r)1206 1455 y Fp(\))257 1556 y(of)14 b Fk(Z)335 1541 y Fl(\002)335 1567 y Fj(p)370 1556 y Fm(\002)9 b Fk(Z)442 1541 y Fl(\002)442 1567 y Fj(p)467 1556 y Fp(.)k(It)h(is)g(then)g(easy)h(to)e(see)j(that) 689 1667 y Fo(f)h Fp(:)11 b Fo(M)16 b Fm(!)11 b Fo(N)r(;)c Fp(\()p Fo(a;)g(b;)g(q)q Fp(\))k Fm(7!)g Fp(\()1127 1639 y Fo(a)p 1127 1658 22 2 v 1129 1696 a(b)1154 1667 y(;)1177 1639 y Fp(1)p 1177 1658 21 2 v 1178 1696 a Fo(b)1203 1667 y(;)c(q)q Fp(\))257 1764 y(is)17 b(an)f(equiv)n(arian)o(t)g (bijection.)f(It)i(therefore)h(su\016ces)g(to)e(coun)o(t)h(the)g (orbits)g(of)f Fo(N)5 b Fp(.)16 b(It)h(is)257 1814 y(ob)o(vious)d(that) h(the)h(isotrop)o(y)e(group)h(of)f(the)h(elemen)o(t)g(\()p Fo(a;)7 b Fp(1)p Fo(;)g Fp(1\))k Fm(2)i Fo(N)20 b Fp(is)14 b(trivial,)f(and)i(an)o(y)257 1864 y(t)o(w)o(o)i(suc)o(h)h(elemen)o(t)f (cannot)g(b)q(elong)g(to)g(the)h(same)e(orbit.)g(Therefore,)i(w)o(e)g (ha)o(v)o(e)f(found)257 1914 y Fo(p)9 b Fm(\000)h Fp(1)k(distinct)g (orbits)g(of)f(length)h(\()p Fo(p)9 b Fm(\000)h Fp(1\))931 1899 y Fn(2)949 1914 y Fp(.)257 1997 y(No)o(w)g(\014x)h(an)f(elemen)o (t)g Fo(a)630 2003 y Fn(0)660 1997 y Fm(2)h Fk(Z)730 1982 y Fl(\002)730 2007 y Fj(p)765 1997 y Fp(that)g(is)f(not)g(a)g (square.)h(Then)g(the)g(elemen)o(ts)f(\(1)p Fo(;)d Fp(1)p Fo(;)g Fp(0\))j Fm(2)i Fo(N)257 2047 y Fp(and)17 b(\()p Fo(a)379 2053 y Fn(0)398 2047 y Fo(;)7 b Fp(1)p Fo(;)g Fp(0\))15 b Fm(2)h Fo(N)22 b Fp(are)c(not)f(conjugate)g(under)g(the)h (action)f(of)f Fk(Z)1353 2032 y Fl(\002)1353 2057 y Fj(p)1389 2047 y Fm(\002)c Fk(Z)1463 2032 y Fl(\002)1463 2057 y Fj(p)1488 2047 y Fp(.)17 b(If)f(\()p Fo(r)o(;)7 b(t)p Fp(\))17 b(is)257 2097 y(con)o(tained)c(in)g(the)g(isotrop)o(y)g(group) f(of)h(\()p Fo(a)922 2103 y Fn(0)940 2097 y Fo(;)7 b Fp(1)p Fo(;)g Fp(0\),)k(w)o(e)i(ha)o(v)o(e)g Fo(t)1229 2082 y Fn(2)1248 2097 y Fo(a)1270 2103 y Fn(0)1300 2097 y Fp(=)f Fo(a)1366 2103 y Fn(0)1397 2097 y Fp(and)h Fo(r)q(t)e Fp(=)h(1,)g(i.)g(e.,)257 2146 y Fo(t)g Fp(=)g Fm(\006)p Fp(1)h(and)g Fo(r)f Fp(=)g Fo(t)p Fp(.)g(Therefore,)i(the)g(isotrop)o (y)f(group)f(of)h(\()p Fo(a)1218 2152 y Fn(0)1237 2146 y Fo(;)7 b Fp(1)p Fo(;)g Fp(0\))k(consists)j(of)f(t)o(w)o(o,)f(the)257 2196 y(orbit)k(of)g(\()p Fo(a)449 2202 y Fn(0)467 2196 y Fo(;)7 b Fp(1)p Fo(;)g Fp(0\))15 b(of)632 2180 y Fn(1)p 632 2187 17 2 v 632 2211 a(2)654 2196 y Fp(\()p Fo(p)10 b Fm(\000)h Fp(1\))781 2181 y Fn(2)816 2196 y Fp(elemen)o(ts.)k(Since)i (the)f(same)g(applies)f(to)h(\(1)p Fo(;)7 b Fp(1)p Fo(;)g Fp(0\),)14 b(w)o(e)257 2246 y(ha)o(v)o(e)e(found)f(t)o(w)o(o)g (additional)e(orbits)j(of)f(length)1024 2230 y Fn(1)p 1024 2237 V 1024 2260 a(2)1045 2246 y Fp(\()p Fo(p)t Fm(\000)t Fp(1\))1159 2231 y Fn(2)1179 2246 y Fp(.)g(Since)h(the)g(com) o(bined)e(lengths)257 2296 y(of)k(these)h(orbits)f(satisfy)578 2406 y(\()p Fo(p)9 b Fm(\000)g Fp(1\))702 2389 y Fn(3)730 2406 y Fp(+)h(2)798 2378 y(\()p Fo(p)f Fm(\000)g Fp(1\))922 2363 y Fn(2)p 797 2396 144 2 v 859 2434 a Fp(2)957 2406 y(=)j(\()p Fo(p)d Fm(\000)h Fp(1\))1126 2389 y Fn(2)1144 2406 y Fo(p)i Fp(=)g(card\()p Fo(N)5 b Fp(\))257 2503 y(these)16 b(are)e(already)g(all)e(orbits,)i(whic)o(h)f(means)h(that)f (w)o(e)i(ha)o(v)o(e)e Fo(p)c Fp(+)h(1)j(orbits)h(in)g(total.)f Fi(2)942 2628 y Fp(120)p eop %%Page: 121 121 121 120 bop 257 262 a Fp(W)m(e)14 b(summarize)e(the)i(obtained)g (results)h(in)e(the)i(follo)o(wing)c(theorem:)257 355 y Fq(Theorem)36 b Fp(Supp)q(ose)12 b(that)f Fo(K)k Fp(is)c(an)g (algebraically)f(closed)i(\014eld)f(of)g(c)o(haracteristic)i(zero)257 405 y(and)h(that)g Fo(p)g Fp(is)f(an)h(o)q(dd)g(prime.)308 512 y(1.)20 b(Ev)o(ery)f(semisimple,)c(noncomm)o(utativ)n(e,)g(nonco)q (comm)o(utativ)o(e)h(Hopf)h(algebra)h(of)361 561 y(dimension)10 b Fo(p)576 546 y Fn(3)605 561 y Fp(is)h(isomorphic)e(to)i(a)g(Hopf)g (algebra)f Fo(B)1203 567 y Fj(p)1223 561 y Fp(\()p Fo(a;)d(b;)g(q)q Fp(\))j(for)h(some)f Fo(a;)d(b)k Fm(2)g Fk(Z)1664 546 y Fl(\002)1664 572 y Fj(p)361 611 y Fp(and)j(a)f(normalized)g(2-co)q (cycle)h Fo(q)f Fm(2)e Fo(Z)968 596 y Fn(2)987 611 y Fp(\()p Fk(Z)1034 617 y Fj(p)1050 611 y Fo(;)c Fk(Z)1099 617 y Fj(p)1115 611 y Fp(\).)308 690 y(2.)20 b Fo(B)392 696 y Fj(p)412 690 y Fp(\()p Fo(a;)7 b(b;)g(q)q Fp(\))i(and)i Fo(B)660 696 y Fj(p)680 690 y Fp(\()p Fo(a)718 675 y Fl(0)729 690 y Fo(;)c(b)766 675 y Fl(0)777 690 y Fo(;)g(q)816 675 y Fl(0)827 690 y Fp(\))k(are)g(isomorphic)f(if)f(and)i(only)f(if)g (there)i(are)f Fo(r)o(;)c(t)k Fm(2)g Fk(Z)1664 675 y Fl(\002)1664 701 y Fj(p)361 747 y Fp(suc)o(h)k(that)f Fo(a)567 732 y Fl(0)590 747 y Fp(=)639 731 y Fj(ta)p 639 738 31 2 v 646 761 a(r)674 747 y Fp(,)g Fo(b)718 732 y Fl(0)741 747 y Fp(=)796 731 y Fj(b)p 789 738 29 2 v 789 761 a(r)q(t)823 747 y Fp(,)g(and)f Fo(q)i Fp(and)f Fo(r)q(q)1084 732 y Fl(0)1109 747 y Fp(are)g(cohomologous.)308 826 y(3.)20 b(There)14 b(are)e Fo(p)6 b Fp(+)g(1)13 b(isomorphism)c (classes)14 b(of)e(semisimple,)d(noncomm)o(utativ)n(e,)h(non-)361 876 y(co)q(comm)o(utativ)o(e)h(Hopf)j(algebras)g(of)f(dimension)f Fo(p)1177 861 y Fn(3)1196 876 y Fp(.)257 970 y Fq(Pro)q(of.)36 b Fp(The)18 b(\014rst)f(result)h(w)o(as)f(obtained)f(in)h(P)o(aragraph) f(8.5.)g(One)i(part)f(of)f(the)h(sec-)257 1019 y(ond)22 b(assertion)g(w)o(as)f(pro)o(v)o(ed)h(in)f(Prop)q(osition)g(8.7,)f(the) i(other)g(part)g(w)o(as)f(pro)o(v)o(ed)h(at)257 1069 y(the)c(b)q(eginning)f(of)f(this)i(paragraph.)e(Since)i(w)o(e)f(kno)o 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1561 y(dimensional)e(Y)m(etter-Drinfel'd)h(Hopf)g (algebras)h(o)o(v)o(er)g Fo(K)s Fp([)p Fk(Z)1228 1567 y Fn(2)1244 1561 y Fp(])f(that)h(are)g(non)o(trivial,)d(co-)257 1611 y(comm)o(utativ)o(e,)c(and)i(cosemisimple:)f(There)j(are)f(t)o(w)o (o)f(isomorphism)e(t)o(yp)q(es)k(that)f(are)g(rep-)257 1661 y(resen)o(ted)22 b(b)o(y)d(the)h(t)o(w)o(o)f(nonisomorphic)e(Y)m (etter-Drinfel'd)i(Hopf)g(algebras)g(that)h(w)o(ere)257 1710 y(denoted)13 b Fo(A)443 1716 y Fn(+)483 1710 y Fp(and)f Fo(A)593 1716 y Fl(\000)621 1710 y Fp(.)f(W)m(e)h(therefore)h(can)f(sa) o(y)g(that)g(an)o(y)f(eigh)o(t-dimensional,)e(noncom-)257 1760 y(m)o(utativ)o(e,)h(nonco)q(comm)o(utativ)o(e,)g(semisimple)f (Hopf)j(algebra)g Fo(B)j Fp(o)o(v)o(er)d(the)h(algebraically)257 1810 y(closed)h(\014eld)f Fo(K)k Fp(of)12 b(c)o(haracteristic)j(zero)f (is)f(isomorphic)f(either)i(to)f(the)g(Radford)g(bipro)q(d-)257 1860 y(uct)21 b Fo(B)366 1866 y Fn(+)415 1860 y Fp(:=)g Fo(A)511 1866 y Fn(+)551 1860 y Fm(\012)14 b Fo(H)22 b Fp(or)e(to)f(the)i(Radford)d(bipro)q(duct)i Fo(B)1238 1866 y Fl(\000)1288 1860 y Fp(:=)h Fo(A)1384 1866 y Fl(\000)1425 1860 y Fm(\012)14 b Fo(H)s Fp(.)19 b(W)m(e)g(no)o(w)257 1910 y(in)o(tro)q(duce)d(a)e(basis)h(for)g(these)h(algebras)f(and)f (describ)q(e)j(their)e(structure)i(elemen)o(ts)e(with)257 1959 y(resp)q(ect)i(to)c(this)h(basis:)257 2053 y Fq(De\014nition)33 b Fp(In)14 b(the)g(Hopf)g(algebra)f Fo(B)902 2059 y Fn(+)942 2053 y Fp(:=)e Fo(A)1028 2059 y Fn(+)1065 2053 y Fm(\012)f Fo(H)s Fp(,)j(w)o(e)h(in)o(tro)q(duce)h(the)f(basis)801 2135 y Fo(b)819 2117 y Fn(+)819 2147 y Fj(ij)r(k)878 2135 y Fp(:=)e Fo(e)953 2141 y Fj(i)976 2135 y Fm(\012)d Fo(c)1035 2141 y Fj(j)1062 2135 y Fm(\012)h Fo(d)1126 2141 y Fj(k)257 2216 y Fp(where)15 b Fo(d)399 2222 y Fj(k)433 2216 y Fp(is)f(the)h(idemp)q(oten)o(t)797 2335 y Fo(d)819 2341 y Fj(k)850 2335 y Fp(:=)911 2307 y(1)p 911 2325 21 2 v 911 2363 a(2)965 2283 y Fn(1)943 2295 y Fg(X)947 2385 y Fj(l)p Fn(=0)1003 2335 y Fp(\()p Fm(\000)p Fp(1\))1088 2317 y Fj(k)q(l)1120 2335 y Fo(c)1138 2341 y Fj(l)257 2453 y Fp(constructed)i(in)c(analogy)g(to)h(De\014nition)f (8.5.)f(Here)k(the)e(indices)h Fo(i;)7 b(j;)g(k)14 b Fp(tak)o(e)g(the)h(v)n(alues)257 2503 y(0)f(and)g(1.)f(W)m(e)g(in)o (tro)q(duce)i(an)f(analogous)e(basis)i Fo(b)1044 2485 y Fl(\000)1044 2516 y Fj(ij)r(k)1105 2503 y Fp(in)g Fo(B)1185 2509 y Fl(\000)1213 2503 y Fp(.)942 2628 y(121)p eop %%Page: 122 122 122 121 bop 257 262 a Fp(As)15 b(in)e(P)o(aragraph)h(3.6,)e(w)o(e)i (de\014ne)h(co)q(cycles)h Fo(q)1008 268 y Fn(+)1035 262 y Fo(;)7 b(q)1073 268 y Fl(\000)1111 262 y Fm(2)k Fo(Z)1181 246 y Fn(2)1200 262 y Fp(\()p Fk(Z)1247 268 y Fn(2)1263 262 y Fo(;)c Fk(Z)1312 268 y Fn(4)1327 262 y Fp(\))14 b(b)o(y:)311 390 y Fo(q)330 396 y Fn(+)358 390 y Fp(\()p Fo(i;)7 b(j)r Fp(\))12 b(:=)509 319 y Fg(\()542 362 y Fp(0)42 b(if)13 b Fo(i)f Fp(=)f(0)j(or)g Fo(j)g Fp(=)e(0)542 422 y(1)42 b(if)13 b Fo(i)f Fp(=)f(1)j(and)g Fo(j)g Fp(=)d(1)1018 390 y Fo(q)1037 396 y Fl(\000)1065 390 y Fp(\()p Fo(i;)c(j)r Fp(\))12 b(:=)1216 319 y Fg(\()1250 362 y Fp(0)41 b(if)13 b Fo(i)f Fp(=)g(0)h(or)h Fo(j)g Fp(=)e(0)1250 422 y(3)41 b(if)13 b Fo(i)f Fp(=)g(1)h(and)h Fo(j)g Fp(=)e(1)257 517 y(Also)i(as)g(in)f(P)o(aragraph)h(3.6,)e(w)o(e)j(use)f(the)h (notation)657 608 y Fo(\033)682 590 y Fl(\006)681 619 y Fn(0)710 608 y Fp(\()p Fo(i;)7 b(j)r Fp(\))12 b(:=)g(1)83 b Fo(\033)991 590 y Fl(\006)990 619 y Fn(1)1019 608 y Fp(\()p Fo(i;)7 b(j)r Fp(\))12 b(:=)f Fo(\023)1185 591 y Fj(q)1200 595 y Fh(\006)1225 591 y Fn(\()p Fj(i;j)r Fn(\))257 699 y Fp(With)i(this)h(notation,)e(the)i(structure)i(maps)c (of)h Fo(B)1057 705 y Fl(\006)1099 699 y Fp(can)h(b)q(e)g(expressed)i (with)d(resp)q(ect)j(to)257 749 y(this)e(basis:)308 876 y(1.)20 b(Multiplication:)12 b Fo(b)662 858 y Fl(\006)662 889 y Fj(ij)r(k)709 876 y Fo(b)727 858 y Fl(\006)727 889 y Fj(lmn)801 876 y Fp(=)g Fo(\016)863 882 y Fj(k)q Fl(\000)p Fj(n;lm)980 876 y Fo(\016)998 882 y Fj(il)1023 876 y Fo(\033)1048 858 y Fl(\006)1047 888 y Fj(i)1076 876 y Fp(\()p Fo(j;)7 b(m)p Fp(\))p Fo(b)1198 858 y Fl(\006)1198 888 y Fj(i;j)r Fn(+)p Fj(m;n)308 971 y Fp(2.)20 b(Unit:)13 b(1)f(=)544 940 y Fg(P)588 950 y Fn(1)588 984 y Fj(i;k)q Fn(=0)679 971 y Fo(b)697 953 y Fl(\006)697 984 y Fj(i;)p Fn(0)p Fj(;k)308 1066 y Fp(3.)20 b(Com)o(ultiplicatio)o(n:)11 b(\001\()p Fo(b)759 1049 y Fl(\006)759 1079 y Fj(ij)r(k)806 1066 y Fp(\))h(=)877 1035 y Fg(P)921 1046 y Fn(1)921 1079 y Fj(l;m)p Fn(=0)1015 1066 y Fp(\()p Fm(\000)p Fp(1\))1100 1051 y Fn(\()p Fj(i)p Fl(\000)p Fj(l)p Fn(\))p Fj(j)r(m)1222 1066 y Fo(b)1240 1049 y Fl(\006)1240 1079 y Fj(lj)r(m)1307 1066 y Fm(\012)d Fo(b)1366 1049 y Fl(\006)1366 1079 y Fj(i)p Fl(\000)p Fj(l;j;k)q Fl(\000)p Fj(m)308 1158 y Fp(4.)20 b(Counit:)13 b Fo(\017)p Fp(\()p Fo(b)562 1140 y Fl(\006)562 1170 y Fj(ij)r(k)609 1158 y Fp(\))f(=)g Fo(\016)699 1164 y Fj(i)p Fn(0)730 1158 y Fo(\016)748 1164 y Fj(k)q Fn(0)308 1249 y Fp(5.)20 b(An)o(tip)q(o)q(de:)14 b Fo(S)r Fp(\()p Fo(b)616 1255 y Fj(ij)r(k)664 1249 y Fp(\))e(=)g(\()p Fm(\000)p Fp(1\))821 1234 y Fj(ij)r(k)869 1249 y Fo(\033)894 1231 y Fl(\006)893 1260 y Fj(i)922 1249 y Fp(\()p Fo(j;)7 b Fm(\000)p Fo(j)r Fp(\))1041 1234 y Fl(\000)p Fn(1)1086 1249 y Fo(b)1104 1231 y Fl(\006)1104 1261 y(\000)p Fj(i;)p Fl(\000)p Fj(j;)p Fl(\000)p Fj(k)q Fl(\000)p Fj(ij)257 1376 y Fp(The)14 b(remaining)e(op)q(en)i(question)g (in)f(our)h(treatmen)o(t)f(of)g(eigh)o(t-dimensional)e(semisimple)257 1426 y(Hopf)k(algebras)h(is)f(whether)i Fo(B)759 1432 y Fn(+)802 1426 y Fp(and)e Fo(B)915 1432 y Fl(\000)959 1426 y Fp(are)h(isomorphic.)d(W)m(e)i(ha)o(v)o(e)h(seen)g(in)f(Prop)q (o-)257 1476 y(sition)i(4.9)e(that)i Fo(A)567 1482 y Fn(+)612 1476 y Fp(and)f Fo(A)726 1482 y Fl(\000)771 1476 y Fp(are)h(not)g(isomorphic.)e(Nev)o(ertheless,)j Fo(B)1429 1482 y Fn(+)1474 1476 y Fp(and)f Fo(B)1589 1482 y Fl(\000)1634 1476 y Fp(are)257 1525 y(isomorphic:)257 1625 y Fq(Prop)q(osition)33 b Fp(The)15 b(linear)e(map)643 1716 y Fo(f)k Fp(:)11 b Fo(B)734 1722 y Fn(+)773 1716 y Fm(!)g Fo(B)857 1722 y Fl(\000)886 1716 y Fo(;)18 b(b)934 1699 y Fn(+)934 1729 y Fj(ij)r(k)993 1716 y Fm(7!)11 b Fo(\033)1071 1699 y Fn(+)1070 1727 y(1)1099 1716 y Fp(\()p Fo(i;)c(j)r Fp(\))p Fo(b)1201 1699 y Fl(\000)1201 1729 y Fj(i;j;k)q Fn(+)p Fj(i)257 1808 y Fp(is)14 b(a)g(Hopf)f(algebra) h(isomorphism)o(.)257 1907 y Fq(Pro)q(of.)36 b Fp(It)14 b(is)g(easy)g(to)g(v)o(erify)f(the)i(equations:)355 1999 y Fo(\033)380 1981 y Fl(\000)379 2010 y Fj(i)409 1999 y Fp(\()p Fo(j;)7 b(k)q Fp(\))k(=)h(\()p Fm(\000)p Fp(1\))640 1982 y Fj(ij)r(k)688 1999 y Fo(\033)713 1981 y Fn(+)712 2010 y Fj(i)740 1999 y Fp(\()p Fo(j;)7 b(k)q Fp(\))83 b Fo(\033)939 1981 y Fn(+)938 2010 y(1)967 1999 y Fp(\()p Fo(i;)7 b(j)12 b Fp(+)d Fo(k)q Fp(\))j(=)g(\()p Fm(\000)p Fp(1\))1266 1982 y Fj(ij)r(k)1313 1999 y Fo(\033)1338 1981 y Fn(+)1337 2010 y(1)1366 1999 y Fp(\()p Fo(i;)7 b(j)r Fp(\))p Fo(\033)1475 1981 y Fn(+)1474 2010 y(1)1504 1999 y Fp(\()p Fo(i;)g(k)q Fp(\))257 2090 y(Therefore,)15 b Fo(f)k Fp(is)13 b(an)h(algebra)g(homom)o(orphism)o(:)297 2181 y Fo(f)t Fp(\()p Fo(b)355 2164 y Fn(+)355 2194 y Fj(ij)r(k)403 2181 y Fo(b)421 2164 y Fn(+)421 2194 y Fj(lmn)484 2181 y Fp(\))d(=)h Fo(\016)573 2187 y Fj(k)q Fl(\000)p Fj(n;lm)691 2181 y Fo(\016)709 2187 y Fj(il)733 2181 y Fo(\033)758 2164 y Fn(+)757 2193 y Fj(i)786 2181 y Fp(\()p Fo(j;)7 b(m)p Fp(\))p Fo(f)t Fp(\()p Fo(b)948 2164 y Fn(+)948 2193 y Fj(i;j)r Fn(+)p Fj(m;n)1074 2181 y Fp(\))511 2252 y(=)12 b Fo(\016)573 2258 y Fj(k)q Fl(\000)p Fj(n;lm)691 2252 y Fo(\016)709 2258 y Fj(il)733 2252 y Fo(\033)758 2234 y Fn(+)757 2263 y Fj(i)786 2252 y Fp(\()p Fo(j;)7 b(m)p Fp(\))p Fo(\033)915 2234 y Fn(+)914 2263 y(1)943 2252 y Fp(\()p Fo(i;)g(j)12 b Fp(+)d Fo(m)p Fp(\))p Fo(b)1132 2234 y Fl(\000)1132 2263 y Fj(i;j)r Fn(+)p Fj(m;n)p Fn(+)p Fj(i)511 2322 y Fp(=)j Fo(\016)573 2328 y Fj(k)q Fl(\000)p Fj(n;lm)691 2322 y Fo(\016)709 2328 y Fj(il)733 2322 y Fp(\()p Fm(\000)p Fp(1\))818 2305 y Fj(ij)r(m)877 2322 y Fp(\()p Fm(\000)p Fp(1\))962 2305 y Fj(ij)r(m)1021 2322 y Fo(\033)1046 2304 y Fl(\000)1045 2334 y Fj(i)1075 2322 y Fp(\()p Fo(j;)7 b(m)p Fp(\))p Fo(\033)1204 2304 y Fn(+)1203 2333 y(1)1232 2322 y Fp(\()p Fo(i;)g(j)r Fp(\))p Fo(\033)1341 2304 y Fn(+)1340 2333 y(1)1369 2322 y Fp(\()p Fo(i;)g(m)p Fp(\))p Fo(b)1488 2304 y Fl(\000)1488 2334 y Fj(i;j)r Fn(+)p Fj(m;n)p Fn(+)p Fj(i)511 2392 y Fp(=)12 b Fo(\016)573 2398 y Fj(k)q Fl(\000)p Fj(n;lm)691 2392 y Fo(\016)709 2398 y Fj(il)733 2392 y Fo(\033)758 2374 y Fl(\000)757 2403 y Fj(i)787 2392 y Fp(\()p Fo(j;)7 b(m)p Fp(\))p Fo(\033)916 2374 y Fn(+)915 2403 y(1)944 2392 y Fp(\()p Fo(i;)g(j)r Fp(\))p Fo(\033)1053 2374 y Fn(+)1052 2403 y(1)1081 2392 y Fp(\()p Fo(l)q(;)g(m)p Fp(\))p Fo(b)1199 2374 y Fl(\000)1199 2403 y Fj(i;j)r Fn(+)p Fj(m;n)p Fn(+)p Fj(i)511 2461 y Fp(=)12 b Fo(\033)580 2443 y Fn(+)579 2472 y(1)608 2461 y Fp(\()p Fo(i;)7 b(j)r Fp(\))p Fo(\033)717 2443 y Fn(+)716 2472 y(1)745 2461 y Fp(\()p Fo(l)q(;)g(m)p Fp(\))p Fo(b)863 2443 y Fl(\000)863 2474 y Fj(i;j;k)q Fn(+)p Fj(i)966 2461 y Fo(b)984 2443 y Fl(\000)984 2474 y Fj(l;m;n)p Fn(+)p Fj(l)1114 2461 y Fp(=)12 b Fo(f)t Fp(\()p Fo(b)1216 2443 y Fn(+)1216 2474 y Fj(ij)r(k)1264 2461 y Fp(\))p Fo(f)t Fp(\()p Fo(b)1338 2443 y Fn(+)1338 2474 y Fj(lmn)1402 2461 y Fp(\))942 2628 y(122)p eop %%Page: 123 123 123 122 bop 257 262 a Fp(W)m(e)13 b(can)h(also)e(deduce)j(from)c(these) k(equations)e(that)h Fo(f)k Fp(is)13 b(a)g(coalgebra)g(homom)o(orphism) o(:)281 386 y(\()p Fo(f)h Fm(\012)9 b Fo(f)t Fp(\)\001\()p Fo(b)481 368 y Fn(+)481 398 y Fj(ij)r(k)529 386 y Fp(\))j(=)639 334 y Fn(1)617 346 y Fg(X)601 436 y Fj(l;m)p Fn(=0)693 386 y Fp(\()p Fm(\000)p Fp(1\))778 368 y Fn(\()p Fj(i)p Fl(\000)p Fj(l)p Fn(\))p Fj(j)r(m)899 386 y Fo(\033)924 368 y Fn(+)923 397 y(1)952 386 y Fp(\()p Fo(l)q(;)7 b(j)r Fp(\))p Fo(\033)1060 368 y Fn(+)1059 397 y(1)1088 386 y Fp(\()p Fo(i)j Fm(\000)g Fo(l)q(;)d(j)r Fp(\))p Fo(b)1255 368 y Fl(\000)1255 398 y Fj(l;j;m)p Fn(+)p Fj(l)1376 386 y Fm(\012)i Fo(b)1435 368 y Fl(\000)1435 398 y Fj(i)p Fl(\000)p Fj(l;j;k)q Fl(\000)p Fj(m)p Fn(+)p Fj(i)p Fl(\000)p Fj(l)557 537 y Fp(=)639 485 y Fn(1)617 497 y Fg(X)601 587 y Fj(l;m)p Fn(=0)693 537 y Fp(\()p Fm(\000)p Fp(1\))778 520 y Fn(\()p Fj(i)p Fl(\000)p Fj(l)p Fn(\))p Fj(j)r Fn(\()p Fj(m)p Fl(\000)p Fj(l)p Fn(\))962 537 y Fo(\033)987 519 y Fn(+)986 548 y(1)1015 537 y Fp(\()p Fo(l)q(;)e(j)r Fp(\))p Fo(\033)1123 519 y Fn(+)1122 548 y(1)1151 537 y Fp(\()p Fo(i)j Fm(\000)f Fo(l)q(;)e(j)r Fp(\))p Fo(b)1317 519 y Fl(\000)1317 549 y Fj(l;j;m)1402 537 y Fm(\012)i Fo(b)1461 519 y Fl(\000)1461 549 y Fj(i)p Fl(\000)p Fj(l;j;k)q Fl(\000)p Fj(m)p Fn(+)p Fj(i)557 688 y Fp(=)639 636 y Fn(1)617 648 y Fg(X)601 738 y Fj(l;m)p Fn(=0)693 688 y Fp(\()p Fm(\000)p Fp(1\))778 671 y Fn(\()p Fj(i)p Fl(\000)p Fj(l)p Fn(\))p Fj(j)r(m)899 688 y Fo(\033)924 670 y Fn(+)923 699 y(1)952 688 y Fp(\()p Fo(i;)e(j)r Fp(\))p Fo(b)1054 670 y Fl(\000)1054 700 y Fj(lj)r(m)1121 688 y Fm(\012)j Fo(b)1181 670 y Fl(\000)1181 700 y Fj(i)p Fl(\000)p Fj(l;j;k)q Fl(\000)p Fj(m)p Fn(+)p Fj(i)557 800 y Fp(=)i Fo(\033)626 782 y Fn(+)625 811 y(1)654 800 y Fp(\()p Fo(i;)7 b(j)r Fp(\)\001\()p Fo(b)807 782 y Fl(\000)807 812 y Fj(i;j;k)q Fn(+)p Fj(i)909 800 y Fp(\))12 b(=)g(\001\()p Fo(f)t Fp(\()p Fo(b)1090 782 y Fn(+)1090 812 y Fj(ij)r(k)1138 800 y Fp(\)\))257 887 y(It)18 b(is)f(easy)h(to)g(see)h(that)e Fo(f)23 b Fp(preserv)o(es)d(the)e(unit)f(and)g(the)i(counit,)e(and)g (therefore)i(also)257 937 y(comm)o(utes)12 b(with)i(the)h(an)o(tip)q(o) q(des)f(\(cf.)f([84)o(],)g(Lem.)g(4.0.4,)e(p.)j(81\).)f Fi(2)257 1052 y Fp(W)m(e)f(therefore)h(see)f(that,)g(in)f(dimension)f (eigh)o(t,)h(there)i(is)e(only)g(one)h(isomorphism)c(class)k(of)257 1102 y(noncomm)o(utativ)o(e,)g(nonco)q(comm)o(utativ)o(e,)g(semisimple) g(Hopf)j(algebras.)f(This)h(w)o(as)g(\014rst)257 1152 y(pro)o(v)o(ed)f(b)o(y)g(R.)e(William)o(s)f(and)j(A.)f(Masuok)n(a)g (\(cf.)g([87)o(],)g([47)o(],)g(Thm.)f(2.13,)f(p.)j(371\),)e(who,)257 1202 y(ho)o(w)o(ev)o(er,)f(used)h(comparativ)o(ely)d(di\013eren)o(t)j (metho)q(ds.)e(The)h(Hopf)g(algebra)f(itself)h(w)o(as)g(\014rst)257 1251 y(constructed)16 b(b)o(y)e(G.)f(I.)g(Kac)i(and)e(V.)h(G.)f(P)o (aljutkin)f(\(cf.)i([31)o(],)f([32)o(]\).)257 1385 y Fq(8.10)48 b Fp(W)m(e)14 b(ha)o(v)o(e)g(describ)q(ed)i(ab)q(o)o(v)o(e)e (the)g(semisimple)e(Hopf)i(algebras)g(of)f(dimension)g Fo(p)1671 1370 y Fn(3)257 1435 y Fp(that)k(are)g(neither)g(comm)o (utativ)o(e)d(nor)i(co)q(comm)o(utativ)o(e.)d(W)m(e)j(no)o(w)g(describ) q(e)j(the)e(semi-)257 1485 y(simple)c(Hopf)g(algebras)h(of)f(dimension) f Fo(p)914 1470 y Fn(3)946 1485 y Fp(that)i(are)g(comm)o(utativ)o(e)d (or)j(co)q(comm)o(utativ)o(e.)257 1534 y(W)m(e)19 b(main)o(tain)d(our)i (assumption)g(that)h(the)g(base)g(\014eld)g Fo(K)j Fp(is)d (algebraically)e(closed)i(of)257 1584 y(c)o(haracteristic)d(zero.)e(In) g(this)g(case,)g(\014nite-dimensional)e(Hopf)i(algebras)g(that)g(are)g (com-)257 1634 y(m)o(utativ)o(e)h(or)h(co)q(comm)o(utativ)o(e)d(are)k (automatically)12 b(semisimple;)i(this)i(follo)o(ws,)e(for)h(ex-)257 1684 y(ample,)h(from)f(an)i(ev)o(en)h(more)f(general)g(result)h(of)f (R.)g(G.)f(Larson)i(and)f(D.)f(E.)h(Radford)257 1734 y(\(cf.)c([37)o(],)e(Cor.)h(2.6,)f(p.)h(275\).)g(By)g(a)h(result)g(of)f (D.)f(K.)i(Harrison)f(and)h(P)m(.)e(Cartier)i(\(cf.)f([35)o(],)257 1783 y(Thm.)j(3.2,)f(p.)i(354,)f([7)o(],)g(p.)h(102,)f([57)o(],)g(Thm.) g(2.3.1,)f(p.)h(22\),)h(a)g(\014nite-dimensional)d(co-)257 1833 y(comm)o(utativ)o(e)e(Hopf)i(algebra)g(is)h(isomorphic)e(to)h(a)g (group)h(ring.)e(The)i(description)h(of)e(all)257 1883 y(comm)o(utativ)o(e)c(or)i(co)q(comm)o(utativ)o(e)e(Hopf)h(algebras)i (of)e(dimension)g Fo(p)1354 1868 y Fn(3)1384 1883 y Fp(therefore)i (reduces)257 1933 y(to)j(the)g(description)g(of)f(all)f(groups)i(of)f (order)h Fo(p)1013 1918 y Fn(3)1032 1933 y Fp(.)f(This)g(is,)g(of)g (course,)h(w)o(ell)f(kno)o(wn,)g(and)257 1983 y(here)j(w)o(e)e(shall)g (only)f(describ)q(e)j(the)f(results)g(\(cf.)g([25)o(],)e(Kap.)h(I,)f (Satz)i(14.10,)d(p.)i(93,)f([34)o(],)257 2033 y(Kap.)k(IV,)f Fm(x)h Fp(1,)f(p.)g(61/62,)f([83)o(],)h(Chap.)g(4,)g(\(4.13\),)f(p.)i (67\);)f(nev)o(ertheless,)i(it)f(is)f(not)h(a)257 2082 y(totally)e(easy)h(task.)g(By)g(the)g(structure)i(theorem)d(for)h (\014nite)g(ab)q(elian)f(groups)h(\(cf.)f([82)o(],)257 2132 y(Chap.)10 b(2,)f(Thm.)g(5.2,)f(p.)i(145,)f([34)o(],)g(Kap.)h(I)q (I,)g Fm(x)g Fp(2,)g(p.)g(30\),)f(the)i(ab)q(elian)e(groups)h(of)g (order)h Fo(p)1671 2117 y Fn(3)257 2182 y Fp(are:)658 2232 y Fk(Z)688 2239 y Fj(p)705 2231 y Fd(3)804 2232 y Fk(Z)834 2239 y Fj(p)851 2231 y Fd(2)876 2232 y Fm(\002)e Fk(Z)948 2238 y Fj(p)1047 2232 y Fk(Z)1078 2238 y Fj(p)1103 2232 y Fm(\002)h Fk(Z)1176 2238 y Fj(p)1201 2232 y Fm(\002)f Fk(Z)1273 2238 y Fj(p)257 2304 y Fp(The)k(nonab)q(elian)e(groups)i(of)e (order)i(8)f(are)g(the)h(dihedral)f(group)g Fo(D)1315 2310 y Fn(4)1346 2304 y Fp(and)g(the)h(quaternion)257 2354 y(group)f Fo(Q)p Fp(.)g(No)o(w)g(supp)q(ose)h(that)g Fo(p)e Fm(6)p Fp(=)h(2)g(and)g(that)g Fo(G)g Fp(is)g(a)g(nonab)q(elian) g(group)g(of)f(order)i Fo(p)1659 2339 y Fn(3)1678 2354 y Fp(.)257 2403 y(It)21 b(is)g(easy)g(to)g(see)h(that)f(the)g(cen)o (ter)i(of)d Fo(G)g Fp(is)h(of)f(order)h Fo(p)g Fp(and)g(that)g Fo(G)f Fp(con)o(tains)h(a)257 2453 y(subgroup)16 b Fo(N)k Fp(of)14 b(order)i Fo(p)671 2438 y Fn(2)689 2453 y Fp(,)f(whic)o(h)g (is)g(therefore)h(normal.)d(No)o(w)i(t)o(w)o(o)f(cases)j(ma)o(y)c(o)q (ccur:)257 2503 y(Either)f Fo(G)e Fp(con)o(tains)g(an)h(elemen)o(t)f (of)f(order)j Fo(p)959 2488 y Fn(2)988 2503 y Fp(or)e(all)g(non)o (trivial)e(elemen)o(ts)j(are)g(of)f(order)h Fo(p)p Fp(,)942 2628 y(123)p eop %%Page: 124 124 124 123 bop 257 262 a Fp(i.)17 b(e.,)g(w)o(e)g(ha)o(v)o(e)g Fo(N)575 250 y Fm(\030)575 264 y Fp(=)625 262 y Fk(Z)656 269 y Fj(p)673 261 y Fd(2)705 262 y Fp(or)h Fo(N)815 250 y Fm(\030)815 264 y Fp(=)864 262 y Fk(Z)895 268 y Fj(p)923 262 y Fm(\002)11 b Fk(Z)997 268 y Fj(p)1013 262 y Fp(.)17 b(In)g(an)o(y)g(case,)h(w)o(e)g(ha)o(v)o(e)f(a)g(short)h (exact)257 311 y(sequence:)848 361 y Fo(N)e Fe(\032)c Fo(G)f Fe(\020)g Fk(Z)1083 367 y Fj(p)257 429 y Fp(The)i(non)o(trivial) e(task)h(is)h(to)f(pro)o(v)o(e)g(that)h(this)f(sequence)j(is)d(split,)f (i.)h(e.,)f Fo(G)h Fp(is)h(a)f(semidirect)257 479 y(pro)q(duct)k(in)e (b)q(oth)h(cases.)h(Since)f(card)q(\(Aut\()p Fk(Z)1000 486 y Fj(p)1017 478 y Fd(2)1033 479 y Fp(\)\))e(=)g Fo(p)p Fp(\()p Fo(p)d Fm(\000)g Fp(1\),)k(there)i(is)f(an)g(essen)o(tially)257 529 y(unique)j(automorphism)c(of)j(order)h Fo(p)f Fp(in)g(the)h (\014rst)g(case,)g(whic)o(h)g(can)f(b)q(e)h(c)o(hosen)h(to)e(b)q(e)257 579 y(m)o(ultiplication)10 b(b)o(y)j Fo(p)7 b Fp(+)g(1.)13 b(In)g(the)g(second)h(case,)g Fo(N)k Fp(is)12 b(elemen)o(tary)h(ab)q (elian,)f(and)g(there-)257 629 y(fore)i(group)g(homom)o(orphism)o(s)d (are)j(linear)f(maps)g(o)o(v)o(er)g(the)h(\014eld)g(with)g Fo(p)f Fp(elemen)o(ts.)g(The)257 678 y(automorphism)e(in)i(the)i (semidirect)e(pro)q(duct)i(can)e(b)q(e)i(c)o(hosen)f(in)g(suc)o(h)g(a)g (w)o(a)o(y)f(that)g(it)h(is)257 728 y(describ)q(ed)i(b)o(y)e(the)g (matrix)901 773 y Fg(\022)932 806 y Fp(1)41 b(1)932 856 y(0)g(1)1015 773 y Fg(\023)257 935 y Fp(whic)o(h)14 b(is)g(of)f(order)i Fo(p)p Fp(.)e(This)h(yields:)257 1028 y Fq(Prop)q(osition)33 b Fp(Supp)q(ose)15 b(that)f Fo(p)g Fp(is)f(an)h(o)q(dd)g(prime.)e(Then) j(the)f(semidirect)g(pro)q(duct)834 1108 y Fo(G)867 1114 y Fn(1)897 1108 y Fp(:=)d Fk(Z)983 1115 y Fj(p)1000 1107 y Fd(2)1024 1108 y Fk(o)f(Z)1097 1114 y Fj(p)257 1189 y Fp(with)16 b(resp)q(ect)i(to)e(the)h(action)e(of)g Fk(Z)831 1195 y Fj(p)863 1189 y Fp(on)h Fk(Z)954 1196 y Fj(p)971 1188 y Fd(2)1002 1189 y Fp(giv)o(en)f(b)o(y)h(m)o (ultiplicatio)o(n)d(b)o(y)j Fo(p)11 b Fp(+)f(1,)16 b(and)257 1239 y(the)f(semidirect)f(pro)q(duct)777 1288 y Fo(G)810 1294 y Fn(2)840 1288 y Fp(:=)d(\()p Fk(Z)942 1294 y Fj(p)968 1288 y Fm(\002)e Fk(Z)1040 1294 y Fj(p)1056 1288 y Fp(\))g Fk(o)h(Z)1153 1294 y Fj(p)257 1357 y Fp(with)k(resp)q(ect)i(to)e(the)g (action)g(of)f Fk(Z)819 1363 y Fj(p)849 1357 y Fp(on)g Fk(Z)937 1363 y Fj(p)963 1357 y Fm(\002)c Fk(Z)1035 1363 y Fj(p)1065 1357 y Fp(giv)o(en)k(b)o(y)h(the)g(matrix)901 1405 y Fg(\022)932 1438 y Fp(1)41 b(1)932 1488 y(0)g(1)1015 1405 y Fg(\023)257 1572 y Fp(are,)14 b(up)g(to)g(isomorphism,)c(the)15 b(only)e(nonab)q(elian)g(groups)h(of)f(order)i Fo(p)1388 1557 y Fn(3)1406 1572 y Fp(.)257 1683 y(W)m(e)f(therefore)h(ha)o(v)o(e) f(the)g(follo)o(wing)e(corollary:)257 1776 y Fq(Corollary)35 b Fp(Supp)q(ose)21 b(that)f Fo(K)j Fp(is)c(an)h(algebraically)e(closed) i(\014eld)g(of)f(c)o(haracteristic)257 1825 y(zero.)12 b(Supp)q(ose)g(that)g Fo(p)f Fp(is)g(a)g(prime)g(n)o(um)o(b)q(er)f(and) i(that)f Fo(B)j Fp(is)d(a)g(semisimple)e(Hopf)i(algebra)257 1875 y(of)j(dimension)e Fo(p)522 1860 y Fn(3)540 1875 y Fp(.)308 1981 y(1.)20 b(Supp)q(ose)11 b(that)g Fo(B)i Fp(is)d(comm)o(utativ)n(e)e(and)i(co)q(comm)o(utativ)o(e.)d(Then)k Fo(B)i Fp(is)d(isomorphic)361 2031 y(to)k Fo(K)s Fp([)p Fk(Z)492 2038 y Fj(p)509 2030 y Fd(3)525 2031 y Fp(],)f Fo(K)s Fp([)p Fk(Z)642 2038 y Fj(p)659 2030 y Fd(2)684 2031 y Fm(\002)c Fk(Z)756 2037 y Fj(p)772 2031 y Fp(],)k(or)h Fo(K)s Fp([)p Fk(Z)940 2037 y Fj(p)966 2031 y Fm(\002)9 b Fk(Z)1038 2037 y Fj(p)1063 2031 y Fm(\002)h Fk(Z)1135 2037 y Fj(p)1152 2031 y Fp(].)308 2109 y(2.)20 b(Supp)q(ose)d(that)e Fo(B)j Fp(is)d(co)q(comm)o(utativ)o(e,)e(but)i(not)h(comm)o(utativ)n (e.)d(Then)j Fo(B)i Fp(is)d(iso-)361 2159 y(morphic)e(to)g Fo(K)s Fp([)p Fo(D)657 2165 y Fn(4)676 2159 y Fp(])h(or)f Fo(K)s Fp([)p Fo(Q)p Fp(])g(if)g Fo(p)f Fp(=)g(2,)h(and)g(to)h Fo(K)s Fp([)p Fo(G)1235 2165 y Fn(1)1253 2159 y Fp(])g(or)f Fo(K)s Fp([)p Fo(G)1412 2165 y Fn(2)1430 2159 y Fp(])h(if)f Fo(p)e Fm(6)p Fp(=)h(2.)308 2238 y(3.)20 b(Supp)q(ose)d(that)e Fo(B)j Fp(is)d(comm)o(utativ)o(e,)d(but)k(not)f(co)q(comm)o(utativ)o (e.)e(Then)j Fo(B)i Fp(is)d(iso-)361 2288 y(morphic)i(to)h Fo(K)620 2273 y Fj(D)647 2277 y Fd(4)683 2288 y Fp(or)g Fo(K)776 2273 y Fj(Q)823 2288 y Fp(if)f Fo(p)h Fp(=)h(2,)e(and)h(to)g Fo(K)1183 2273 y Fj(G)1209 2277 y Fd(1)1245 2288 y Fp(or)g Fo(K)1338 2273 y Fj(G)1364 2277 y Fd(2)1401 2288 y Fp(if)f Fo(p)h Fm(6)p Fp(=)h(2,)e(where)361 2337 y Fo(K)399 2322 y Fj(G)439 2337 y Fp(:=)11 b(Map\()p Fo(G;)c(K)s Fp(\))13 b(is)h(the)h(Hopf)e(algebra)h(of)f(functions)h(on)f(the)i(group)f Fo(G)p Fp(.)257 2443 y(The)j(reader)h(should)f(note)f(that)h(the)g (notation)f Fo(G)1073 2449 y Fn(1)1108 2443 y Fp(and)g Fo(G)1224 2449 y Fn(2)1259 2443 y Fp(is)g(in)o(terc)o(hanged)i(in)e (com-)257 2493 y(parison)e(to)g([50)o(],)f(Example)f(2.6,)g(p.)i(796.) 942 2628 y(124)p eop %%Page: 125 125 125 124 bop 257 262 a Fr(9)67 b(Semisimple)24 b(Hopf)e(algebras)g(of)g (dimension)h Fb(pq)257 613 y Fq(9.1)48 b Fp(In)14 b(this)g(section,)h (w)o(e)f(assume)g(that)g Fo(K)j Fp(is)d(an)g(algebraically)e(closed)j (\014eld)f(of)f(c)o(har-)257 662 y(acteristic)k(zero,)f(and)f(that)h Fo(p)f Fp(and)g Fo(q)i Fp(are)e(t)o(w)o(o)g(distinct)h(prime)e(n)o(um)o (b)q(ers.)h Fo(B)j Fp(denotes)f(a)257 712 y(semisimple)d(Hopf)i (algebra)h(of)e(dimension)g Fo(pq)j Fp(o)o(v)o(er)e Fo(K)k Fp(that)d(is)f(neither)h(comm)o(utativ)o(e)257 762 y(nor)e(co)q(comm)o (utativ)o(e.)d(Note)j(that)g Fo(B)j Fp(is)c(then)i(also)e(cosemisimple) e(\(cf.)j([37)o(],)f(Thm.)f(3.3,)257 812 y(p.)h(276\).)f(W)m(e)g (assume)h(that)g Fo(B)i Fp(con)o(tains)e(a)g(non)o(trivial)e(grouplik)o (e)h(elemen)o(t)h Fo(g)e Fm(6)p Fp(=)g(1)1580 818 y Fj(B)1623 812 y Fp(and)257 862 y(that)17 b Fo(B)383 847 y Fl(\003)419 862 y Fp(con)o(tains)g(a)f(non)o(trivial)f(grouplik)o(e)h(elemen)o(t)g Fo(\015)j Fm(6)p Fp(=)d Fo(\017)1258 868 y Fj(B)1287 862 y Fp(.)g(Our)h(goal)e(is)i(to)f(pro)o(v)o(e)257 911 y(that)f(these)g(assumptions)e(are)i(con)o(tradictory)m(.)e Fo(H)k Fp(denotes)f(the)f(Hopf)e(subalgebra)h(of)g Fo(B)257 961 y Fp(spanned)i(b)o(y)e(the)h(p)q(o)o(w)o(ers)g(of)f Fo(g)q Fp(.)g(Observ)o(e)i(that,)e(b)o(y)g(the)h(Nic)o(hols-Zo)q(eller) f(theorem,)g(the)257 1011 y(orders)k(of)d Fo(g)j Fp(and)e Fo(\015)j Fp(m)o(ust)c(divide)g(dim)5 b Fo(B)r Fp(.)17 b(Since)f Fo(B)j Fp(is)d(neither)h(comm)o(utativ)o(e)c(nor)j(co-)257 1061 y(comm)o(utativ)o(e,)c(these)17 b(orders)f(cannot)f(b)q(e)h(equal) f(to)g Fo(pq)q Fp(,)g(and)g(therefore)h(are)g(equal)f(to)g Fo(p)257 1111 y Fp(or)f Fo(q)q Fp(.)g(By)g(con)o(v)o(en)o(tion,)f(w)o (e)h(supp)q(ose)h(that)f(the)h(order)f(of)f Fo(g)j Fp(is)d Fo(p)p Fp(.)257 1282 y Fq(9.2)48 b Fp(The)11 b(\014rst)g(step)g(in)e (this)i(in)o(v)o(estigation)d(is)i(similar)e(to)i(the)h(\014rst)g(step) g(of)e(the)i(previous)257 1331 y(section)k(in)e(P)o(aragraph)h(8.2.)e (It)i(do)q(es)h(not)f(rely)g(on)f(the)i(existence)h(of)d Fo(\015)r Fp(.)257 1455 y Fq(Prop)q(osition)33 b Fo(g)15 b Fp(is)f(not)g(cen)o(tral.)257 1554 y Fq(Pro)q(of.)36 b Fp(Assume)17 b(on)f(the)i(con)o(trary)f(that)f Fo(g)i Fp(is)f(cen)o(tral.)g(Then)g Fo(H)i Fp(is)e(a)f(normal)f(Hopf)257 1604 y(subalgebra)e(of)f Fo(B)r Fp(.)h(The)h(corresp)q(onding)f(Hopf)g (algebra)f(quotien)o(t)h Fo(B)r(=B)r(H)1451 1589 y Fn(+)1492 1604 y Fp(has)g(dimen-)257 1654 y(sion)i Fo(q)g Fp(b)o(y)f(the)i (normal)c(basis)j(theorem)f(\(cf.)g([72)o(],)g(Thm.)f(2.4,)g(p.)h(300,) f([57)o(],)h(Cor.)g(8.4.7,)257 1704 y(p.)g(142\).)f(By)i(Zh)o(u's)f (theorem)g(\(cf.)g([90)o(],)g(Thm.)e(2,)h(p.)h(57\),)g(this)g(quotien)o (t)g(is)g(isomorphic)257 1753 y(to)j(the)g(group)g(ring)f(of)g(the)h (cyclic)g(group)g(of)f(order)i Fo(q)q Fp(.)e(By)h(a)f(di\013eren)o(t)i (v)o(ersion)f(of)f(the)257 1803 y(normal)e(basis)i(theorem)f(\(cf.)g ([72)o(],)g(Thm.)f(2.2,)g(p.)h(299,)g([57)o(],)f(Thm.)g(8.4.6,)f(p.)j (141\),)e(w)o(e)257 1853 y(can)k(conclude)g(that,)f(as)h(an)f(algebra,) f Fo(B)k Fp(is)d(isomorphic)f(to)h(a)g(crossed)j(pro)q(duct)e(of)f Fo(H)257 1903 y Fp(and)h(the)h(group)f(ring)g Fo(K)s Fp([)p Fk(Z)711 1909 y Fj(q)726 1903 y Fp(],)f(since)i(the)g(set)h(of)d (coin)o(v)n(arian)o(t)g(elemen)o(ts)h(with)g(resp)q(ect)257 1953 y(to)13 b Fo(B)r(=B)r(H)432 1938 y Fn(+)472 1953 y Fp(is)f(precisely)i Fo(H)h Fp(\(cf.)d([57)o(],)f(Prop.)h(3.4.3,)e(p.) i(34\).)g(The)h(fact)f(that)g Fo(g)i Fp(is)e(cen)o(tral)257 2003 y(implies)g(that)i(the)h(corresp)q(onding)g(action)e(is)h(trivial) e(\(cf.)i([57)o(],)f(Prop.)h(7.2.3,)d(p.)j(106\).)257 2112 y(No)o(w)21 b(w)o(e)f(kno)o(w)g(from)f(Prop)q(osition)h(1.13.2)f (that)h Fo(H)1137 2097 y Fn(2)1156 2112 y Fp(\()p Fk(Z)1202 2118 y Fj(q)1218 2112 y Fo(;)7 b(U)e Fp(\()p Fo(H)s Fp(\)\))1378 2101 y Fm(\030)1378 2114 y Fp(=)1433 2112 y Fo(U)g Fp(\()p Fo(H)s Fp(\))p Fo(=U)g Fp(\()p Fo(H)s Fp(\))1660 2097 y Fj(q)1678 2112 y Fp(,)257 2162 y(where)20 b Fo(U)5 b Fp(\()p Fo(H)s Fp(\))19 b(denotes)h(the)g(group)e(of)h(units)g(of)f Fo(H)s Fp(.)g(Since)h Fo(H)1311 2150 y Fm(\030)1311 2164 y Fp(=)1363 2162 y Fo(K)1401 2146 y Fj(p)1439 2162 y Fp(as)g(an)g(algebra)257 2211 y(and)14 b(since)h Fo(K)i Fp(is)d(algebraically)f(closed,)h(ev)o(ery)h(unit)e(is)h(a)g Fo(q)q Fp(-th)g(p)q(o)o(w)o(er,)g(and)g(w)o(e)g(see)h(that)257 2261 y Fo(H)295 2246 y Fn(2)314 2261 y Fp(\()p Fk(Z)361 2267 y Fj(q)376 2261 y Fo(;)7 b(U)e Fp(\()p Fo(H)s Fp(\)\))14 b(=)h Fm(f)p Fp(1)p Fm(g)p Fp(.)f(The)i(co)q(cycle)g(in)o(v)o(olv)o(ed) f(in)g(the)h(crossed)h(pro)q(duct)g(is)e(therefore)257 2311 y(trivial.)g(This)i(implies)e(that)i Fo(B)780 2300 y Fm(\030)780 2313 y Fp(=)829 2311 y Fo(H)d Fm(\012)d Fo(K)s Fp([)p Fk(Z)1002 2317 y Fj(q)1017 2311 y Fp(])16 b(as)h(an)g(algebra)f(\(cf.)h([57)o(],)f(Thm.)f(7.3.4,)257 2361 y(p.)f(113\);)f(in)g(particular,)g Fo(B)k Fp(is)c(comm)o(utativ)o (e.)e(This)j(is)f(a)h(con)o(tradiction.)f Fi(2)942 2628 y Fp(125)p eop %%Page: 126 126 126 125 bop 257 262 a Fq(9.3)48 b Fp(W)m(e)14 b(no)o(w)f(rule)h(out)g (the)h(case)f(that)g(the)h(orders)g(of)e Fo(g)i Fp(and)f Fo(\015)j Fp(are)d(di\013eren)o(t.)257 361 y Fq(Prop)q(osition)33 b Fo(g)15 b Fp(and)f Fo(\015)j Fp(ha)o(v)o(e)c(the)i(same)e(order)i Fo(p)p Fp(.)257 461 y Fq(Pro)q(of.)36 b Fp(\(1\))21 b(Supp)q(ose)11 b(that)f(the)g(order)h(of)e Fo(\015)k Fp(is)c Fo(q)q Fp(.)h(By)g(exc)o(hanging)f Fo(B)k Fp(and)d Fo(B)1495 446 y Fl(\003)1514 461 y Fp(,)g(w)o(e)g(could)257 511 y(exc)o(hange)15 b Fo(g)g Fp(and)e Fo(\015)r Fp(,)h(and)f(therefore)j (w)o(e)d(can)h(assume)g(that)f Fo(p)f(<)g(q)q Fp(.)h(Since)h(then)g Fo(B)1580 496 y Fl(\003)1613 511 y Fp(con-)257 560 y(tains)c(a)f (grouplik)o(e)g(elemen)o(t)g(of)g(order)h Fo(q)q Fp(,)f(w)o(e)h(get)g (a)f(Hopf)g(algebra)g(injection)g Fo(K)s Fp([)p Fk(Z)1534 566 y Fj(q)1549 560 y Fp(])i Fm(!)g Fo(B)1658 545 y Fl(\003)1678 560 y Fp(.)257 610 y(Since)16 b Fo(K)s Fp([)p Fk(Z)448 616 y Fj(q)463 610 y Fp(])f(is)g(self-dual)g(\(cf.)g(P)o(aragraph)g (1.10\),)f(w)o(e)i(get)g(b)o(y)f(dualization)f(a)h(Hopf)g(al-)257 660 y(gebra)f(surjection)841 710 y Fo(\031)f Fp(:)e Fo(B)j Fm(!)d Fo(K)s Fp([)p Fk(Z)1079 716 y Fj(q)1094 710 y Fp(])257 785 y(Then)17 b Fo(\031)q Fp(\()p Fo(g)q Fp(\))f(is)g(a)f (grouplik)o(e)g(elemen)o(t)g(whose)h(order)h(sim)o(ultaneously)c (divides)j Fo(p)f Fp(and)h Fo(q)q Fp(.)257 834 y(Therefore,)f(w)o(e)f (ha)o(v)o(e)g Fo(\031)q Fp(\()p Fo(g)q Fp(\))e(=)g(1.)h(Consider)h(the) h(space)g(of)e(coin)o(v)n(arian)o(t)g(elemen)o(ts:)626 926 y Fo(A)f Fp(:=)f Fm(f)p Fo(b)h Fm(2)f Fo(B)j Fm(j)d Fp(\(id)933 932 y Fj(B)968 926 y Fm(\012)p Fo(\031)q Fp(\)\001)1076 932 y Fj(B)1105 926 y Fp(\()p Fo(b)p Fp(\))h(=)f Fo(b)e Fm(\012)h Fp(1)p Fm(g)257 1017 y Fp(By)k(the)g(normal)d(basis)j (theorem,)e(w)o(e)i(kno)o(w)f(that)g(dim)5 b Fo(A)12 b Fp(=)g Fo(p)p Fp(.)g(Since)i Fo(A)g Fp(con)o(tains)f Fo(g)i Fp(and)257 1067 y(its)10 b(p)q(o)o(w)o(ers,)g(w)o(e)g(see)h (that)f Fo(A)g Fp(is)g(precisely)h(the)f(v)o(ector)h(space)f(spanned)h (b)o(y)f(the)g(p)q(o)o(w)o(ers)g(of)g Fo(g)q Fp(,)257 1117 y(i.)j(e.,)h Fo(A)g Fp(is)f(isomorphic)g(to)g Fo(K)s Fp([)p Fk(Z)778 1123 y Fj(p)794 1117 y Fp(])g(as)h(an)g(algebra.)f(In)h (particular,)f Fo(A)h Fp(is)g(comm)o(utati)o(v)o(e.)257 1200 y(\(2\))21 b(By)10 b(the)h(second)f(v)o(ersion)g(of)f(the)i (normal)d(basis)h(theorem)h(men)o(tioned)e(in)i(the)g(previous)257 1249 y(paragraph,)j(w)o(e)g(kno)o(w)f(that)h Fo(B)j Fp(is)c(isomorphic) g(to)g(a)h(crossed)i(pro)q(duct)f(of)e Fo(A)h Fp(and)g Fo(K)s Fp([)p Fk(Z)1651 1255 y Fj(q)1666 1249 y Fp(].)257 1299 y(Since)k Fo(A)f Fp(is)g(comm)o(utativ)o(e)d(and)j Fo(K)s Fp([)p Fk(Z)873 1305 y Fj(q)888 1299 y Fp(])g(is)g(co)q(comm)o (utativ)o(e,)d Fo(A)j Fp(is)g(not)g(only)f(a)h(t)o(wisted)257 1349 y Fo(K)s Fp([)p Fk(Z)338 1355 y Fj(q)353 1349 y Fp(]-mo)q(dule)8 b(\(cf.)h([57)o(],)g(Lem.)g(7.1.2,)e(p.)i(101\),)g (but)h(rather)h(an)e(ordinary)h Fo(K)s Fp([)p Fk(Z)1516 1355 y Fj(q)1532 1349 y Fp(]-mo)q(dule)257 1399 y(algebra.)i(As)g(in)g (P)o(aragraph)g(6.1,)e Fk(Z)827 1405 y Fj(q)854 1399 y Fp(acts)j(b)o(y)f(p)q(erm)o(utations)f(on)h(the)h(set)g Fo(E)h Fp(of)e(primitiv)o(e)257 1449 y(idemp)q(oten)o(ts)j(of)g Fo(A)p Fp(.)g(The)h(orbits)g(of)e(this)i(action)f(ha)o(v)o(e)g(length)h (1)f(or)g Fo(q)q Fp(.)g(Since)h(dim)5 b Fo(A)14 b Fp(=)257 1499 y(card)q(\()p Fo(E)r Fp(\))21 b(=)h Fo(p)e(<)i(q)q Fp(,)d(orbits)h(of)f(length)g Fo(q)i Fp(cannot)e(o)q(ccur.)i (Therefore,)f(all)e(orbits)i(are)257 1548 y(of)f(length)g(1,)g(whic)o (h)g(means)g(that)g(the)h(action)f(is)g(trivial.)e(No)o(w)i(the)h (argumen)o(t)f(used)257 1598 y(in)c(the)h(preceding)h(paragraph)e(sho)o (ws)g(that)h Fo(B)1028 1587 y Fm(\030)1028 1600 y Fp(=)1074 1598 y Fo(A)11 b Fm(\012)f Fo(K)s Fp([)p Fk(Z)1239 1604 y Fj(q)1254 1598 y Fp(])15 b(as)g(an)g(algebra)g(\(cf.)g([57)o(],)257 1648 y(Thm.)10 b(7.3.4,)f(p.)i(113\);)f(in)h(particular,)f Fo(B)k Fp(is)d(comm)o(utativ)o(e,)d(whic)o(h)j(is)g(a)g(con)o (tradiction.)f Fi(2)257 1783 y Fq(9.4)48 b Fp(Supp)q(ose)18 b(that)e Fo(\025)648 1789 y Fj(B)693 1783 y Fm(2)f Fo(B)769 1768 y Fl(\003)805 1783 y Fp(is)i(an)f(in)o(tegral)g(that)g (satis\014es)i Fo(\025)1336 1789 y Fj(B)1365 1783 y Fp(\(1)1402 1789 y Fj(B)1430 1783 y Fp(\))e(=)g(1.)g(In)g(con-)257 1833 y(sistency)h(with)e(our)g(con)o(v)o(en)o(tions)g(in)g(P)o (aragraph)g(1.12,)e(w)o(e)j(denote)g(the)f(c)o(haracter)i(ring)257 1883 y(of)e Fo(B)j Fp(b)o(y)d(Ch\()p Fo(B)r Fp(\).)h(If)f Fo(e)h Fp(is)f(a)g(primitiv)o(e)f(idemp)q(oten)o(t)g(in)h(Ch\()p Fo(B)r Fp(\),)h(the)g(class)g(equation)f(of)257 1933 y(G.)d(I.)f(Kac)i(and)f(Y.)g(Zh)o(u)g(\(cf.)g([33)o(],)f(Thm.)g(2,)g (p.)h(158,)f([90)o(],)g(Thm.)f(1,)i(p.)f(56,)g([42)o(],)h(p.)f(2842\)) 257 1983 y(sa)o(ys)j(that)g(dim)5 b Fo(B)544 1968 y Fl(\003)564 1983 y Fo(e)14 b Fp(divides)f(dim)5 b Fo(B)r Fp(.)14 b(Since)g(dim)5 b Fo(B)1088 1968 y Fl(\003)1108 1983 y Fo(e)14 b Fp(cannot)g(b)q(e)g(equal)g(to)f Fo(pq)q Fp(,)g(it)g(m)o(ust)257 2032 y(b)q(e)i(1,)e Fo(p)p Fp(,)g(or)h Fo(q)q Fp(.)257 2118 y(The)h(left)e(coregular)h(represen)o(tation)i(of) d Fo(B)k Fp(induces)d(an)g(isomorphism)699 2209 y Fo(Z)s Fp(\()p Fo(B)r Fp(\))f Fm(!)e Fp(Ch\()p Fo(B)r Fp(\))p Fo(;)c(b)k Fm(7!)g Fp(\()p Fo(b)h Fm(!)f Fo(\025)1203 2215 y Fj(B)1232 2209 y Fp(\))257 2301 y(whic)o(h)16 b(restricts)h(to)e(a)g(bijection)g(b)q(et)o(w)o(een)i(the)f(cen)o(tral) g(grouplik)o(e)f(elemen)o(ts)g(of)g Fo(B)j Fp(and)257 2350 y(those)e(idemp)q(oten)o(ts)f Fo(e)h Fp(of)e(Ch\()p Fo(B)r Fp(\))i(that)f(generate)i(one-dimensional)c(left)i(ideals)g(of)f Fo(B)1658 2335 y Fl(\003)1678 2350 y Fp(,)257 2400 y(i.)f(e.,)h (satisfy)g(dim)5 b Fo(B)589 2385 y Fl(\003)608 2400 y Fo(e)12 b Fp(=)g(1)i(\(cf.)g([71)o(],)f(Lem.)g(4.14,)f(p.)h(50,)g([66)o (],)g(Prop.)h(6,)g(p.)f(598\).)g(Note)257 2450 y(that,)d(b)o(y)h(the)g (W)m(edderburn)g(structure)i(theorem,)c(all)g(one-dimensional)g(left)h (ideals)g(of)g Fo(B)1670 2435 y Fl(\003)257 2500 y Fp(are)k(already)f (t)o(w)o(o-sided)g(ideals)g(and)g(con)o(tain)g(a)g(unique)g(idemp)q (oten)o(t,)f(whic)o(h)h(is)g(cen)o(tral.)942 2628 y(126)p eop %%Page: 127 127 127 126 bop 257 262 a Fp(Since,)13 b(b)o(y)e(Prop)q(osition)h(9.2,)f 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y(b)o(y)g(p)q(erforming)d(an)j(in)o(v)o (erse)g(discrete)h(F)m(ourier)e(transform)f(as)i(in)f(P)o(aragraph)g (7.6.)f(F)m(or)257 1778 y Fo(j)e Fp(=)e(0)p Fo(;)7 b(:)g(:)g(:)12 b(;)7 b(p)h Fm(\000)i Fp(1,)j(w)o(e)h(in)o(tro)q(duce)h(the)f(elemen)o (ts:)812 1908 y Fo(e)831 1914 y Fj(j)861 1908 y Fp(:=)921 1880 y(1)p 921 1899 21 2 v 921 1937 a Fo(p)954 1854 y Fj(p)p Fl(\000)p Fn(1)954 1869 y Fg(X)957 1957 y Fj(i)p Fn(=0)1021 1908 y Fo(\020)1042 1891 y Fl(\000)p Fj(ij)1097 1908 y Fo(\015)1120 1891 y Fj(i)257 2036 y Fp(where)h Fo(\020)i Fp(is)d(a)g(primitiv)o(e)d Fo(p)p Fp(-th)j(ro)q(ot)g(of)f (unit)o(y.)257 2127 y Fq(Prop)q(osition)308 2176 y Fp(1.)20 b Fo(p)14 b Fp(divides)g Fo(q)c Fm(\000)f Fp(1.)308 2259 y(2.)20 b(The)14 b(unique)f(nonnegativ)o(e)h(in)o(tegers)g Fo(n)989 2265 y Fj(p)1021 2259 y Fp(and)g Fo(n)1127 2265 y Fj(q)1158 2259 y Fp(satisfying)f(1)c(+)g Fo(pn)1460 2265 y Fj(p)1488 2259 y Fp(+)h Fo(q)q(n)1575 2265 y Fj(q)1605 2259 y Fp(=)i Fo(pq)361 2309 y Fp(are)i Fo(n)455 2315 y 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1174 y(assertion.)257 1257 y(\(2\))21 b(Since)15 b Fo(\017)457 1263 y Fj(B)483 1255 y Fh(\003)502 1257 y Fp(\()p Fo(e)537 1263 y Fn(0)556 1257 y Fp(\))d(=)g(1,)h(w)o(e)h(ha)o(v)o(e)g Fo(\025)855 1263 y Fj(B)883 1257 y Fo(e)902 1263 y Fn(0)933 1257 y Fp(=)e Fo(\025)1001 1263 y Fj(B)1030 1257 y Fp(.)h(Therefore)i Fo(e)1262 1263 y Fn(0)1293 1257 y Fp(=)c Fo(\025)1360 1263 y Fj(B)1398 1257 y Fp(+)f(\()p Fo(e)1475 1263 y Fn(0)1503 1257 y Fm(\000)f Fo(\025)1568 1263 y Fj(B)1597 1257 y Fp(\))14 b(is)g(a)257 1307 y(non)o(trivial)g(decomp)q(osition)f (of)i Fo(e)790 1313 y Fn(0)823 1307 y Fp(in)o(to)g(orthogonal)e(idemp)q (oten)o(ts,)h(whic)o(h)h(means)f(that)257 1357 y Fo(e)276 1363 y Fn(0)309 1357 y Fp(cannot)g(b)q(e)h(primitiv)o(e.)c(Supp)q(ose)k (that)824 1479 y Fo(e)843 1485 y Fn(0)871 1479 y Fm(\000)10 b Fo(\025)937 1485 y Fj(B)977 1479 y Fp(=)1036 1427 y Fj(m)1021 1440 y Fg(X)1024 1528 y Fj(i)p Fn(=1)1088 1479 y Fo(d)1110 1485 y Fj(i)257 1609 y Fp(is)17 b(a)f(decomp)q(osition)g (of)g 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Fp(,)g(w)o(e)h(m)o(ust)f(ha)o(v)o(e)h(dim)5 b Fo(B)1264 1877 y Fl(\003)1284 1892 y Fo(d)1306 1898 y Fj(i)1331 1892 y Fp(=)12 b Fo(p)g Fp(and)h Fo(p)f(<)f(q)q Fp(.)i(No)o(w)257 1942 y(w)o(e)h(ha)o(v)o(e:)581 2015 y Fo(q)f Fp(=)f(dim)5 b Fo(B)766 1998 y Fl(\003)785 2015 y Fo(e)804 2021 y Fn(0)835 2015 y Fp(=)12 b(1)d(+)966 1963 y Fj(m)950 1975 y Fg(X)953 2064 y Fj(i)p Fn(=1)1017 2015 y Fp(dim)c Fo(B)1126 1998 y Fl(\003)1146 2015 y Fo(d)1168 2021 y Fj(i)1193 2015 y Fp(=)12 b(1)d(+)g Fo(mp)257 2128 y Fp(W)m(e)14 b(therefore)h(see)g(that)f Fo(p)e Fm(j)f Fo(pm)h Fp(=)f Fo(q)g Fm(\000)e Fp(1.)k(Since)i(w)o(e)f(ha)o(v)o (e:)731 2239 y Fo(pq)e Fp(=)g(1)d(+)903 2211 y Fo(q)i Fm(\000)e Fp(1)p 903 2229 92 2 v 939 2267 a Fo(p)1000 2239 y(p)g Fp(+)h(\()p Fo(p)f Fm(\000)g Fp(1\))p Fo(q)257 2358 y Fp(w)o(e)14 b(m)o(ust)f(ha)o(v)o(e)h Fo(n)542 2364 y Fj(p)572 2358 y Fp(=)e Fo(m)g Fp(=)713 2340 y Fj(q)q Fl(\000)p Fn(1)p 713 2349 59 2 v 734 2373 a Fj(p)790 2358 y Fp(and)i Fo(n)896 2364 y Fj(q)925 2358 y Fp(=)e Fo(p)d Fm(\000)g Fp(1.)k(This)h(pro)o(v)o(es)g(the)g(\014rst,)g (second,)h(and)257 2408 y(\014fth)f(assertion.)942 2628 y(128)p eop %%Page: 129 129 129 128 bop 257 262 a Fp(\(3\))21 b(It)15 b(remains)f(to)h(pro)o(v)o(e) g(the)h(fourth)e(assertion.)i(Assume)f(on)f(the)i(con)o(trary)f(that,)g (for)257 311 y(some)h Fo(i)h Fm(2)f(f)p Fp(1)p Fo(;)7 b(:)g(:)g(:)k(;)c(p)k Fm(\000)g Fp(1)p Fm(g)p Fp(,)16 b Fo(e)744 317 y Fj(i)775 311 y Fp(w)o(ere)i(not)e(primitiv)o(e.)f (Then)i(it)f(could)h(b)q(e)g(decomp)q(osed)257 361 y(in)o(to)d (primitiv)o(e)d(idemp)q(oten)o(ts:)877 442 y Fo(e)896 448 y Fj(i)922 442 y Fp(=)986 390 y Fj(n)966 402 y Fg(X)967 491 y Fj(j)r Fn(=1)1033 442 y Fo(e)1052 425 y Fl(0)1052 452 y Fj(j)257 558 y Fp(This)j(w)o(ould)f(lead)h(to)g(the)g(v)o(ector)h (space)g(decomp)q(osition:)823 678 y Fo(B)856 661 y Fl(\003)876 678 y Fo(e)895 684 y Fj(i)921 678 y Fp(=)986 626 y Fj(n)964 639 y Fg(M)967 727 y Fj(j)r Fn(=1)1034 678 y Fo(B)1067 661 y Fl(\003)1087 678 y Fo(e)1106 661 y 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Fp(1,)e(w)o(e)i(ha)o(v)o(e)361 1932 y Fo(\037)p Fp(\()p Fo(g)q Fp(\))12 b(=)g(0)i(and)f Fo(p)f Fm(j)f Fo(n)p Fp(.)257 2031 y Fq(Pro)q(of.)36 b Fp(\(1\))21 b(The)12 b(pro)q(of)f(of)g(the)h(\014rst)g(statemen)o(t)f(follo)o(ws)f(the)i (pro)q(of)f(of)f([47)o(],)h(Prop.)g(1.2,)257 2081 y(p.)h(362:)e(In)i (an)o(y)f(case,)i Fo(\015)r Fp(\()p Fo(g)q Fp(\))g(is)f(a)f Fo(p)p Fp(-th)h(ro)q(ot)g(of)f(unit)o(y;)g(w)o(e)h(ha)o(v)o(e)f(to)h (sho)o(w)g(that)g(it)f(is)h(primi-)257 2131 y(tiv)o(e.)e(So)g(assume)g (on)g(the)h(con)o(trary)f(that)h Fo(\015)r Fp(\()p Fo(g)q Fp(\))i(=)f(1.)d(As)i(in)f(the)h(pro)q(of)e(of)h(Prop)q(osition)g(9.3,) 257 2180 y(w)o(e)17 b(ha)o(v)o(e)f(a)f(Hopf)h(algebra)f(injection)h Fo(K)s Fp([)p Fk(Z)959 2186 y Fj(p)975 2180 y Fp(])e Fm(!)h Fo(B)1091 2165 y Fl(\003)1111 2180 y Fp(.)g(Dually)m(,)f(w)o(e)i (get)h(a)e(Hopf)h(algebra)257 2230 y(surjection:)856 2280 y Fo(\031)d Fp(:)e Fo(B)j Fm(!)d Fo(K)1052 2263 y Fc(Z)1074 2267 y Ff(p)257 2354 y Fp(W)m(e)16 b(can)h(iden)o(tify)e Fo(K)601 2339 y Fc(Z)624 2343 y Ff(p)656 2354 y Fp(with)h Fo(K)791 2339 y Fj(p)827 2354 y Fp(b)o(y)g(mapping)e(the)j(basis)f(of)g (primitiv)o(e)e(idemp)q(oten)o(ts)257 2403 y(to)f(the)g(canonical)f (basis)g(of)g Fo(K)742 2388 y Fj(p)762 2403 y Fp(.)g(Under)h(this)g (iden)o(ti\014cation,)e(the)i(mapping)d Fo(\031)k Fp(tak)o(es)f(the)257 2453 y(form:)588 2503 y Fo(\031)g Fp(:)e Fo(B)j Fm(!)d Fo(K)784 2486 y Fj(p)803 2503 y Fo(;)c(b)k Fm(7!)g Fp(\()p Fo(\017)937 2509 y Fj(B)966 2503 y Fp(\()p Fo(b)p Fp(\))p Fo(;)c(\015)r Fp(\()p Fo(b)p Fp(\))p Fo(;)g(:)g(:)g(:)12 b(;)7 b(\015)1231 2486 y Fj(p)p Fl(\000)p Fn(1)1293 2503 y Fp(\()p Fo(b)p Fp(\)\))942 2628 y(129)p eop %%Page: 130 130 130 129 bop 257 262 a Fp(Therefore,)15 b(w)o(e)f(ha)o(v)o(e)g Fo(\031)q Fp(\()p Fo(g)q Fp(\))e(=)g(1.)h(Consider)h(the)h(space)g(of)e (coin)o(v)n(arian)o(t)g(elemen)o(ts:)626 346 y Fo(A)f Fp(:=)f Fm(f)p Fo(b)h Fm(2)f Fo(B)j Fm(j)d Fp(\(id)933 352 y Fj(B)968 346 y Fm(\012)p Fo(\031)q Fp(\)\001)1076 352 y Fj(B)1105 346 y Fp(\()p Fo(b)p Fp(\))h(=)f Fo(b)e Fm(\012)h Fp(1)p Fm(g)257 430 y Fp(By)16 b(the)g(normal)d(basis)i (theorem,)f(w)o(e)i(kno)o(w)f(that)g(dim)5 b Fo(A)14 b Fp(=)g Fo(q)q Fp(.)h(Since)g Fo(A)h Fp(con)o(tains)f Fo(g)q Fp(,)g(it)257 480 y(is)g(a)g(left)f(relativ)o(e)h Fo(K)s Fp([)p Fk(Z)638 486 y Fj(p)655 480 y Fp(]-)p Fo(B)r Fp(-Hopf)f(mo)q(dule)f(with)i(resp)q(ect)i(to)e(the)g(action)g(of)f Fo(g)i Fp(and)f(the)257 530 y(regular)c(coaction)g(of)f Fo(B)r Fp(.)g(By)h(the)h(Nic)o(hols-Zo)q(eller)e(theorem,)g Fo(A)h Fp(is)g(a)f(free)h Fo(K)s Fp([)p Fk(Z)1504 536 y Fj(p)1520 530 y Fp(]-mo)q(dule)n(;)257 580 y(in)j(particular,)f Fo(p)h Fp(divides)f Fo(q)q Fp(,)h(whic)o(h)g(is)f(a)h(con)o (tradiction.)257 660 y(\(2\))21 b(Recall)16 b(from)f(P)o(aragraph)i (9.5)e(the)j(de\014nition)e(of)g(the)i(idemp)q(oten)o(ts)e Fo(e)1479 666 y Fn(0)1498 660 y Fo(;)7 b(:)g(:)g(:)12 b(;)7 b(e)1617 666 y Fj(p)p Fl(\000)p Fn(1)1678 660 y Fp(,)257 710 y(where)15 b(w)o(e)g(de\014ne)f Fo(\020)h Fp(:=)d Fo(\015)r Fp(\()p Fo(g)q Fp(\).)i(The)h(mappings)726 794 y Fo(\030)744 800 y Fj(j)773 794 y Fp(:)c(Ch\()p Fo(B)r Fp(\))h Fm(!)f Fo(K)q(;)c(\037)k Fm(7!)g Fo(\037)p Fp(\()p Fo(g)1187 777 y Fj(j)1205 794 y Fp(\))257 879 y(are)18 b(one-dimensional)d(c)o(haracters)k(of)d(the)i(c)o(haracter)g (ring)f(that)g(satisfy)g Fo(\030)1481 885 y Fj(j)1499 879 y Fp(\()p Fo(e)1534 885 y Fj(i)1548 879 y Fp(\))g(=)g Fo(\016)1648 885 y Fj(j)r(i)1678 879 y Fp(.)257 929 y(Since)c(the)g (idemp)q(oten)o(ts)f Fo(e)687 935 y Fn(1)706 929 y Fo(;)7 b(:)g(:)g(:)k(;)c(e)824 935 y Fj(p)p Fl(\000)p Fn(1)898 929 y Fp(are)12 b(primitiv)o(e)e(b)o(y)i(Prop)q(osition)g(9.5,)f(these) i(m)o(ust)257 979 y(b)q(e)19 b(the)f(cen)o(tral)h(idemp)q(oten)o(ts)e (of)h(Ch\()p Fo(B)r Fp(\))h(that)f(generate)h(the)f(one-dimensional)e (t)o(w)o(o-)257 1028 y(sided)f(ideals)e(corresp)q(onding)i(to)f Fo(\030)815 1034 y Fn(1)834 1028 y Fo(;)7 b(:)g(:)g(:)k(;)c(\030)951 1034 y Fj(p)p Fl(\000)p Fn(1)1013 1028 y Fp(.)257 1093 y(On)17 b(the)g(other)g(hand,)f(w)o(e)g(also)g(ha)o(v)o(e)g(a)g (di\013eren)o(t)i(description)f(of)f(these)h(idemp)q(oten)o(ts.)257 1143 y(If)i Fm(f)p Fo(\037)351 1149 y Fn(1)369 1143 y Fo(;)7 b(:)g(:)g(:)12 b(;)7 b(\037)495 1149 y Fj(k)515 1143 y Fm(g)19 b Fp(is)g(the)g(set)h(of)e(irreducible)i(c)o(haracters)h (of)d Fo(B)r Fp(,)h(w)o(e)h(kno)o(w)e(from)f([80)o(],)257 1199 y(Prop.)d(3.5,)e(p.)h(211)g(that)662 1168 y Fg(P)706 1178 y Fj(k)706 1211 y(i)p Fn(=1)769 1199 y Fo(\037)795 1205 y Fj(i)817 1199 y Fm(\012)c Fo(\037)882 1201 y Fn(\026)884 1208 y Fj(i)912 1199 y Fp(is)k(a)g(Casimir)f(elemen)o(t)h(of)g(the)h(F) m(rob)q(enius)g(alge-)257 1249 y(bra)g(Ch\()p Fo(B)r Fp(\),)h(i.)d(e.,)i(w)o(e)g(ha)o(v)o(e:)743 1322 y Fj(k)723 1335 y Fg(X)726 1423 y Fj(i)p Fn(=1)789 1374 y Fo(\037\037)841 1380 y Fj(i)864 1374 y Fm(\012)c Fo(\037)930 1376 y Fn(\026)932 1383 y Fj(i)957 1374 y Fp(=)1022 1322 y Fj(k)1001 1335 y Fg(X)1004 1423 y Fj(i)p Fn(=1)1068 1374 y Fo(\037)1094 1380 y Fj(i)1117 1374 y Fm(\012)g Fo(\037)1183 1376 y Fn(\026)1185 1383 y Fj(i)1198 1374 y Fo(\037)257 1497 y Fp(for)k(all)f Fo(\037)e Fm(2)g Fp(Ch\()p Fo(B)r Fp(\).)k(This)e (implies)f(that)i(w)o(e)g(ha)o(v)o(e:)689 1571 y Fj(k)668 1583 y Fg(X)671 1672 y Fj(i)p Fn(=1)735 1623 y Fo(\037\037)787 1629 y Fj(i)801 1623 y Fo(\030)819 1629 y Fj(j)836 1623 y Fp(\()p Fo(\037)876 1625 y Fn(\026)878 1632 y Fj(i)892 1623 y Fp(\))e(=)g Fo(\030)982 1629 y Fj(j)999 1623 y Fp(\()p Fo(\037)p Fp(\))1085 1571 y Fj(k)1065 1583 y Fg(X)1068 1672 y Fj(i)p Fn(=1)1131 1623 y Fo(\037)1157 1629 y Fj(i)1171 1623 y Fo(\030)1189 1629 y Fj(j)1207 1623 y Fp(\()p Fo(\037)1247 1625 y Fn(\026)1249 1632 y Fj(i)1263 1623 y Fp(\))257 1771 y(In)i(particular,)f(since)i Fo(\037)640 1777 y Fj(A)679 1771 y Fp(=)722 1740 y Fg(P)766 1750 y Fj(k)766 1783 y(i)p Fn(=1)829 1771 y Fo(\037)855 1777 y Fj(i)869 1771 y Fo(\037)893 1773 y Fn(\026)895 1780 y Fj(i)923 1771 y Fp(\(cf.)e([80)o(],)g(P)o(ar.)h(3.3,)e(p.)h (208\),)g(w)o(e)h(ha)o(v)o(e:)361 1898 y(\()398 1846 y Fj(k)377 1859 y Fg(X)380 1947 y Fj(i)p Fn(=1)444 1898 y Fo(\037)470 1904 y Fj(i)484 1898 y Fo(\030)502 1904 y Fj(j)519 1898 y Fp(\()p Fo(\037)559 1900 y Fn(\026)561 1907 y Fj(i)575 1898 y Fp(\)\))607 1881 y Fn(2)638 1898 y Fp(=)e Fo(\030)700 1904 y Fj(j)717 1898 y Fp(\()754 1846 y Fj(k)733 1859 y Fg(X)736 1947 y Fj(i)p Fn(=1)800 1898 y Fo(\037)826 1904 y Fj(i)840 1898 y Fo(\030)858 1904 y Fj(j)876 1898 y Fp(\()p Fo(\037)916 1900 y Fn(\026)918 1907 y Fj(i)932 1898 y Fp(\)\))992 1846 y Fj(k)971 1859 y Fg(X)974 1947 y Fj(i)p Fn(=1)1038 1898 y Fo(\037)1064 1904 y Fj(i)1077 1898 y Fo(\030)1095 1904 y Fj(j)1113 1898 y Fp(\()p Fo(\037)1153 1900 y Fn(\026)1155 1907 y Fj(i)1169 1898 y Fp(\))g(=)g Fo(\037)1267 1904 y Fj(A)1294 1898 y Fp(\()p Fo(g)1331 1881 y Fj(j)1349 1898 y Fp(\))1392 1846 y Fj(k)1372 1859 y Fg(X)1375 1947 y Fj(i)p Fn(=1)1438 1898 y Fo(\037)1464 1904 y Fj(i)1478 1898 y Fo(\030)1496 1904 y Fj(j)1514 1898 y Fp(\()p Fo(\037)1554 1900 y Fn(\026)1556 1907 y Fj(i)1570 1898 y Fp(\))257 2024 y(Since)i Fo(\037)391 2030 y Fj(A)432 2024 y Fp(is)f(in)o(v)o(ertible)g(\(cf.)g([80)o(],)g (Thm.)e(3.8,)h(p.)h(215\),)f Fo(\037)1185 2030 y Fj(A)1212 2024 y Fp(\()p Fo(g)1249 2009 y Fj(j)1267 2024 y Fp(\))i(is)f(nonzero.) h(Therefore,)257 2080 y(for)c Fo(j)k Fp(=)e(1)p Fo(;)7 b(:)g(:)g(:)k(;)c(p)q Fm(\000)q Fp(1,)h(the)j(elemen)o(t)873 2064 y Fn(1)p 830 2071 104 2 v 830 2094 a Fj(\037)851 2098 y Ff(A)874 2094 y Fn(\()p Fj(g)904 2086 y Ff(j)920 2094 y Fn(\))945 2049 y Fg(P)989 2059 y Fj(k)989 2093 y(i)p Fn(=1)1051 2080 y Fo(\037)1077 2086 y Fj(i)1091 2080 y Fo(\030)1109 2086 y Fj(j)1127 2080 y Fp(\()p Fo(\037)1167 2082 y Fn(\026)1169 2089 y Fj(i)1183 2080 y Fp(\))f(is)f(an)h(idemp)q (oten)o(t)f(that)g(m)o(ust)257 2135 y(coincide)14 b(with)f(the)i(idemp) q(oten)o(t)d Fo(e)819 2141 y Fj(j)837 2135 y Fp(.)h(This)h(means)e (that)i(w)o(e)g(ha)o(v)o(e)f(for)g Fo(j)h Fp(=)e(1)p Fo(;)7 b(:)g(:)g(:)12 b(;)7 b(p)g Fm(\000)i Fp(1:)715 2234 y(1)p 664 2252 125 2 v 664 2290 a Fo(\037)690 2296 y Fj(A)717 2290 y Fp(\()p Fo(g)754 2279 y Fj(j)772 2290 y Fp(\))821 2210 y Fj(k)800 2223 y Fg(X)803 2311 y Fj(i)p Fn(=1)867 2262 y Fo(\037)893 2268 y Fj(i)906 2262 y Fo(\030)924 2268 y Fj(j)942 2262 y Fp(\()p Fo(\037)982 2264 y Fn(\026)984 2271 y Fj(i)998 2262 y Fp(\))j(=)1075 2234 y(1)p 1074 2252 21 2 v 1074 2290 a Fo(p)1107 2208 y Fj(p)p Fl(\000)p Fn(1)1107 2223 y Fg(X)1110 2311 y Fj(i)p Fn(=0)1174 2262 y Fo(\020)1195 2245 y Fl(\000)p Fj(ij)1251 2262 y Fo(\015)1274 2245 y Fj(i)257 2403 y Fp(Since)23 b(the)f(grouplik)o(e)f(elemen)o(ts)h Fo(\015)842 2388 y Fj(i)879 2403 y Fp(app)q(ear)g(among)e(all)g(c)o (haracters,)k(w)o(e)e(can)g(com-)257 2453 y(pare)f(co)q(e\016cien)o (ts.)f(W)m(e)f(then)i(get)f(that)f Fo(\037)957 2459 y Fj(A)984 2453 y Fp(\()p Fo(g)1021 2438 y Fj(j)1039 2453 y Fp(\))j(=)f Fo(p)f Fp(and)f Fo(\030)1275 2459 y Fj(j)1293 2453 y Fp(\()p Fo(\037)1335 2459 y Fj(i)1349 2453 y Fp(\))i(=)g Fo(\037)1465 2459 y Fj(i)1479 2453 y Fp(\()p Fo(g)1516 2438 y Fj(j)1534 2453 y Fp(\))g(=)h(0)d(if)257 2503 y Fo(\037)283 2509 y Fj(i)313 2503 y Fo(=)-25 b Fm(2)11 b(f)p Fo(\017)386 2509 y Fj(B)414 2503 y Fo(;)c(\015)r(;)g(:)g(:)g(:)12 b(;)7 b(\015)579 2488 y Fj(p)p Fl(\000)p Fn(1)641 2503 y Fm(g)p Fp(,)13 b(i.)g(e.,)g(if)g(the)i(degree)g(of)e Fo(\037)1091 2509 y Fj(i)1119 2503 y Fp(is)h(greater)h(than)f(1.)942 2628 y(130)p eop %%Page: 131 131 131 130 bop 257 262 a Fp(\(3\))21 b(If)d Fo(V)28 b Fp(is)18 b(a)g(simple)f(mo)q(dule)g(of)h(dimension)f(greater)i(than)f(1)g(with)h (corresp)q(onding)257 311 y(c)o(haracter)d Fo(\037)p Fp(,)d(w)o(e)h(lo)q(ok)f(at)h(the)g(eigenspaces)i(of)d Fo(g)q Fp(:)744 397 y Fo(V)778 380 y Fj(j)807 397 y Fp(:=)e Fm(f)p Fo(v)i Fm(2)e Fo(V)21 b Fm(j)11 b Fo(g)q(v)j Fp(=)d Fo(\020)1142 380 y Fj(j)1160 397 y Fo(v)q Fm(g)257 482 y Fp(Then)k(w)o(e)f(ha)o(v)o(e:)739 510 y Fj(p)p Fl(\000)p Fn(1)739 524 y Fg(X)740 612 y Fj(j)r Fn(=0)799 563 y Fp(\(dim)5 b Fo(V)925 546 y Fj(j)942 563 y Fp(\))p Fo(\020)979 546 y Fj(j)1008 563 y Fp(=)12 b Fo(\037)p Fp(\()p Fo(g)q Fp(\))g(=)g(0)257 687 y Fo(\020)j Fp(is)c(therefore)h(a)f(zero)h(of)e (the)i(p)q(olynomial)942 656 y Fg(P)985 666 y Fj(p)p Fl(\000)p Fn(1)985 700 y Fj(j)r Fn(=0)1047 687 y Fp(\(dim)5 b Fo(V)1173 672 y Fj(j)1190 687 y Fp(\))p Fo(t)1221 672 y Fj(j)1250 687 y Fm(2)12 b Fk(Q)p Fp([)o Fo(t)p Fp(],)7 b(and)k(therefore)i(the)257 749 y(minim)o(um)e(p)q(olynomial)661 718 y Fg(P)704 728 y Fj(p)p Fl(\000)p Fn(1)704 762 y Fj(j)r Fn(=0)773 749 y Fo(t)788 734 y Fj(j)821 749 y Fp(of)k Fo(\020)j Fp(divides)d(this)h(p)q(olynomial)c(\(cf.)j([28)o(],) g(Sec.)g(I)q(I)q(I.1,)257 799 y(p.)h(112\).)e(Since)i(b)q(oth)g(p)q (olynomial)o(s)d(ha)o(v)o(e)j(the)g(same)f(degree,)h(they)g(m)o(ust)f (b)q(e)h(equal)f(up)257 849 y(to)f(a)g(scalar,)f(and)h(w)o(e)g(get:)652 934 y(dim)6 b Fo(V)762 917 y Fn(0)792 934 y Fp(=)12 b(dim)5 b Fo(V)945 917 y Fn(1)976 934 y Fp(=)11 b Fo(:)c(:)g(:)j Fp(=)i(dim)5 b Fo(V)1233 917 y Fj(p)p Fl(\000)p Fn(1)257 1020 y Fp(Therefore,)19 b Fo(p)e Fp(divides)g(dim)5 b Fo(V)27 b Fp(=)18 b Fo(p)7 b Fp(dim)t Fo(V)956 1005 y Fn(0)992 1020 y Fp(\(cf.)17 b([2)o(],)g(Lem.)e(2.6,)h(p.)h(433)g(for)g (a)g(similar)257 1070 y(argumen)o(t\).)c Fi(2)257 1203 y Fq(9.7)48 b Fp(W)m(e)12 b(are)g(no)o(w)g(in)g(a)g(p)q(osition)f(that) h(allo)o(ws)f(us)i(to)f(determine)g(precisely)h(the)g(dimen-)257 1252 y(sions)h(of)g(the)g(simple)e(mo)q(dules)h(of)h Fo(B)r Fp(.)257 1340 y Fq(Prop)q(osition)308 1390 y Fp(1.)20 b(Up)12 b(to)g(isomorphism)o(,)d Fo(B)14 b Fp(has)e Fo(p)f Fp(simple)f(mo)q(dules)h(of)g(dimension)f(1)i(corresp)q(onding)361 1446 y(to)k(the)h(c)o(haracters)i Fo(\017)706 1452 y Fj(B)734 1446 y Fo(;)7 b(\015)r(;)g(\015)818 1431 y Fn(2)837 1446 y Fo(;)g(:)g(:)g(:)12 b(;)7 b(\015)960 1431 y Fj(p)p Fl(\000)p Fn(1)1022 1446 y Fp(,)16 b(and)1138 1428 y Fj(q)q Fl(\000)p Fn(1)p 1138 1436 59 2 v 1159 1460 a Fj(p)1218 1446 y Fp(simple)f(mo)q(dules)g(of)h(dimen-)361 1496 y(sion)e Fo(p)p Fp(.)308 1577 y(2.)20 b(The)14 b(c)o(haracter)i (ring)d(Ch\()p Fo(B)r Fp(\))i(is)f(comm)o(utativ)n(e.)257 1673 y Fq(Pro)q(of.)36 b Fo(B)19 b Fp(has)d Fo(p)f Fp(one-dimensional)f (represen)o(tations.)j(Denote)f(the)h(n)o(um)o(b)q(er)e(of)g(irre-)257 1723 y(ducible)i(represen)o(tations)h(that)e(are)h(not)f (one-dimensional)e(b)o(y)j Fo(m)p Fp(.)f(Since)h(these)g(are)g(of)257 1772 y(dimension)g(at)h(least)h Fo(p)p Fp(,)e(w)o(e)i(ha)o(v)o(e)f Fo(pq)i Fp(=)f(dim)5 b Fo(B)21 b Fm(\025)f Fo(p)12 b Fp(+)g Fo(p)1220 1757 y Fn(2)1239 1772 y Fo(m)p Fp(,)18 b(and)g(equalit)o(y)f(holds)h(if)257 1822 y(and)c(only)e(if)h(all)f (represen)o(tations)j(are)f(of)f(dimension)f(1)h(or)g Fo(p)p Fp(.)g(This)h(implies)d Fo(q)i Fm(\025)e Fp(1)e(+)f Fo(pm)257 1872 y Fp(resp.)15 b Fo(m)d Fm(\024)454 1853 y Fj(q)q Fl(\000)p Fn(1)p 454 1862 V 474 1886 a Fj(p)517 1872 y Fp(.)i(Therefore,)g(w)o(e)h(ha)o(v)o(e:)680 1987 y(dim)5 b(Ch\()p Fo(B)r Fp(\))13 b(=)e Fo(p)e Fp(+)h Fo(m)i Fm(\024)g Fo(p)d Fp(+)1170 1959 y Fo(q)i Fm(\000)e Fp(1)p 1170 1977 92 2 v 1206 2015 a Fo(p)257 2125 y Fp(On)18 b(the)f(other)g(hand,)g(w)o(e)g(kno)o(w)f(from)f(Prop)q(osition)i(9.5)f (that)h(a)f(complete)g(system)h(of)257 2175 y(primitiv)o(e,)11 b(orthogonal)i(idemp)q(oten)o(ts)g(of)h(the)g(c)o(haracter)h(ring)e (con)o(tains)h(1)9 b(+)g Fo(n)1533 2181 y Fj(p)1561 2175 y Fp(+)g Fo(n)1627 2181 y Fj(q)1657 2175 y Fp(=)257 2225 y(1)g(+)334 2206 y Fj(q)q Fl(\000)p Fn(1)p 334 2215 59 2 v 355 2239 a Fj(p)407 2225 y Fp(+)g Fo(p)g Fm(\000)h Fp(1)k(elemen)o(ts.)f(Therefore,)h(w)o(e)h(ha)o(v)o(e:)762 2340 y(dim)5 b(Ch\()p Fo(B)r Fp(\))12 b Fm(\025)g Fo(p)d Fp(+)1089 2312 y Fo(q)h Fm(\000)f Fp(1)p 1089 2330 92 2 v 1124 2368 a Fo(p)257 2453 y Fp(This)14 b(implies)e(that)i(equalit)o (y)f(holds,)g(i.)g(e.,)g(w)o(e)h(ha)o(v)o(e)g Fo(m)e Fp(=)1207 2435 y Fj(q)q Fl(\000)p Fn(1)p 1207 2444 59 2 v 1227 2468 a Fj(p)1284 2453 y Fp(irreducible)j(represen)o(ta-)257 2503 y(tions)10 b(that)g(are)h(not)f(one-dimensional,)d(and)j(all)f(of) g(these)j(are)e(of)g(dimension)e Fo(p)p Fp(.)h(Moreo)o(v)o(er,)942 2628 y(131)p eop %%Page: 132 132 132 131 bop 257 262 a Fp(w)o(e)15 b(see)h(that)f(the)g(dimension)e(of)h (the)h(c)o(haracter)h(ring)f(is)f(equal)g(to)h(the)g(cardinalit)o(y)f (of)g(a)257 311 y(complete)d(system)g(of)g(primitiv)o(e,)e(orthogonal)h (idemp)q(oten)o(ts.)g(F)m(or)h(a)g(semisimple)e(algebra)257 361 y(o)o(v)o(er)k(an)g(algebraically)e(closed)i(\014eld)g(\(cf.)g([90) o(],)f(Lem.)f(2,)h(p.)g(55,)g([80)o(],)g(Thm.)f(3.8,)g(p.)i(215\),)257 411 y(this)h(is)g(only)f(p)q(ossible)h(if)f(it)h(is)g(comm)o(utati)o(v) o(e.)d Fi(2)257 546 y Fq(9.8)48 b Fp(W)m(e)13 b(ha)o(v)o(e)g(pro)o(v)o (ed)h(in)f(Lemma)d(9.6)j(that)g Fo(\015)r Fp(\()p Fo(g)q Fp(\))i(is)e(a)g(primitiv)o(e)e Fo(p)p Fp(-th)i(ro)q(ot)h(of)e(unit)o (y.)257 596 y(If)588 646 y Fo(\031)h Fp(:)e Fo(B)j Fm(!)d Fo(K)784 629 y Fj(p)803 646 y Fo(;)c(b)k Fm(7!)g Fp(\()p Fo(\017)937 652 y Fj(B)966 646 y Fp(\()p Fo(b)p Fp(\))p Fo(;)c(\015)r Fp(\()p Fo(b)p Fp(\))p Fo(;)g(:)g(:)g(:)12 b(;)7 b(\015)1231 629 y Fj(p)p Fl(\000)p Fn(1)1293 646 y Fp(\()p Fo(b)p Fp(\)\))257 721 y(is)14 b(the)h(Hopf)f(algebra)g (homomo)o(rphism)c(considered)16 b(in)e(the)h(pro)q(of)e(of)h(that)g (result,)h Fo(\031)q Fp(\()p Fo(g)q Fp(\))257 770 y(is)k(a)f(non)o (trivial)f(grouplik)o(e)g(elemen)o(t)h(of)g(order)h Fo(p)p Fp(,)f(and)h(therefore)h(the)f(Hopf)f(subalge-)257 820 y(bra)d Fo(\031)q Fp(\()p Fo(H)s Fp(\))g(is)g(equal)f(to)h Fo(K)685 805 y Fj(p)704 820 y Fp(.)f(Since)h Fo(K)877 805 y Fj(p)912 820 y Fp(is)f(isomorphic)f(to)i Fo(H)s Fp(,)f(w)o(e)h(get)g(b)o(y)f(comp)q(osition)257 870 y(with)i(suc)o(h)i (an)e(isomorphism)d(a)j(Hopf)g(algebra)g(map)f(from)f Fo(B)19 b Fp(to)e Fo(H)i Fp(that)d(restricts)j(to)257 920 y(the)13 b(iden)o(tit)o(y)e(on)g Fo(H)s Fp(.)h(By)g(the)g(Radford)f (pro)r(jection)h(theorem)f(\(cf.)h([65)o(],)e(Thm.)g(3,)h(p.)h(336\),) 257 970 y(the)j(map)752 1020 y Fo(A)9 b Fm(\012)h Fo(H)k Fm(!)d Fo(B)r(;)c(a)i Fm(\012)h Fo(h)h Fm(7!)g Fo(ah)257 1094 y Fp(is)i(an)f(isomorphism)d(from)h(the)j(Radford)f(bipro)q(duct)g Fo(A)6 b Fm(\012)g Fo(H)16 b Fp(to)c Fo(B)r Fp(,)h(where)g(the)g (subspace)257 1144 y(of)h(coin)o(v)n(arian)o(t)e(elemen)o(ts)i(with)f (resp)q(ect)k(to)c Fo(\031)i Fp(is)f(denoted)h(b)o(y)641 1235 y Fo(A)c Fp(:=)h Fm(f)p Fo(b)f Fm(2)g Fo(B)j Fm(j)d Fp(\(id)c Fm(\012)p Fo(\031)q Fp(\)\001)1062 1241 y Fj(B)1091 1235 y Fp(\()p Fo(b)p Fp(\))k(=)h Fo(b)d Fm(\012)h Fp(1)p Fm(g)257 1327 y Fp(and)18 b(is)h(regarded)g(as)f(a)g(Y)m (etter-Drinfel'd)g(Hopf)g(algebra)g(o)o(v)o(er)g Fo(H)j Fp(in)d(a)g(suitable)g(w)o(a)o(y)257 1377 y(\(cf.)c([65)o(],)f(Eq.)g (\(3.3b\),)f(p.)i(337\).)e(Therefore,)j(w)o(e)f(shall)e(assume)i(from)e (no)o(w)h(on)g(that)h Fo(B)g Fp(=)257 1426 y Fo(A)8 b Fm(\012)h Fo(H)16 b Fp(is)d(a)g(Radford)f(bipro)q(duct,)h(where)h Fo(H)h Fp(=)d Fo(K)s Fp([)p Fk(Z)1117 1432 y Fj(p)1133 1426 y Fp(])h(is)g(the)h(group)f(ring)g(of)f(the)i(cyclic)257 1476 y(group)g(of)f(order)i Fo(p)f Fp(and)f Fo(A)h Fp(is)g(a)g(left)f (Y)m(etter-Drinfel'd)h(Hopf)f(algebra)h(o)o(v)o(er)g Fo(H)s Fp(.)257 1562 y(As)h(in)e(P)o(aragraph)h(8.4,)e(w)o(e)i(ha)o(v)o (e)g(that)g Fo(A)g Fp(is)g(semisimple.)c(W)m(e)k(no)o(w)f(pro)o(v)o(e:) 257 1661 y Fq(Prop)q(osition)33 b Fo(A)14 b Fp(is)g(comm)o(utativ)o(e.) 257 1761 y Fq(Pro)q(of.)36 b Fp(Supp)q(ose)14 b(that)e Fo(W)19 b Fp(is)12 b(a)h(simple)e Fo(A)p Fp(-mo)q(dule)g(whic)o(h)h(is) h(not)f(the)h(trivial)e(mo)q(dule,)257 1811 y(and)k(denote)g(the)g (corresp)q(onding)g(cen)o(trally)f(primitiv)o(e)f(idemp)q(oten)o(t)g(b) o(y)h Fo(e)p Fp(.)g(W)m(e)g(ha)o(v)o(e)h(to)257 1861 y(pro)o(v)o(e)i(that)g(dim)6 b Fo(W)22 b Fp(=)17 b(1.)g(By)g(Corollary) e(2.3,)h(there)i(is)f(a)g(simple)e Fo(B)r Fp(-mo)q(dule)h Fo(V)27 b Fp(suc)o(h)257 1910 y(that)18 b Fo(e)f Fm(2)g Fo(\024)p Fp(\()p Fo(V)9 b Fp(\).)17 b(Denote)h(the)g(c)o(haracter)h (of)d Fo(V)27 b Fp(b)o(y)17 b Fo(\037)1144 1916 y Fj(V)1173 1910 y Fp(.)g(By)g(Prop)q(osition)g(9.7,)f Fo(V)26 b Fp(has)257 1960 y(dimension)10 b(1)h(or)h Fo(p)p Fp(.)f(Since)h Fo(e)f Fm(2)h Fo(\024)p Fp(\()p Fo(V)d Fp(\),)i Fo(W)18 b Fp(is)11 b(a)g(submo)q(dule)g(of)f(the)i(restriction)h(of)e Fo(V)21 b Fp(to)11 b Fo(A)p Fp(,)257 2010 y(and)g(therefore)h(dim)5 b Fo(V)21 b Fp(=)12 b(1)f(implies)d(that)j(dim)5 b Fo(W)18 b Fp(=)12 b(1.)e(W)m(e)g(therefore)i(ma)o(y)d(assume)h(that)257 2060 y(the)k(dimension)e(of)g Fo(V)23 b Fp(is)13 b Fo(p)p Fp(.)f(The)i Fo(B)r Fp(-mo)q(dule)e Fo(V)1015 2045 y Fl(\003)1041 2060 y Fm(\012)c Fo(V)23 b Fp(can)13 b(b)q(e)h(decomp)q (osed)g(in)o(to)e(simple)257 2110 y(mo)q(dules)j(of)f(dimension)g(1)h (or)g Fo(p)p Fp(.)g(If)g Fo(m)883 2116 y Fn(1)917 2110 y Fp(is)g(the)h(n)o(um)o(b)q(er)f(of)f(one-dimensional)g(mo)q(dules)257 2159 y(and)c Fo(m)370 2165 y Fj(p)400 2159 y Fp(is)g(the)h(n)o(um)o(b)q (er)f(of)f Fo(p)p Fp(-dimensional)f(mo)q(dules)h(o)q(ccurring)i(in)f (this)g(decomp)q(osition,)257 2209 y(w)o(e)k(ha)o(v)o(e:)835 2259 y Fo(p)856 2242 y Fn(2)886 2259 y Fp(=)e Fo(m)966 2265 y Fn(1)994 2259 y Fp(+)e Fo(pm)1093 2265 y Fj(p)257 2334 y Fp(Since)15 b(b)o(y)e(Sc)o(h)o(ur's)h(lemma)d(the)j(trivial)e (mo)q(dule)h(app)q(ears)h(exactly)g(once)g(in)g(this)f(decom-)257 2384 y(p)q(osition,)j(w)o(e)i(ha)o(v)o(e)f Fo(m)630 2390 y Fn(1)666 2384 y Fm(\025)g Fp(1,)f(and)h(since)h Fo(m)989 2390 y Fn(1)1026 2384 y Fp(is)f(divisible)f(b)o(y)h Fo(p)p Fp(,)f(this)h(implies)f(that)h(a)257 2433 y(non)o(trivial)d (one-dimensional)f(mo)q(dule)h(app)q(ears)i(in)f(this)g(decomp)q (osition.)e(As)j(noted)g(in)257 2483 y(P)o(aragraph)d(1.12,)f(this)h (means)g(that)g(w)o(e)h(ha)o(v)o(e)f Fo(\037)1036 2489 y Fj(V)1064 2483 y Fp(\()p Fo(\017)1097 2489 y Fj(A)1133 2483 y Fm(\012)8 b Fo(\015)1196 2468 y Fl(0)1208 2483 y Fp(\))k(=)g Fo(\037)1306 2489 y Fj(V)1348 2483 y Fp(for)h(some)f(non) o(trivial)942 2628 y(132)p eop %%Page: 133 133 133 132 bop 257 262 a Fp(elemen)o(t)19 b Fo(\015)439 246 y Fl(0)471 262 y Fm(2)522 251 y Fp(^)519 262 y Fk(Z)550 268 y Fj(p)566 262 y Fp(,)f(i.)g(e.,)h(the)g(isotrop)o(y)g(group)g Fo(\024)1087 246 y Fl(\003)1106 262 y Fp(\()p Fo(V)9 b Fp(\))19 b(is)g(non)o(trivial,)e(and)i(therefore)257 317 y(equal)g(to)431 306 y(^)427 317 y Fk(Z)458 323 y Fj(p)474 317 y Fp(.)f(This)h(implies)e(b)o(y)h(Corollary)g(2.6)f(that)i (the)g Fk(Z)1274 323 y Fj(p)1290 317 y Fp(-orbit)f Fo(\024)p Fp(\()p Fo(V)10 b Fp(\))19 b(that)f(con-)257 367 y(tains)12 b Fo(e)h Fp(has)f(length)g Fo(p)p Fp(,)g(whic)o(h)g(means)g(that)g Fo(V)21 b Fp(is)13 b(purely)f(unstable.)g(Therefore,)h(w)o(e)g(ha)o(v)o (e)257 417 y(b)o(y)h(Prop)q(osition)f(2.4)g(that)h Fo(p)e Fp(=)f(dim)5 b Fo(V)21 b Fp(=)12 b Fo(p)7 b Fp(dim)e Fo(W)h Fp(.)13 b(This)h(implies)e(that)i(dim)5 b Fo(W)18 b Fp(=)11 b(1.)i Fi(2)257 547 y Fp(As)i(in)e(P)o(aragraph)h(8.4,)e(w)o (e)i(ha)o(v)o(e)g(the)g(follo)o(wing)e(corollary:)257 647 y Fq(Corollary)35 b Fo(A)14 b Fp(is)g(co)q(comm)o(utativ)o(e.)257 782 y Fq(9.9)48 b Fp(W)m(e)14 b(summarize)d(the)k(results)g(of)e(this)h (section)h(in)e(the)i(follo)o(wing)c(theorem:)257 906 y Fq(Theorem)36 b Fp(Supp)q(ose)11 b(that)f Fo(K)k Fp(is)c(an)g (algebraically)e(closed)j(\014eld)f(of)f(c)o(haracteristic)j(zero.)257 955 y(Supp)q(ose)f(that)f Fo(p)f Fp(and)h Fo(q)g Fp(are)g(distinct)g (prime)f(n)o(um)o(b)q(ers.)g(Supp)q(ose)h(that)g Fo(B)i Fp(is)e(a)f(semisimple)257 1005 y(Hopf)h(algebra)g(o)o(v)o(er)g Fo(K)j Fp(of)d(dimension)e Fo(pq)j Fp(suc)o(h)g(that)g(b)q(oth)f Fo(B)j Fp(and)d Fo(B)1342 990 y Fl(\003)1371 1005 y Fp(con)o(tain)g (non)o(trivial)257 1055 y(grouplik)o(e)15 b(elemen)o(ts,)h(i.)f(e,)g (grouplik)o(e)g(elemen)o(ts)h(di\013eren)o(t)h(from)d(the)j(unit.)e (Then)h Fo(B)j Fp(is)257 1105 y(comm)o(utativ)o(e)11 b(or)j(co)q(comm)o(utativ)o(e.)257 1204 y Fq(Pro)q(of.)36 b Fp(W)m(e)9 b(ha)o(v)o(e)h(sho)o(wn)f(that,)g(if)g(this)h(is)f(not)g (the)i(case,)f Fo(B)i Fp(is)d(isomorphic)f(to)h(a)g(Radford)257 1254 y(bipro)q(duct)j Fo(A)s Fm(\012)s Fo(H)s Fp(,)e(where)i Fo(A)e Fp(is)h(a)f(semisimple,)e(comm)o(utativ)o(e,)f(co)q(comm)o (utativ)o(e)i(Y)m(etter-)257 1304 y(Drinfel'd)16 b(Hopf)g(algebra)h(o)o (v)o(er)g Fo(H)i Fp(=)e Fo(K)s Fp([)p Fk(Z)962 1310 y Fj(p)978 1304 y Fp(].)f(Since)i Fo(p)1151 1289 y Fn(2)1186 1304 y Fp(do)q(es)f(not)g(divide)g(dim)5 b Fo(A)16 b Fp(=)h Fo(q)q Fp(,)257 1354 y(w)o(e)c(get)h(from)d(Theorem)h(7.9,)f(or) i(already)g(from)e(Corollary)g(6.7,)h(that)h Fo(A)g Fp(is)f(trivial.)f (F)m(rom)257 1404 y(Prop)q(osition)16 b(1.11,)f(w)o(e)i(therefore)g (see)h(that)e(either)h(the)g(action)f(or)h(the)g(coaction)f(of)g Fo(H)257 1453 y Fp(on)e Fo(A)g Fp(is)f(trivial.)f(Therefore,)j(the)f (Radford)f(bipro)q(duct)h Fo(A)9 b Fm(\012)g Fo(H)17 b Fp(is)c(either)i(as)e(an)h(algebra)257 1503 y(or)g(as)g(a)f (coalgebra)g(the)i(ordinary)e(tensor)h(pro)q(duct)h(of)e Fo(A)h Fp(and)f Fo(H)s Fp(.)g(This)h(implies)d(that)j Fo(B)257 1553 y Fp(is)g(comm)o(utativ)o(e)d(or)j(co)q(comm)o(utativ)o (e,)d(whic)o(h)i(is)h(a)g(con)o(tradiction.)f Fi(2)257 1745 y Fq(9.10)48 b Fp(W)m(e)14 b(ha)o(v)o(e)f(already)h(explained)g (in)f(P)o(aragraph)h(8.10)f(wh)o(y)h(the)g(determination)f(of)257 1795 y(Hopf)19 b(algebras)g(of)f(a)g(giv)o(en)h(dimension)e(that)i(are) g(comm)o(utativ)o(e)d(or)j(co)q(comm)o(utativ)o(e)257 1845 y(reduces)d(to)d(the)h(determination)e(of)h(\014nite)g(groups)h (of)e(that)i(order.)f(Ho)o(w)o(ev)o(er,)g(the)h(deter-)257 1895 y(mination)f(of)i(groups)g(of)g(order)h Fo(pq)h Fp(is)e(considerably)g(simpler)f(than)i(the)g(determination)257 1945 y(of)h(groups)g(of)f(order)i Fo(p)628 1930 y Fn(3)646 1945 y Fp(.)e(Supp)q(ose)i(that)f Fo(G)g Fp(is)f(a)h(group)g(of)f (order)i Fo(pq)f Fp(for)g(t)o(w)o(o)f(distinct)257 1994 y(primes)f Fo(p)h Fp(and)f Fo(q)q Fp(,)g(where)i Fo(p)d(<)h(q)q Fp(.)g(The)h(Sylo)o(w)f(subgroups)h Fo(G)1250 2000 y Fj(p)1284 1994 y Fp(and)g Fo(G)1400 2000 y Fj(q)1433 1994 y Fp(are)g(cyclic;)g(the)257 2044 y(n)o(um)o(b)q(er)d(of)g Fo(q)q Fp(-Sylo)o(w)g(subgroups)h(is)f(1)h(or)f Fo(p)p Fp(,)g(and)g(is)h(congruen)o(t)g(to)g(1)f(mo)q(dulo)e Fo(q)q Fp(.)i(There-)257 2094 y(fore,)h Fo(G)384 2100 y Fj(q)416 2094 y Fp(is)h(normal)d(\(cf.)i([34)o(],)f(Kap.)h(I)q(I)q (I.3,)f(Aufg.)h(4,)f(p.)h(47\),)g(and)g Fo(G)g Fp(is)g(the)h (semidirect)257 2144 y(pro)q(duct)f(of)e Fo(G)491 2150 y Fj(p)523 2144 y Fp(and)h Fo(G)636 2150 y Fj(q)654 2144 y Fp(.)f(Since)h(Aut\()p Fo(G)904 2150 y Fj(q)923 2144 y Fp(\))g(is)f(cyclic)h(of)f(order)i Fo(q)8 b Fm(\000)g Fp(1,)k(Aut\()p Fo(G)1490 2150 y Fj(q)1508 2144 y Fp(\))h(do)q(es)h (not)257 2194 y(con)o(tain)g(elemen)o(ts)h(of)e(order)i Fo(p)g Fp(if)e Fo(p)h Fp(do)q(es)i(not)e(divide)g Fo(q)c Fm(\000)g Fp(1,)k(and)g(therefore)i(the)f(action)257 2243 y(determining)d(the)i(semidirect)f(pro)q(duct)h(is)e(trivial)g(in) h(this)g(case.)g(If)g Fo(p)g Fp(do)q(es)g(divide)g Fo(q)8 b Fm(\000)g Fp(1,)257 2293 y(Aut)q(\()p Fo(G)377 2299 y Fj(q)395 2293 y Fp(\))i(con)o(tains)g(a)h(unique)f(subgroup)h(of)e (order)i Fo(p)p Fp(,)f(and)g(the)h(action)f(can)h(b)q(e)g(non)o (trivial.)257 2343 y(A)k(detailed)g(analysis)g(of)f(the)i(isomorphism) 11 b(classes)17 b(arising)d(in)g(this)h(latter)h(case)g(yields)257 2393 y(\(cf.)g([3)o(],)e(Chap.)h(3,)g(Exerc.)h(4,)f(p.)g(33,)f([25)o (],)h(Kap.)g(I,)g(Satz)h(8.10,)d(p.)i(40,)g([34)o(],)f(Kap.)i(I)q(I.3,) 942 2628 y(133)p eop %%Page: 134 134 134 133 bop 257 262 a Fp(Aufg.)13 b(1,)g(p.)h(35,)f([82)o(],)g(Chap.)g (2,)g Fm(x)h Fp(2,)g(Example)e(1,)h(p.)g(103\):)257 361 y Fq(Prop)q(osition)33 b Fp(Supp)q(ose)15 b(that)f Fo(p)g Fp(and)f Fo(q)i Fp(are)g(distinct)f(prime)e(n)o(um)o(b)q(ers)i(with)g Fo(p)d(<)h(q)q Fp(.)308 480 y(1.)20 b(If)14 b Fo(p)d Fk(-)g Fo(q)f Fm(\000)g Fp(1,)j(then)i(all)d(groups)i(of)g(order)g Fo(pq)h Fp(are)f(isomorphic)e(to)i Fk(Z)1433 486 y Fj(pq)1465 480 y Fp(.)308 563 y(2.)20 b(If)15 b Fo(p)e Fm(j)g Fo(q)e Fm(\000)f Fp(1,)15 b(then)g(b)q(esides)i Fk(Z)852 569 y Fj(pq)900 563 y Fp(there)f(is)f(a)g(unique)g(isomorphism)d(t)o(yp)q (e)j(of)g(non-)361 613 y(ab)q(elian)e(groups)h(of)g(order)g Fo(pq)q Fp(,)g(whic)o(h)f(is)h(a)g(semidirect)f(pro)q(duct)i Fk(Z)1442 619 y Fj(p)1467 613 y Fk(n)10 b(Z)1539 619 y Fj(q)1555 613 y Fp(.)257 732 y(As)15 b(in)e(P)o(aragraph)h(8.10,)e(w) o(e)i(ha)o(v)o(e)g(the)g(follo)o(wing)d(corollary:)257 831 y Fq(Corollary)35 b Fp(Supp)q(ose)21 b(that)f Fo(K)j Fp(is)c(an)h(algebraically)e(closed)i(\014eld)g(of)f(c)o(haracteristic) 257 881 y(zero.)d(Supp)q(ose)f(that)g Fo(p)g Fp(and)g Fo(q)h Fp(are)f(distinct)g(prime)f(n)o(um)o(b)q(ers)g(with)h Fo(p)e(<)g(q)j Fp(and)f(that)g Fo(B)257 931 y Fp(is)f(a)g(semisimple)d (Hopf)i(algebra)h(of)f(dimension)f Fo(pq)q Fp(.)308 1050 y(1.)20 b(Supp)q(ose)11 b(that)g Fo(B)i Fp(is)d(comm)o(utativ)n(e)e (and)i(co)q(comm)o(utativ)o(e.)d(Then)k Fo(B)i Fp(is)d(isomorphic)361 1099 y(to)k Fo(K)s Fp([)p Fk(Z)492 1105 y Fj(pq)525 1099 y Fp(].)308 1182 y(2.)20 b(Supp)q(ose)e(that)e Fo(B)j Fp(is)d(co)q(comm)o(utativ)o(e,)e(but)j(not)f(comm)o(utativ)o(e.)d (Then)k(w)o(e)g(ha)o(v)o(e)361 1232 y Fo(q)c Fm(\021)f Fp(1)f(mo)q(d)f Fo(p)p Fp(,)20 b(and)g Fo(B)j Fp(is)d(isomorphic)f(to)h Fo(K)s Fp([)p Fk(Z)1156 1238 y Fj(p)1186 1232 y Fk(n)14 b(Z)1263 1238 y Fj(q)1278 1232 y Fp(],)19 b(where)j Fk(Z)1479 1238 y Fj(p)1508 1232 y Fk(n)14 b(Z)1585 1238 y Fj(q)1621 1232 y Fp(is)20 b(a)361 1282 y(nonab)q(elian)c(semidirect)h(pro)q(duct) g(of)f Fk(Z)1016 1288 y Fj(p)1049 1282 y Fp(and)g Fk(Z)1163 1288 y Fj(q)1178 1282 y Fp(,)h(whic)o(h)f(is)h(unique)g(up)f(to)h(iso-) 361 1332 y(morphism.)308 1415 y(3.)j(Supp)q(ose)e(that)e Fo(B)j Fp(is)d(comm)o(utativ)o(e,)d(but)k(not)g(co)q(comm)o(utativ)o (e.)c(Then)k(w)o(e)g(ha)o(v)o(e)361 1465 y Fo(q)c Fm(\021)f Fp(1)f(mo)q(d)f Fo(p)p Fp(,)h(and)h Fo(B)i Fp(is)e(isomorphic)e(to)h Fo(K)1062 1450 y Fc(Z)1085 1454 y Ff(p)1099 1450 y Fc(nZ)1148 1454 y Ff(q)1163 1465 y Fp(,)g(where)i Fk(Z)1335 1471 y Fj(p)1356 1465 y Fk(n)5 b(Z)1424 1471 y Fj(q)1451 1465 y Fp(is)12 b(the)g(semidi-)361 1514 y(rect)j(pro)q(duct)g(ab)q(o)o(v)o (e.)942 2628 y(134)p eop %%Page: 135 135 135 134 bop 257 262 a Fr(10)67 b(Applications)257 577 y Fq(10.1)48 b Fp(In)12 b(this)h(section,)f(w)o(e)g(assume)g(that)h Fo(K)i Fp(is)d(an)g(algebraically)f(closed)h(\014eld)h(of)e(c)o(har-) 257 627 y(acteristic)j(zero,)g(and)e(that)h Fo(p)g Fp(and)g Fo(q)h Fp(are)f(t)o(w)o(o)f(distinct)i(prime)d(n)o(um)o(b)q(ers.)i (Since)g(the)g(case)257 677 y(of)g(the)i(ev)o(en)f(prime)f(2)g(has)h(b) q(een)h(treated)g(b)o(y)e(A.)h(Masuok)n(a)f(\(cf.)g([48)o(]\),)g(w)o(e) h(shall)f(assume)257 727 y(that)18 b Fo(p)g Fp(and)g Fo(q)h Fp(are)f(o)q(dd.)g Fo(B)i Fp(denotes)f(a)f(semisimple)d(Hopf)j (algebra)f(of)g(dimension)g Fo(pq)257 776 y Fp(o)o(v)o(er)i Fo(K)k Fp(that)c(is)g(neither)h(comm)o(utativ)n(e)d(nor)i(co)q(comm)o (utativ)o(e.)d(Our)k(goal)d(is)i(to)g(\014nd)257 826 y(su\016cien)o(t)d(conditions)f(for)g(the)h(existence)i(of)c(non)o (trivial)g(grouplik)o(e)h(elemen)o(ts,)f(thereb)o(y)257 876 y(arriving)e(at)h(a)g(con)o(tradiction)g(to)g(Theorem)f(9.9.)g (Among)f(the)j(su\016cien)o(t)g(conditions)e(ex-)257 926 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1818 y Fp(+)e(139.)i Fi(2)257 1931 y Fp(The)18 b(ab)q(o)o(v)o(e)e(prop)q (osition)g(can)h(b)q(e)g(used)h(to)f(determine)f(certain)i(cases)g(in)e (whic)o(h)h(semi-)257 1981 y(simple)c(Hopf)g(algebras)h(of)f(dimension) g(5)p Fo(p)g Fp(are)h(comm)o(utativ)o(e)d(or)j(co)q(comm)o(utativ)o(e:) 257 2076 y Fq(Corollary)35 b Fp(Supp)q(ose)15 b(that)e Fo(B)i Fp(is)e(a)g(semisimple)e(Hopf)h(algebra)h(of)f(dimension)g(5)p Fo(p)g Fp(o)o(v)o(er)257 2126 y(an)i(algebraically)e(closed)j(\014eld)f (of)f(c)o(haracteristic)i(zero,)f(where)h Fo(p)f Fp(is)g(an)g(o)q(dd)f (prime.)308 2235 y(1.)20 b(If)15 b Fo(p)f Fm(\021)g Fp(2)d(mo)q(d)f(5) 15 b(or)h Fo(p)e Fm(\021)g Fp(4)d(mo)q(d)f(5,)15 b(then)h Fo(B)h Fp(is)f(comm)o(utati)o(v)o(e)d(and)i(co)q(comm)o(uta-)361 2285 y(tiv)o(e.)308 2365 y(2.)20 b(If)12 b Fo(p)f Fm(2)g(f)p Fp(3)p Fo(;)c Fp(13)p Fo(;)g Fp(23)p Fo(;)g Fp(43)p Fo(;)f Fp(53)p Fo(;)g Fp(73)p Fo(;)g Fp(83)p Fo(;)g Fp(103)o Fo(;)h Fp(1)o(13)p Fo(;)f Fp(163)p Fo(;)g Fp(173)o Fo(;)h Fp(19)o(3)p Fo(;)g Fp(2)o(23)p Fo(;)f Fp(233)p Fo(;)g Fp(263)o Fm(g)p Fp(,)i(then)13 b Fo(B)361 2415 y Fp(is)h(comm)o(utativ) o(e)d(and)j(co)q(comm)o(utativ)o(e.)308 2495 y(3.)20 b(If)14 b Fo(p)d Fp(=)h(11,)h(then)h Fo(B)j Fp(is)c(comm)o(utativ)o(e)e (or)j(co)q(comm)o(utativ)o(e.)942 2628 y(137)p eop %%Page: 138 138 138 137 bop 257 262 a Fq(Pro)q(of.)36 b Fp(Supp)q(ose)18 b(that)f Fo(B)j Fp(do)q(es)e(not)f(con)o(tain)g(a)f(non)o(trivial)g (grouplik)o(e)g(elemen)o(t.)g(By)257 311 y(Lemma)d(9.4,)h(w)o(e)i(kno)o (w)f(that)h Fo(pn)798 317 y Fj(p)832 311 y Fm(\021)e(\000)p Fp(1)e(mo)q(d)e(5.)15 b(By)h(the)g(preceding)h(prop)q(osition,)d(w)o(e) 257 361 y(therefore)i(ha)o(v)o(e)d(the)i(follo)o(wing)c(cases:)352 452 y Fo(p)g Fm(\021)h Fp(1)f(mo)q(d)f(5)i Fm(\))f Fo(n)662 458 y Fj(p)692 452 y Fp(=)h(4)f Fm(\))h Fp(64)p Fo(p)e Fm(\025)i Fp(1039)f Fm(\))g Fo(p)g Fm(6)p Fp(=)h(11)352 515 y Fo(p)f Fm(\021)h Fp(2)f(mo)q(d)f(5)i Fm(\))f Fo(n)662 521 y Fj(p)692 515 y Fp(=)h(2)f Fm(\))h(\000)p Fp(58)p Fo(p)f Fm(\025)h Fp(589)352 577 y Fo(p)f Fm(\021)h Fp(3)f(mo)q(d)f(5)i Fm(\))f Fo(n)662 583 y Fj(p)692 577 y Fp(=)h(3)f Fm(\))h Fp(3)p Fo(p)f Fm(\025)g Fp(814)g Fm(\))456 639 y Fo(p)16 b(=)-26 b Fm(2)11 b(f)p Fp(3)p Fo(;)c Fp(13)p Fo(;)g Fp(23)p Fo(;)g Fp(43)p Fo(;)f Fp(53)p Fo(;)g Fp(73)p Fo(;)g Fp(83)p Fo(;)g Fp(103)o Fo(;)h Fp(11)o(3)p Fo(;)f Fp(163)p Fo(;)g Fp(173)o Fo(;)h Fp(19)o(3)p Fo(;)g Fp(2)o(23)p Fo(;)f Fp(233)p Fo(;)g Fp(263)o Fm(g)352 702 y Fo(p)11 b Fm(\021)h Fp(4)f(mo)q(d)f(5)i Fm(\))f Fo(n)662 708 y Fj(p)692 702 y Fp(=)h(1)f Fm(\))h(\000)p Fp(119)p Fo(p)e Fm(\025)i Fp(364)257 793 y(In)j(the)h(cases)g(stated)g(ab)q(o)o(v)o(e,) e Fo(B)j Fp(therefore)g(con)o(tains)e(a)f(non)o(trivial)f(grouplik)o(e) h(elemen)o(t.)257 843 y(Dually)m(,)e Fo(B)433 828 y Fl(\003)467 843 y Fp(con)o(tains)i(a)g(non)o(trivial)e(grouplik)o(e)i(elemen)o(t,)f (and)h(therefore)i Fo(B)g Fp(is)e(comm)o(u-)257 892 y(tativ)o(e)g(or)g (co)q(comm)o(utativ)o(e)d(b)o(y)i(Theorem)h(9.9.)e Fo(B)k Fp(is)e(therefore)h(a)f(group)f(ring)h(or)g(a)f(dual)257 942 y(group)i(ring.)f(Since)i(groups)f(of)g(these)h(orders)g(are)g (comm)o(utativ)n(e)d(b)o(y)i(Prop)q(osition)f(9.10,)257 992 y(except)i(for)d(the)i(case)g Fo(p)c Fp(=)h(11,)h(the)h(assertion)h (follo)o(ws.)d Fi(2)257 1127 y Fq(10.4)48 b Fp(W)m(e)15 b(no)o(w)g(consider)h(the)g(case)g Fo(q)f Fp(=)g(7.)f(In)i(this)f (case,)h(the)g(inequalit)o(y)e(deriv)o(ed)i(in)257 1177 y(Prop)q(osition)e(10.2)e(yields)i(the)h(follo)o(wing)c(results:)257 1277 y Fq(Corollary)35 b Fp(Supp)q(ose)15 b(that)e Fo(B)i Fp(is)e(a)g(semisimple)e(Hopf)h(algebra)h(of)f(dimension)g(7)p Fo(p)g Fp(o)o(v)o(er)257 1327 y(an)i(algebraically)e(closed)j(\014eld)f (of)f(c)o(haracteristic)i(zero,)f(where)h Fo(p)f Fp(is)g(an)g(o)q(dd)f (prime.)308 1446 y(1.)20 b(If)14 b Fo(p)d Fm(\021)h Fp(6)f(mo)q(d)f(7,) j(then)i Fo(B)h Fp(is)e(comm)o(utativ)o(e)d(and)j(co)q(comm)o(utativ)o (e.)308 1529 y(2.)20 b(If)14 b Fo(p)d Fm(2)g(f)p Fp(5)p Fo(;)c Fp(11)p Fo(;)g Fp(17)p Fo(;)g 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2076 y Fp(=)h(2)f Fm(\))g Fp(4)p Fo(p)h Fm(\025)f Fp(285)g Fm(\))g Fo(p)16 b(=)-25 b Fm(2)11 b(f)p Fp(3)p Fo(;)c Fp(17)p Fo(;)g Fp(31)p Fo(;)g Fp(59)o Fm(g)470 2139 y Fo(p)k Fm(\021)h Fp(4)f(mo)q(d)f(7)i Fm(\))f Fo(n)780 2145 y Fj(p)810 2139 y Fp(=)h(5)f Fm(\))g Fp(31)p Fo(p)g Fm(\025)h Fp(474)f Fm(\))g Fo(p)g Fm(6)p Fp(=)h(11)470 2201 y Fo(p)f Fm(\021)h Fp(5)f(mo)q(d)f(7)i Fm(\))f Fo(n)780 2207 y Fj(p)810 2201 y Fp(=)h(4)f Fm(\))g Fp(22)p Fo(p)g Fm(\025)h Fp(411)f Fm(\))g Fo(p)g Fm(6)p Fp(=)h(5)470 2263 y Fo(p)f Fm(\021)h Fp(6)f(mo)q(d)f(7)i Fm(\))f Fo(n)780 2269 y Fj(p)810 2263 y Fp(=)h(1)f Fm(\))g(\000)p Fp(5)p Fo(p)h Fm(\025)g Fp(222)257 2355 y(In)i(the)f(cases)i(stated)f (ab)q(o)o(v)o(e,)e Fo(B)768 2340 y Fl(\003)801 2355 y Fp(therefore)j(con)o(tains)e(a)g(non)o(trivial)e(grouplik)o(e)h(elemen) o(t.)257 2404 y(The)j(assertions)g(no)o(w)e(follo)o(w)f(as)i(in)f(the)i (preceding)g(paragraph.)e Fi(2)942 2628 y Fp(138)p eop %%Page: 139 139 139 138 bop 257 262 a Fq(10.5)48 b Fp(W)m(e)18 b(no)o(w)g(lo)q(ok)g(at) g(the)i(situation)e(where)h(the)g(dimensions)f(of)g(the)h(simple)e Fo(B)r Fp(-)257 311 y(mo)q(dules)h(divide)g(the)h(dimension)e(of)h Fo(B)r Fp(.)h(These)h(dimensions)d(then)i(m)o(ust)f(b)q(e)h(1,)f Fo(p)p Fp(,)g Fo(q)q Fp(,)257 361 y(or)e Fo(pq)q Fp(.)f(It)h(is)f(ob)o (viously)g(imp)q(ossible)f(that)h Fo(B)k Fp(has)c(a)h(simple)e(mo)q (dule)g(of)h(dimension)f Fo(pq)q Fp(,)257 411 y(b)q(ecause)22 b(then)e(it)f(w)o(ould)g(con)o(tain)g(a)g(matrix)e(ring)i(of)g (dimension)f Fo(p)1390 396 y Fn(2)1409 411 y Fo(q)1429 396 y Fn(2)1447 411 y Fp(.)h(It)h(therefore)257 461 y(follo)o(ws)15 b(directly)i(from)e(the)i(result)g(of)f(S.)g(Zh)o(u)h(already)f(used)h (ab)q(o)o(v)o(e,)f(or)g(alternativ)o(ely)257 511 y(from)j(the)i(fact)f (that)h(w)o(e)f(cannot)h(ha)o(v)o(e)f Fo(pq)j Fp(=)f(1)14 b(+)g Fo(m)1171 517 y Fj(p)1190 511 y Fo(p)1211 496 y Fn(2)1243 511 y Fp(+)g Fo(m)1325 517 y Fj(q)1344 511 y Fo(q)1364 496 y Fn(2)1403 511 y Fp(for)20 b(nonnegativ)o(e)257 560 y(in)o(tegers)12 b Fo(m)444 566 y Fj(p)474 560 y Fp(and)e Fo(m)587 566 y Fj(q)606 560 y Fp(,)g(that)h Fo(B)748 545 y Fl(\003)778 560 y Fp(con)o(tains)f(a)g(non)o(trivial)f (grouplik)o(e)h(elemen)o(t.)f(Com)o(bining)257 610 y(this)14 b(with)g(Theorem)f(9.9,)g(w)o(e)h(get:)257 710 y Fq(Corollary)35 b Fp(Supp)q(ose)15 b(that)e Fo(B)i Fp(is)e(a)g(semisimple)e(Hopf)i (algebra)f(of)h(dimension)e Fo(pq)j Fp(o)o(v)o(er)257 760 y(an)k(algebraically)d(closed)k(\014eld)e(of)g(c)o(haracteristic)i (zero,)f(where)h Fo(p)e Fp(and)h Fo(q)g Fp(are)g(distinct)257 809 y(primes.)12 b(If)g(the)h(dimensions)e(of)h(the)h(simple)e Fo(B)r Fp(-mo)q(dules)h(and)h(the)g(simple)e Fo(B)r Fp(-como)q(dules) 257 859 y(divide)j(the)g(dimension)e(of)i Fo(B)r Fp(,)g(then)g Fo(B)j Fp(is)c(comm)o(utativ)o(e)e(or)j(co)q(comm)o(utativ)o(e.)257 978 y(As)d(w)o(e)f(already)g(noted)g(at)g(the)h(b)q(eginning)e(of)h (this)g(section,)g(the)h(premise)f(of)f(this)h(corollary)257 1028 y(has)k(b)q(een)g(recen)o(tly)g(established)g(b)o(y)f(P)m(.)f (Etingof)g(and)h(S.)g(Gelaki)e(\(cf.)i([18)o(]\).)g(Their)g(pro)q(of) 257 1078 y(has)j(b)q(een)g(simpli\014ed)d(b)o(y)i(Y.)g(Tsang)g(and)g (Y.)g(Zh)o(u)g(\(cf.)g([85)o(]\),)g(and)g(b)o(y)g(H.-J.)f(Sc)o(hneider) 257 1127 y(\(cf.)g([74)o(]\).)257 1862 y Fq(Ac)o(kno)o(wledgemen)o(t)c Fp(The)k(author)g(thanks)f(N.)h(Andruskiewitsc)o(h,)g(S.)f(Natale,)f (B.)i(P)o(a-)257 1911 y(reigis,)f(and)h(P)m(.)f(Sc)o(hauen)o(burg)h (for)g(in)o(teresting)g(discussions.)g(He)g(thanks)g(M.)f(Lorenz)i(for) 257 1961 y(helpful)d(commen)o(ts)f(on)h(the)h(literature,)g(in)f (particular)g(for)h(p)q(oin)o(ting)e(out)h(reference)j([40)o(].)257 2011 y(He)f(also)f(thanks)g(N.)g(Andruskiewitsc)o(h)i(for)e(the)h(kind) f(p)q(ermission)f(to)h(use)h(his)f(approac)o(h)257 2061 y(to)h(the)h(construction)f(of)g(Y)m(etter-Drinfel'd)f(Hopf)h(algebras) g(\(cf.)f([1]\).)257 2146 y(P)o(art)j(of)g(the)g(results)h(w)o(ere)g (presen)o(ted)h(at)d(the)i(conference)h(`Hopf)d(algebras)g(and)h(quan-) 257 2196 y(tum)f(groups',)g(June)h(15-18)o(,)f(1998,)g(Brussels,)i (Belgium.)c(The)j(structure)i(theorem)e(w)o(as)257 2246 y(\014rst)e(presen)o(ted)i(at)d(the)h(`In)o(ternational)e(conference)j (on)e(algebra)g(and)g(its)g(applications',)257 2296 y(Marc)o(h)20 b(25-28,)e(1999,)g(A)o(thens,)j(United)e(States.)i(The)f(author)f (thanks)h(the)g(Lautrac)o(h)257 2346 y(foundation)h(for)h(\014nancial)g (supp)q(ort)h(making)d(the)j(visit)e(to)i(Brussels)h(p)q(ossible,)e (and)257 2395 y(the)17 b(`Graduiertenk)o(olleg:)e(Mathematik)f(im)h (Bereic)o(h)i(ihrer)g(W)m(ec)o(hselwirkung)f(mit)f(der)257 2445 y(Ph)o(ysik')i(of)f(the)i(Deutsc)o(he)g(F)m(orsc)o(h)o (ungsgemeinsc)o(haft)f(for)g(\014nancial)f(supp)q(ort)i(making)257 2495 y(the)d(visit)e(to)h(A)o(thens)h(p)q(ossible.)942 2628 y(139)p eop %%Page: 140 140 140 139 bop 257 262 a Fr(References)278 369 y Fp([1])20 b(N.)13 b(Andruskiewitsc)o(h:)i(Priv)n(ate)f(comm)o(unicati)o(on,)d (1998)278 452 y([2])20 b(N.)11 b(Andruskiewitsc)o(h/H.-J.)h(Sc)o (hneider:)g(Hopf)f(algebras)g(of)g(order)h Fo(p)1445 437 y Fn(2)1475 452 y Fp(and)f(braided)343 502 y(Hopf)i(algebras)h(of)f (order)i Fo(p)p Fp(,)e(J.)h(Algebra)g(199)f(\(1998\),)f(430-454)278 585 y([3])20 b(M.)f(Asc)o(h)o(bac)o(her:)h(Finite)g(group)f(theory)m(,) g(Cam)o(b.)e(Stud.)j(Adv.)f(Math.,)f(V)m(ol.)g(10,)343 635 y(Cam)o(b.)11 b(Univ.)i(Press,)i(Cam)o(bridge,)d(1993)278 718 y([4])20 b(M.)c(Beattie:)h(On)g(the)g(Blattner-Mon)o(tgomery)f (dualit)o(y)f(theorem)i(for)f(Hopf)g(alge-)343 768 y(bras.)c(In:)g (Azuma)o(y)o(a)e(algebras,)i(actions,)f(and)h(mo)q(dules,)f(Pro)q (ceedings)j(of)d(a)h(confer-)343 817 y(ence)17 b(in)f(honour)g(of)g (Goro)f(Azuma)o(y)o(a's)g(sev)o(en)o(tieth)i(birthda)o(y)m(,)e(Con)o (temp.)f(Math.,)343 867 y(V)m(ol.)e(124,)h(Am.)f(Math.)h(So)q(c.,)g (Pro)o(vidence,)i(1992,)d(23-28)278 950 y([5])20 b(A.)e(Borel/J.)g(P)m (.)g(Serre:)i(Th)o(\023)-20 b(eor)o(\022)g(emes)19 b(de)g(\014nitude)g (en)g(cohomologie)d(galoisienne,)343 1000 y(Commen)o(t.)10 b(Math.)k(Helv.)f(39)h(\(1964\),)e(111-164)278 1083 y([6])20 b(K.)12 b(S.)g(Bro)o(wn:)g(Cohomology)d(of)j(groups,)g(Grad.)f(T)m (exts)i(Math.,)e(V)m(ol.)g(87,)h(Springer,)343 1133 y(Berlin,)h(1982) 278 1216 y([7])20 b(P)m(.)f(Cartier:)h(Group)q(es)h(alg)o(\023)-20 b(ebriques)21 b(et)g(group)q(es)g(formels.)d(In:)i(Collo)q(que)f(sur)i (la)343 1266 y(th)o(\023)-20 b(eorie)14 b(des)h(group)q(es)g(alg)o (\023)-20 b(ebriques,)13 b(Bruxelles,)h(CBRM,)g(1962)278 1349 y([8])20 b(W.)13 b(Chin:)g(Sp)q(ectra)i(of)e(smash)g(pro)q(ducts,) i(Isr.)f(J.)g(Math.)f(72)g(\(1990\),)g(84-98)278 1432 y([9])20 b(A.)f(H.)h(Cli\013ord:)e(Represen)o(tations)j(induced)g(in)e (an)h(in)o(v)n(arian)o(t)e(subgroup,)i(Ann.)343 1482 y(Math.)13 b(38)g(\(1937\),)g(533-550)257 1565 y([10])20 b(E.)13 b(Cline:)g(Stable)h(Cli\013ord)f(theory)m(,)h(J.)f(Algebra)h (22)g(\(1972\),)e(350-364)257 1648 y([11])20 b(K.)11 b(Conrad:)g(The)h(origin)f(of)g(represen)o(tation)i(theory)m(,)e (Enseign.)h(Math.,)e(I)q(I.)i(S)o(\023)-20 b(er.)11 b(44,)343 1697 y(\(1998\),)h(361-392)257 1780 y([12])20 b(C.)10 b(W.)g(Curtis/I.)h(Reiner:)g(Metho)q(ds)g(of)g(represen)o(tation)h (theory)g(with)e(applications)343 1830 y(to)j(\014nite)i(groups)f(and)f (orders,)i(V)m(ol.)d(I,)i(Wiley)m(,)e(New)i(Y)m(ork,)f(1981)257 1913 y([13])20 b(E.)11 b(C.)g(Dade:)g(Comp)q(ounding)e(Cli\013ord's)h (theory)m(,)h(Ann.)h(Math.,)e(I)q(I.)h(Ser.)h(91)f(\(1970\),)343 1963 y(236-290)257 2046 y([14])20 b(E.)k(C.)g(Dade:)h(Cli\013ord)f (theory)h(for)f(group-graded)h(rings,)f(J.)h(Reine)g(Angew.)343 2096 y(Math.)13 b(369)g(\(1986\),)g(40-86)257 2179 y([15])20 b(E.)e(C.)g(Dade:)g(Cli\013ord)g(theory)h(for)g(group-graded)f(rings.)g (I)q(I,)h(J.)f(Reine)h(Angew.)343 2229 y(Math.)13 b(387)g(\(1988\),)g (148-181)257 2312 y([16])20 b(Y.)13 b(Doi:)g(Unifying)f(Hopf)i(mo)q (dules,)e(J.)i(Algebra)f(153)g(\(1992\),)g(373-385)257 2395 y([17])20 b(Y.)10 b(Doi/M.)f(T)m(ak)o(euc)o(hi:)h(Cleft)g(como)q (dule)g(algebras)h(for)f(a)g(bialgebra,)g(Comm)o(un.)d(Al-)343 2445 y(gebra)14 b(14)f(\(1986\),)g(801-817)942 2628 y(140)p eop %%Page: 141 141 141 140 bop 257 262 a Fp([18])20 b(P)m(.)11 b(Etingof/S.)g(Gelaki:)f (Semisimple)g(Hopf)h(algebras)h(of)g(dimension)f Fo(pq)i Fp(are)f(trivial,)343 311 y(J.)h(Algebra)h(210)f(\(1998\),)g(664-669) 257 394 y([19])20 b(M.)12 b(F)m(eth:)g(Erw)o(eiterungen)i(v)o(on)e (Bialgebren)h(und)g(ihre)g(k)o(ohomologi)o(sc)o(he)e(Besc)o(hrei-)343 444 y(bung,)i(Diplomarb)q(eit,)e(M)q(\177)-22 b(unc)o(hen,)14 b(1982)257 527 y([20])20 b(D.)10 b(Fisc)o(hman/S.)f(Mon)o (tgomery/H.-J.)f(Sc)o(hneider:)k(F)m(rob)q(enius)f(extensions)h(of)e (sub-)343 577 y(algebras)j(of)h(Hopf)f(algebras,)h(T)m(rans.)f(Am.)f (Math.)h(So)q(c.)h(349)f(\(1997\),)g(4857-4895)257 660 y([21])20 b(S.)h(Gelaki/S.)f(W)m(estreic)o(h:)i(On)g(semisimple)d(Hopf) j(algebras)g(of)f(dimension)f Fo(pq)q Fp(,)343 710 y(Preprin)o(t,)14 b(Beer)h(Shev)n(a,)f(1998,)e(to)i(app)q(ear)g(in:)f(Pro)q(c.)h(Am.)e (Math.)i(So)q(c.)257 793 y([22])20 b(P)m(.)13 b(J.)g(Hilton/U.)f(Stamm) o(bac)o(h,)f(A)i(course)i(in)f(homologi)o(cal)d(algebra,)h(Grad.)h(T)m (exts)343 843 y(Math.,)g(V)m(ol.)f(4,)h(Springer,)h(Berlin,)f(1971)257 926 y([23])20 b(I.)c(Hofstetter:)h(Erw)o(eiterungen)i(v)o(on)d (Hopf-Algebren)g(und)h(ihre)g(k)o(ohomologi)o(sc)o(he)343 976 y(Besc)o(hreibung,)e(Dissertation,)e(M)q(\177)-22 b(unc)o(hen,)15 b(1990)257 1059 y([24])20 b(I.)12 b(Hofstetter:)i (Extensions)f(of)f(Hopf)h(algebras)f(and)h(their)g(cohomological)c (descrip-)343 1108 y(tion,)k(J.)g(Algebra)h(164)f(\(1994\),)g(264-298) 257 1191 y([25])20 b(B.)i(Hupp)q(ert:)i(Endlic)o(he)f(Grupp)q(en)g(I,)g (Grundlehren)g(Math.)g(Wiss.,)e(V)m(ol.)h(134,)343 1241 y(Springer,)14 b(Berlin,)f(1967)257 1324 y([26])20 b(M.)13 b(Izumi/H.)f(Kosaki:)h(Finite-dimensional)e(Kac)k(algebras)f(arising)f (from)f(certain)343 1374 y(group)h(actions)h(on)g(a)g(factor,)f(In)o (t.)g(Math.)h(Res.)g(Not.)f(8)h(\(1996\),)e(357-370)257 1457 y([27])20 b(N.)13 b(Jacobson:)h(Lectures)h(in)f(abstract)g (algebra)g(I)q(I:)f(Linear)g(algebra,)g(2nd)h(prin)o(ting,)343 1507 y(Grad.)f(T)m(exts)h(Math.,)f(V)m(ol.)f(31,)h(Springer,)h(Berlin,) f(1984)257 1590 y([28])20 b(N.)14 b(Jacobson:)h(Lectures)j(in)c (abstract)i(algebra)f(I)q(I)q(I:)g(Theory)g(of)g(\014elds)g(and)g (Galois)343 1640 y(theory)m(,)e(Grad.)g(T)m(exts)h(Math.,)f(V)m(ol.)g (32,)g(Springer,)g(Berlin,)h(1976)257 1723 y([29])20 b(M.)27 b(Josek:)g(Zerlegung)h(induzierter)h(Darstellungen)e(f)q(\177) -22 b(ur)28 b(Hopf-Galois-Erw)o(ei-)343 1773 y(terungen,)14 b(Algebra-Ber.,)g(V)m(ol.)e(65,)h(V)m(erlag)h(Reinhard)f(Fisc)o(her,)i (M)q(\177)-22 b(unc)o(hen,)14 b(1992)257 1856 y([30])20 b(A.)9 b(Jo)o(y)o(al/R.)f(Street:)k(Braided)e(tensor)h(categories,)g (Adv.)f(Math.)f(102)h(\(1993\),)f(20-78)257 1939 y([31])20 b(G.)f(I.)g(Kac/V.)g(G.)g(P)o(aljutkin:)f(An)i(example)f(of)g(a)g(ring) h(group)f(of)h(order)g(eigh)o(t,)343 1988 y(So)o(v.)13 b(Math.)g(Surv.)h(20)f(\(1965\),)g(268-269)257 2071 y([32])20 b(G.)12 b(I.)h(Kac/V.)g(G.)g(P)o(aljutkin:)f(Finite)h(ring)g(groups,)g (T)m(rans.)g(Mosc.)h(Math.)f(So)q(c.)g(15)343 2121 y(\(1966\),)f (251-294)257 2204 y([33])20 b(G.)9 b(I.)g(Kac:)h(Certain)f(arithmetic)g (prop)q(erties)i(of)e(ring)h(groups,)f(F)m(unct.)h(Anal.)f(Appl.)g(6) 343 2254 y(\(1972\),)j(158-160)257 2337 y([34])20 b(H.)13 b(Kurzw)o(eil:)h(Endlic)o(he)g(Grupp)q(en,)g(Ho)q(c)o(hsc)o(h)o (ultext,)h(Springer,)e(Berlin,)h(1977)257 2420 y([35])20 b(R.)e(G.)h(Larson:)h(Co)q(comm)o(utati)o(v)o(e)d(Hopf)j(algebras,)f (Can.)g(J.)g(Math.)g(19)h(\(1967\),)343 2470 y(350-360)942 2628 y(141)p eop %%Page: 142 142 142 141 bop 257 262 a Fp([36])20 b(R.)13 b(G.)g(Larson:)g(Characters)i (of)f(Hopf)f(algebras,)g(J.)h(Algebra)g(17)f(\(1971\),)g(352-368)257 344 y([37])20 b(R.)15 b(G.)g(Larson/D.)g(E.)h(Radford:)e(Finite)i (dimensional)e(cosemisimple)g(Hopf)h(alge-)343 393 y(bras)f(in)f(c)o (haracteristic)j(0)d(are)i(semisimple,)10 b(J.)k(Algebra)g(117)f (\(1988\),)g(267-289)257 475 y([38])20 b(R.)e(G.)f(Larson/D.)h(E.)h (Radford:)e(Semisimple)f(Hopf)j(algebras,)f(J.)h(Algebra)f(171)343 525 y(\(1995\),)12 b(5-35)257 607 y([39])20 b(F.)12 b(W.)f(Long:)h(The) h(Brauer)g(group)g(of)f(dimo)q(dule)e(algebras,)i(J.)g(Algebra)h(30)f (\(1974\),)343 657 y(559-601)257 739 y([40])20 b(M.)10 b(Lorenz/D.)g(S.)g(P)o(assman:)f(Prime)g(ideals)h(in)g(crossed)j(pro)q (ducts)e(of)f(\014nite)h(groups,)343 789 y(Isr.)j(J.)f(Math.)h(33)f (\(1979\),)g(89-132)257 871 y([41])20 b(M.)d(Lorenz:)i(Represen)o (tations)g(of)f(\014nite-dimensional)e(Hopf)h(algebras,)h(J.)g(Alge-) 343 921 y(bra)c(188)f(\(1997\),)f(476-505)257 1003 y([42])20 b(M.)g(Lorenz:)i(On)f(the)h(class)f(equation)g(for)f(Hopf)h(algebras,)f (Pro)q(c.)i(Am.)d(Math.)343 1053 y(So)q(c.)13 b(126)h(\(1998\),)e (2841-2844)257 1135 y([43])20 b(S.)g(Mac)h(Lane:)g(Homology)l(,)d (Grundlehren)j(Math.)g(Wiss.,)f(V)m(ol.)f(114,)h(Springer,)343 1185 y(Berlin,)13 b(1963)257 1267 y([44])20 b(S.)j(Mac)g(Lane:)g (Categories)h(for)f(the)h(w)o(orking)f(mathematician,)d(Grad.)i(T)m (exts)343 1316 y(Math.,)13 b(V)m(ol.)f(5,)h(Springer,)h(Berlin,)f(1971) 257 1398 y([45])20 b(A.)h(Masuok)n(a:)f(F)m(reeness)j(of)e(Hopf)g (algebras)h(o)o(v)o(er)f(coideal)g(subalgebras,)g(Com-)343 1448 y(m)o(un.)12 b(Algebra)h(20)h(\(1992\),)e(1353-1373)257 1530 y([46])20 b(A.)d(Masuok)n(a:)f(Coideal)h(subalgebras)h(in)f (\014nite)h(Hopf)f(algebras,)g(J.)h(Algebra)f(163)343 1580 y(\(1994\),)12 b(819-831)257 1662 y([47])20 b(A.)14 b(Masuok)n(a:)f(Semisimpl)o(e)f(Hopf)i(algebras)g(of)g(dimension)f(6,)g (8,)h(Isr.)g(J.)g(Math.)g(92)343 1712 y(\(1995\),)e(361-373)257 1794 y([48])20 b(A.)c(Masuok)n(a:)g(Semisimple)f(Hopf)h(algebras)h(of)g (dimension)e(2)p Fo(p)p Fp(,)h(Comm)o(un.)e(Alge-)343 1844 y(bra)g(23)f(\(1995\),)g(1931-1940)257 1926 y([49])20 b(A.)g(Masuok)n(a:)g(The)i Fo(p)712 1911 y Fj(n)734 1926 y Fp(-theorem)f(for)g(semisimple)d(Hopf)j(algebras,)g(Pro)q(c.)g(Am.) 343 1976 y(Math.)13 b(So)q(c.)h(124)f(\(1996\),)g(735-737)257 2058 y([50])20 b(A.)c(Masuok)n(a:)g(Self-dual)g(Hopf)g(algebras)h(of)g (dimension)e Fo(p)1305 2043 y Fn(3)1340 2058 y Fp(obtained)i(b)o(y)g (an)f(ex-)343 2108 y(tension,)d(J.)h(Algebra)g(178)f(\(1995\),)g (791-806)257 2190 y([51])20 b(A.)25 b(Masuok)n(a:)g(Semisimple)e(Hopf)i (algebras)h(of)f(dimension)f(3)p Fo(p)p Fp(,)h(unpublished)343 2239 y(man)o(uscript,)12 b(1997)257 2321 y([52])20 b(A.)c(Masuok)n(a:)g (Some)f(further)j(classi\014cation)e(results)i(on)f(semisimple)d(Hopf)i (alge-)343 2371 y(bras,)e(Comm)n(un.)d(Algebra)j(24)f(\(1996\),)g (307-329)257 2453 y([53])20 b(A.)11 b(Masuok)n(a:)f(Calculations)g(of)h (some)g(groups)h(of)f(Hopf)g(algebra)g(extensions,)h(J.)f(Al-)343 2503 y(gebra)j(191)f(\(1997\),)g(568-588)942 2628 y(142)p eop %%Page: 143 143 143 142 bop 257 262 a Fp([54])20 b(A.)15 b(Masuok)n(a:)f(F)m(aithfully) f(\015at)i(forms)f(and)h(cohomology)d(of)j(Hopf)g(algebra)g(exten-)343 311 y(sions,)e(Comm)o(un.)d(Algebra)k(25)g(\(1997\),)e(1169-1197)257 394 y([55])20 b(A.)f(Masuok)n(a:)g(Quotien)o(t)i(theory)f(of)g(Hopf)g (algebras.)f(In:)h(J.)g(Bergen/S.)h(Mon)o(t-)343 444 y(gomery)15 b(\(ed.\):)h(Adv)n(ances)h(in)f(Hopf)g(algebras,)g(Lect.)h (Notes)g(Pure)h(Appl.)d(Math.,)343 494 y(V)m(ol.)d(158,)h(Dekk)o(er,)h (New)g(Y)m(ork,)f(1994,)f(107-133)257 577 y([56])20 b(A.)14 b(Masuok)n(a/D.)e(Wigner:)i(F)m(aithful)e(\015atness)k(of)d(Hopf)h (algebras,)g(J.)g(Algebra)g(170)343 627 y(\(1994\),)e(156-164)257 710 y([57])20 b(S.)10 b(Mon)o(tgomery:)f(Hopf)h(algebras)h(and)f(their) h(actions)g(on)f(rings,)h(2nd)f(revised)i(prin)o(t-)343 760 y(ing,)g(Reg.)i(Conf.)e(Ser.)j(Math.,)d(V)m(ol.)h(82,)g(Am.)f (Math.)h(So)q(c.,)g(Pro)o(vidence,)i(1997)257 843 y([58])20 b(S.)c(Mon)o(tgomery/S.)e(J.)j(Withersp)q(o)q(on:)f(Irreducible)i (represen)o(tations)h(of)d(crossed)343 892 y(pro)q(ducts,)e(J.)g(Pure)h (Appl.)e(Algebra)h(129)f(\(1998\),)g(315-326)257 976 y([59])20 b(S.)26 b(Natale:)h(On)g(semisimple)e(Hopf)i(algebras)g(of)f (dimension)g Fo(pq)1469 960 y Fn(2)1487 976 y Fp(,)h(Preprin)o(t,)343 1025 y(C\023)-21 b(ordoba,)13 b(1998,)f(to)i(app)q(ear)g(in:)f(J.)h (Algebra)257 1108 y([60])20 b(M.)g(Neuc)o(hl/P)m(.)g(Sc)o(hauen)o (burg:)h(Reconstruction)g(in)f(braided)h(categories)g(and)g(a)343 1158 y(notion)15 b(of)g(comm)o(utativ)o(e)e(bialgebra,)i(J.)h(Pure)h (Appl.)e(Algebra)h(124)g(\(1998\),)e(241-)343 1208 y(259)257 1291 y([61])20 b(W.)e(D.)h(Nic)o(hols/M.)f(B.)h(Zo)q(eller:)g(A)h(Hopf) f(algebra)f(freeness)k(theorem,)c(Am.)g(J.)343 1341 y(Math.)13 b(111)g(\(1989\),)g(381-385)257 1424 y([62])20 b(W.)9 b(D.)g(Nic)o(hols/M.)g(B.)g(Ric)o(hmond:)f(The)i(Grothendiec)o(k)g (group)g(of)f(a)h(Hopf)f(algebra,)343 1474 y(J.)k(Pure)i(Appl.)e (Algebra)h(106)f(\(1996\),)g(297-306)257 1557 y([63])20 b(B.)k(P)o(areigis:)g(Non-additiv)o(e)g(ring)g(and)h(mo)q(dule)e (theory)i(I)q(I.)f Fm(C)r Fp(-categories,)h Fm(C)r Fp(-)343 1606 y(functors)14 b(and)g Fm(C)r Fp(-morphisms,)d(Publ.)i(Math.)h(24)f (\(1977\),)g(351-361)257 1689 y([64])20 b(D.)11 b(S.)g(P)o(assman:)g (In\014nite)h(crossed)i(pro)q(ducts,)f(Pure)g(Appl.)e(Math.,)g(V)m(ol.) g(135,)f(Aca-)343 1739 y(demic)j(Press,)i(San)e(Diego,)g(1989)257 1822 y([65])20 b(D.)10 b(E.)g(Radford:)g(The)h(structure)i(of)d(Hopf)g (algebras)h(with)g(a)f(pro)r(jection,)h(J.)g(Algebra)343 1872 y(92)i(\(1985\),)g(322-347)257 1955 y([66])20 b(D.)9 b(E.)g(Radford:)g(The)h(trace)h(function)e(and)h(Hopf)f(algebras,)g(J.) h(Algebra)g(163)f(\(1994\),)343 2005 y(583-622)257 2088 y([67])20 b(A.)10 b(Rosen)o(b)q(erg:)g(Hopf)g(algebras)g(and)h(Lie)f (algebras)g(in)g(quasisymmetric)e(categories,)343 2138 y(Preprin)o(t,)14 b(Mosco)o(w,)g(1978)257 2221 y([68])20 b(B.)13 b(Sc)o(harfsc)o(h)o(w)o(erdt:)i(Dissertation,)f(M)q(\177)-22 b(unc)o(hen,)14 b(in)g(preparation)257 2304 y([69])20 b(P)m(.)13 b(Sc)o(hauen)o(burg:)h(On)h(the)f(braiding)f(on)h(a)g(Hopf)g (algebra)f(in)h(a)g(braided)g(category)m(,)343 2354 y(New)g(Y)m(ork)g (J.)f(Math.)h(4)f(\(1998\),)g(259-263)257 2437 y([70])20 b(H.-J.)10 b(Sc)o(hneider:)i(Some)f(remarks)g(on)g(exact)h(seqences)i (of)c(quan)o(tum)g(groups,)h(Com-)343 2487 y(m)o(un.)h(Algebra)h(21)h (\(1993\),)e(3337-3357)942 2628 y(143)p eop %%Page: 144 144 144 143 bop 257 262 a Fp([71])20 b(H.-J.)14 b(Sc)o(hneider:)j(Lectures) g(on)e(Hopf)g(algebras,)g(Univ)o(ersidad)g(de)h(C\023)-21 b(ordoba)15 b(T)m(ra-)343 311 y(ba)r(jos)e(de)i(Matematica,)d(Serie)i (\\B",)g(No.)f(31/95,)f(C\023)-21 b(ordoba,)13 b(Argen)o(tina,)g(1995) 257 394 y([72])20 b(H.-J.)11 b(Sc)o(hneider:)i(Normal)d(basis)i(and)g (transitivit)o(y)f(of)h(crossed)i(pro)q(ducts)f(for)f(Hopf)343 444 y(algebras,)h(J.)h(Algebra)f(152)h(\(1992\),)e(289-312)257 527 y([73])20 b(H.-J.)g(Sc)o(hneider:)i(Hopf)f(Galois)f(extensions,)h (crossed)i(pro)q(ducts,)f(and)f(Cli\013ord)343 577 y(theory)m(.)e(In:)h (J.)f(Bergen/S.)i(Mon)o(tgomery)e(\(ed.\):)g(Adv)n(ances)i(in)f(Hopf)f (algebras,)343 627 y(Lect.)14 b(Notes)h(Pure)f(Appl.)f(Math.,)g(V)m (ol.)f(158,)h(Dekk)o(er,)h(New)g(Y)m(ork,)f(1994,)f(267-297)257 710 y([74])20 b(H.-J.)10 b(Sc)o(hneider:)i(Some)e(remarks)h(on)f (factorizable)h(Hopf)g(algebras,)g(submitted)f(to:)343 760 y(Pro)q(c.)k(Am.)e(Math.)h(So)q(c.)257 843 y([75])20 b(J.)12 b(P)m(.)g(Serre:)i(Lo)q(cal)f(\014elds,)f(Grad.)g(T)m(exts)h (Math.,)f(V)m(ol.)f(67,)h(Springer,)h(Berlin,)f(1979)257 926 y([76])20 b(J.)13 b(P)m(.)h(Serre:)g(Galois)f(cohomology)m(,)d (Springer,)k(Berlin,)f(1997)257 1009 y([77])20 b(W.)12 b(M.)h(Singer:)g(Extension)h(theory)f(of)g(connected)i(Hopf)e (algebras,)g(J.)g(Algebra)g(21)343 1059 y(\(1972\),)f(1-16)257 1142 y([78])20 b(Y.)28 b(Sommerh\177)-21 b(auser:)27 b(Deformierte)h(univ)o(erselle)h(Einh)q(\177)-22 b(ullende,)28 b(Diplomarb)q(eit,)343 1191 y(M)q(\177)-22 b(unc)o(hen,)14 b(1994)257 1274 y([79])20 b(Y.)e(Sommerh\177)-21 b(auser:)18 b(Deformed)g(en)o(v)o(eloping)g(algebras,)h(New)g(Y)m(ork)g(J.)g(Math.) g(2)343 1324 y(\(1996\),)12 b(35-58)257 1407 y([80])20 b(Y.)12 b(Sommerh\177)-21 b(auser:)11 b(On)j(Kaplansky's)e(\014fth)h (conjecture,)h(J.)f(Algebra)g(204)f(\(1998\),)343 1457 y(202-224)257 1540 y([81])20 b(Y.)e(Sommerh\177)-21 b(auser:)16 b(Ribb)q(on)i(transformations,)f(in)o(tegrals,)g(and)i(triangular)e (de-)343 1590 y(comp)q(ositions,)11 b(Preprin)o(t)j(gk-mp-9707/5)o(2,)c (M)q(\177)-22 b(unc)o(hen,)14 b(to)f(app)q(ear)g(in:)g(J.)g(Algebra)257 1673 y([82])20 b(M.)14 b(Suzuki:)h(Group)f(theory)h(I,)g(Grundlehren)g (Math.)g(Wiss.,)f(V)m(ol.)f(247,)h(Springer,)343 1723 y(Berlin,)f(1982)257 1806 y([83])20 b(M.)12 b(Suzuki:)h(Group)g(theory) h(I)q(I,)f(Grundlehren)h(Math.)f(Wiss.,)f(V)m(ol.)g(248,)g(Springer,) 343 1856 y(Berlin,)h(1986)257 1939 y([84])20 b(M.)13 b(E.)h(Sw)o(eedler:)g(Hopf)g(algebras,)f(Benjamin,)f(New)i(Y)m(ork,)f (1969)257 2022 y([85])20 b(Y.)10 b(Tsang/Y.)f(Zh)o(u:)h(On)h(the)g (Drinfeld)e(double)h(of)g(a)g(Hopf)g(algebra,)f(Preprin)o(t,)i(Hong)343 2071 y(Kong,)i(1998)257 2154 y([86])20 b(S.)14 b(J.)g(Withersp)q(o)q (on:)g(Cli\013ord)g(corresp)q(ondence)j(for)e(\014nite)f(dimensional)e (Hopf)i(al-)343 2204 y(gebras,)g(Preprin)o(t,)g(T)m(oron)o(to,)e(1997,) h(to)h(app)q(ear)g(in:)f(J.)h(Algebra)257 2287 y([87])20 b(R.)11 b(William)o(s:)d(Finite)k(dimensional)e(Hopf)h(algebras,)g(Ph.) h(D.)f(thesis,)h(Florida)f(State)343 2337 y(Univ)o(ersit)o(y)m(,)i(T)m (allahassee,)g(1988)257 2420 y([88])20 b(D.)12 b(N.)g(Y)m(etter:)h (Quan)o(tum)f(groups)h(and)f(represen)o(tations)j(of)d(monoidal)e (categories,)343 2470 y(Math.)j(Pro)q(c.)h(Cam)o(b.)e(Philos.)h(So)q (c.)h(108)f(\(1990\),)f(261-290)942 2628 y(144)p eop %%Page: 145 145 145 144 bop 257 262 a Fp([89])20 b(S.)c(Zh)o(u:)h(On)g(\014nite)g (dimensional)e(semisimple)f(Hopf)j(algebras,)f(Comm)o(un.)e(Alge-)343 311 y(bra)g(21)f(\(1993\),)g(3871-3885)257 394 y([90])20 b(Y.)14 b(Zh)o(u:)h(Hopf)f(algebras)h(of)f(prime)g(dimension,)e(In)o (t.)j(Math.)f(Res.)h(Not.)f(1)h(\(1994\),)343 444 y(53-59)257 571 y(T)o(yp)q(eset)h(using)d Fm(A)550 580 y(M)595 571 y(S)h Fp(-)g(L)673 563 y Fn(A)691 571 y Fp(T)714 580 y(E)737 571 y(X)942 2628 y(145)p eop %%Page: 146 146 146 145 bop 257 262 a Fr(Sub)t(ject)23 b(index)257 421 y Fp(Action,)11 b(20,)g(28{31,)e(34,)i(35,)f(37{39,)g(43,)423 471 y(46{48,)18 b(70,)h(72,)f(84,)g(86,)h(90,)423 521 y(95,)c(97,)f(101,)g(102,)g(104,)g(105,)423 570 y(116,)c(120,)g(124,)g (125,)g(130,)g Fa(se)n(e)423 620 y(also)24 b Fp(Mo)q(dule,)e(Represen)o (ta-)423 670 y(tion)340 720 y(adjoin)o(t,)13 b Fa(73)p Fp(,)g(74,)g(75)340 770 y(coadjoin)o(t,)g Fa(75)p Fp(,)h(76)340 819 y(diagonal,)e(19)340 869 y(left,)i(35)340 919 y(righ)o(t,)f(34,)g (35,)g(83)340 969 y(trivial,)8 b(23,)h(24,)g(97,)g(112,)f(114,)h(126,) 423 1019 y(133)257 1068 y(Algebra)340 1118 y(opp)q(osite,)14 b(11,)f(13,)g(14)340 1168 y(tensor)18 b(pro)q(duct,)f(15,)f(18,)g(23,)g (31,)423 1218 y(32,)d(82,)g(109,)g(133)257 1268 y(An)o(tip)q(o)q(de,)g (11,)g(14,)f(15,)g(17,)g(23,)h(39,)f(41,)423 1318 y(43,)21 b(45{48,)g(51,)g(67,)f(71{73,)423 1367 y(77,)9 b(79,)f(81,)g(82,)g (101,)h(102,)f(115,)423 1417 y(122,)13 b(123)340 1467 y(in)o(v)o(erse,)h(73,)f(74)257 1558 y(Borel)i(subalgebra,)e(66,)g(77) 257 1650 y(Casimir)f(elemen)o(t,)h(130)257 1699 y(Category)m(,)g (12{14,)f(18,)h(19,)g(22,)g(54,)g(55)340 1749 y Fm(C)r(\000)p Fp(,)h(19)340 1799 y(quasisymmetric,)e(11,)g(12,)h(14,)g(16,)423 1849 y(18)257 1899 y(Cauc)o(h)o(y's)h(theorem,)f(111)257 1949 y(Cen)o(tralizer,)h(52,)f(54)257 1998 y(Character,)g Fa(23)p Fp(,)f(30,)g(32{36,)e(42{45,)h(52,)423 2048 y(54,)19 b(55,)f(58,)h(62,)f(66,)g(88{90,)423 2098 y(92{98,)d(100,)h(107,)f (115,)h(116,)423 2148 y(129,)d(131,)g(132)340 2198 y(irreducible,)19 b(28,)e(29,)h(34,)f(35,)h(90,)423 2247 y(130)340 2297 y(one-dimensional,)12 b(17,)g(93,)h(130)257 2347 y(Character)i(group,)f Fa(22)p Fp(,)f(28,)g(38)257 2397 y(Character)21 b(ring,)d Fa(23)p Fp(,)h(126,)f(127,)g(130{)423 2447 y(132,)13 b(135,)g(136)1046 421 y(Characteristic,)23 b(10,)f(22,)f(23,)h(28,)f (42,)1212 471 y(46,)10 b(68,)g(81,)g(82,)f(88,)h(100,)g(109,)1212 521 y(111,)e(121,)g(123{125,)f(133{135,)1212 570 y(137{139)1046 620 y(Class)14 b(equation,)f(126,)g(135)1046 670 y(Cleft)f(extension,)h Fa(se)n(e)i Fp(Extension,)d(cleft)1046 720 y(Cli\013ord)h(theory)m(,)h (28,)f(89)1046 770 y(Coaction,)h(11,)g(37,)g(39,)g(43,)g(46{48,)f(71,) 1212 819 y(72,)19 b(97,)f(101,)g(102,)g(105,)g Fa(se)n(e)1212 869 y(also)f Fp(Como)q(dule)1129 919 y(left,)c(103)1129 969 y(regular,)h(92,)e(130)1129 1019 y(righ)o(t,)h(103)1129 1068 y(trivial,)f(23,)h(114,)g(133)1046 1118 y(Coalgebra)1129 1168 y(co)q(opp)q(osite,)h(11,)f(13,)g(14,)g(71,)g(83)1129 1218 y(tensor)18 b(pro)q(duct,)f(15,)f(37,)g(39,)f(43,)1212 1268 y(46{48,)9 b(57,)i(63,)f(68,)g(100,)g(109,)1212 1318 y(115,)j(133)1046 1367 y(Cob)q(oundary)m(,)f Fa(24,)j(25)p Fp(,)e(48,)f(61,)h(65,)f(120)1046 1417 y(Co)q(c)o(hain,)j Fa(24,)h(25)p Fp(,)g(50,)f(53,)f(54,)h(56,)f(59,)1212 1467 y(61,)f(62,)g(65,)g(120)1046 1517 y(Co)q(cycle,)19 b Fa(24)p Fp(,)f(25,)f(27,)h(38{47,)e(49,)h(50,)1212 1567 y(52{55,)h(59,)g(60,)h(62,)f(64,)g(70,)1212 1616 y(79,)12 b(80,)g(83,)f(84,)h(86,)g(87,)f(103{)1212 1666 y(105,)i(108,)f(110,)h(122,)g(125)1129 1716 y(cohomologous,)18 b Fa(24,)j(25)p Fp(,)f(48,)f(50,)1212 1766 y(53,)8 b(54,)h(56,)f(59,)g (60,)g(117,)h(119{)1212 1816 y(121)1129 1866 y(normalized,)k Fa(25)p Fp(,)i(27,)f(38,)g(42,)g(44{)1212 1915 y(47,)j(50,)g(52,)f(54,) h(55,)g(58,)f(60,)1212 1965 y(62,)8 b(64,)h(66,)f(81,)g(107,)g(108,)g (110,)1212 2015 y(112,)13 b(114,)f(117,)h(121)1046 2065 y(Cohomology)1129 2115 y(class,)h(27,)f(59)1129 2164 y(group,)g Fa(24,)i(25)p Fp(,)f(27,)f(59)1129 2214 y(nonab)q(elian,)g (24,)g(31,)f(37,)h(49)1129 2264 y(of)g(groups,)h(24,)f(26)1129 2314 y(set,)h Fa(24)1046 2364 y Fp(Coideal,)f(101,)f(102)1046 2413 y(Coin)o(v)n(arian)o(t)7 b(elemen)o(t,)i(70,)f(84,)g(112,)g(113,) 1212 2463 y(125,)13 b(126,)f(130,)h(132)942 2628 y(146)p eop %%Page: 147 147 147 146 bop 257 262 a Fp(Comm)o(utator,)11 b(43,)i(80,)f(85)257 311 y(Como)q(dule,)20 b(11,)h(14,)f(21,)h(22,)g(86,)g(91,)423 361 y(101{103,)9 b Fa(se)n(e)j(also)h Fp(Coaction)340 411 y(algebra,)g(18,)g(91,)g(104)340 461 y(coalgebra,)h(17)340 511 y(co)q(diagonal)f(structure,)i(11)340 560 y(simple,)d(139)340 610 y(trivial,)g(11,)h(104,)g(109)257 660 y(Compatibilit)o(y)d (condition,)i(11,)g(21,)g Fa(38)p Fp(,)423 710 y(40,)h(44,)f(47,)h(50,) f(51,)g(105{107)257 760 y(Com)o(ultiplication,)d(11,)j(15,)f(17,)h(23,) g(67,)423 809 y(71,)i(72,)f(77,)g(79,)g(83,)h(92,)f(114,)423 859 y(115,)g(122)257 909 y(Connecting)h(homomorphism)o(,)c Fa(26)257 959 y Fp(Cosmash)j(pro)q(duct,)h(17)257 1009 y(Counit,)c(11,)g(15,)f(17,)h(39,)f(43,)h(46{48,)f(51,)423 1059 y(67,)h(72,)g(77,)g(81,)g(84,)f(115,)h(122,)423 1108 y(123)257 1158 y(Crossed)16 b(pro)q(duct,)e(28,)f(37,)g(38,)g(42,) g(43,)423 1208 y(46{48,)18 b(68,)h(70,)f(84,)g(86,)h(87,)423 1258 y(100,)8 b(104,)g(106,)g(112,)h(125,)f(126)257 1308 y(Cup)k(pro)q(duct,)f Fa(25)p Fp(,)g(45{47,)f(60{62,)f(107,)423 1357 y(108)257 1449 y(Deformed)k(en)o(v)o(eloping)g(algebra,)g(71)257 1499 y(Dihedral)d(group,)f Fa(se)n(e)14 b Fp(Group,)9 b(dihedral)257 1548 y(Dimo)q(dule,)j(21,)g(22)257 1598 y(Discrete)20 b(F)m(ourier)d(transform,)g(57,)g(63,)423 1648 y(91)340 1698 y(in)o(v)o(erse,)d(58,)f(64,)g(104,)g(114,)g(127)257 1748 y(Divisibilit)o(y)e(theorem,)i(109)257 1839 y(Equiv)n(arian)o(t)f (map,)f(26,)g(36,)h(45,)g(50,)g(52,)423 1889 y(120)257 1939 y(Exact)j(sequence)340 1988 y(long,)e(26)340 2038 y(of)h(algebras,)f(31)340 2088 y(of)h(groups,)f(124)340 2138 y(of)g(Hopf)g(algebras,)g(69,)f(70,)g(83,)g(84)340 2188 y(of)i(mo)q(dules,)e(26)257 2237 y(Extension)340 2287 y(cleft,)i(104)340 2337 y(Hopf)g(Galois,)e(33)340 2387 y(of)i(groups,)f(27)340 2437 y(of)d(Hopf)f(algebras,)g(69,)g(70,)g (83,)g(111)1046 262 y(Field,)14 b(10,)f(11,)h(17,)f(23,)h(30,)f(36,)h (37,)f(42,)1212 311 y(46,)g(49,)g(90,)g(105,)f(107)1129 361 y(algebraically)g(closed,)h(12,)g(28,)f(88,)1212 411 y(100,)c(109,)g(111,)g(121,)g(123{125,)1212 461 y(132{135,)k (137{139)1129 511 y(\014nite,)i(37,)f(45,)f(102,)h(113,)g(124)1046 560 y(F)m(rob)q(enius)h(algebra,)f(21,)g(89,)g(130)1046 610 y(F)m(rob)q(enius)h(homomorphism)o(,)c(21,)j(89)1046 660 y(F)m(unctor,)h(18,)f(49,)g(54,)f(55)1129 710 y(monoidal,)f(19)1129 760 y(quasisymmetric,)g(13,)i(14)1046 851 y(Group,)20 b(16,)g(17,)f(24,)h(25,)g(27,)g(28,)f(38,)1212 901 y(100,)13 b(101,)f(112,)h(124)1129 951 y(additiv)o(e,)g(37,)g(42,)h(43,)f(52,)g (54,)g(55,)1212 1000 y(66,)g(83)1129 1050 y(cyclic,)h(26,)f(27,)g(109) 1129 1100 y(dihedral,)g(123)1129 1150 y(elemen)o(tary)f(ab)q(elian,)g (9,)g(113,)g(124)1129 1200 y(factor,)i(24,)e(25,)h(49,)g(111)1129 1249 y(\014nite,)f(37,)f(42,)h(45,)f(46,)g(49,)g(52,)g(54,)1212 1299 y(55,)i(60,)g(62,)g(66,)g(67)1129 1349 y(\014nite)h(ab)q(elian,)f (21,)g(22)1129 1399 y(isotrop)o(y)m(,)g Fa(se)n(e)k Fp(Isotrop)o(y)d (group)1129 1449 y(m)o(ultiplicativ)o(e,)c(28,)j(102)1129 1499 y(of)g(order)i Fo(p)1306 1483 y Fn(2)1325 1499 y Fp(,)e(111,)g(123)1129 1548 y(of)g(order)i Fo(p)1306 1533 y Fn(3)1325 1548 y Fp(,)e(123,)g(124)1129 1598 y(of)g(order)i Fo(pq)q Fp(,)e(133,)g(134,)g(138)1129 1648 y(of)k(prime)f(order,)i(22,) f(23,)f(28,)g(37,)1212 1698 y(45,)e(88,)h(100,)f(111,)g(112,)g(125,) 1212 1748 y(132)1129 1797 y(of)h(units,)g(16,)f(17,)g(31,)h(33,)f(37,)g (38,)1212 1847 y(42,)f(43,)g(50,)g(52,)g(54,)g(66,)f(108)1129 1897 y(opp)q(osite,)i(83)1129 1947 y(quaternion,)f(123)1129 1997 y(Sylo)o(w,)g Fa(se)n(e)k Fp(Sylo)o(w)12 b(subgroup)1129 2046 y(symmetric,)g(43)1046 2096 y(Group)18 b(ring,)f(12,)g(21,)g(22,)g (28,)g(37,)g(42,)1212 2146 y(43,)g(49,)g(52,)f(55,)h(66,)g(68,)f(82,) 1212 2196 y(83,)e(88,)h(100,)f(101,)g(103,)g(109,)1212 2246 y(111,)8 b(112,)g(114,)g(123,)g(125,)g(128,)1212 2296 y(132,)13 b(138)1129 2345 y(dual,)g(37,)g(43,)h(49,)f(57,)g(63,)g (66,)g(82,)1212 2395 y(83,)g(100,)g(138)1046 2445 y(Grouplik)o(e)19 b(elemen)o(t,)h(16,)f(22,)h(28,)f(57,)1212 2495 y(63,)14 b(88,)f(100{103,)f(111,)h(125{)942 2628 y(147)p eop %%Page: 148 148 148 147 bop 423 262 a Fp(127,)8 b(130,)g(132,)g(133,)h(135,)f(136,)423 311 y(138,)13 b(139)340 361 y(cen)o(tral,)h(116,)e(117,)h(126,)f(135,)h (137)340 411 y(coin)o(v)n(arian)o(t,)8 b(16,)g(39,)g(100{103,)f(115,) 423 461 y(116)340 511 y(in)o(v)n(arian)o(t,)k(16,)h(39,)g(100{103,)f (115,)423 560 y(116)257 652 y(Heyneman-Sw)o(eedler)f(sigma)d(notation,) 423 702 y(11,)13 b(73,)g(74)257 751 y(Hopf)h(algebra,)g(11,)f(12,)g (15{18,)g(20,)g(23,)423 801 y(28,)k(35,)g(51,)g(55,)f(66,)h(69,)g(71,) 423 851 y(76,)c(81,)g(82,)g(101,)g(111)340 901 y(co)q(comm)o(utativ)o (e,)e(101,)i(123)340 951 y(comm)o(utativ)o(e,)e(123,)h(133)340 1000 y(co)q(opp)q(osite,)i(11,)f(13,)g(14,)g(83)340 1050 y(dual,)g(17,)g(28)340 1100 y(\014nite-dimensional,)f(19)340 1150 y(in)i(a)f(category)m(,)h(12,)f(14,)g(16)340 1200 y(of)h(dimension)e(3)p Fo(p)p Fp(,)h(136)340 1249 y(of)h(dimension)e(5) p Fo(p)p Fp(,)h(136,)g(137)340 1299 y(of)h(dimension)e(7)p Fo(p)p Fp(,)h(138)340 1349 y(of)h(dimension)e Fo(p)605 1334 y Fn(2)623 1349 y Fp(,)i(111)340 1399 y(of)g(dimension)f Fo(p)606 1384 y Fn(3)625 1399 y Fp(,)h(115,)f(121,)g(123,)423 1449 y(124)340 1499 y(of)i(dimension)f Fo(pq)q Fp(,)h(125,)f(133{135,) 423 1548 y(139)340 1598 y(of)g(dimension)e(8,)h(121{123)340 1648 y(opp)q(osite,)h(11,)f(13,)g(14)340 1698 y(quotien)o(t,)h(69,)f (111,)f(125)340 1748 y(subalgebra,)i(19,)f(69,)g(125,)f(132)382 1797 y(normal,)f(111,)i(125)340 1847 y(tensor)i(pro)q(duct,)g(14)340 1897 y(Y)m(etter-Drinfel'd,)f Fa(se)n(e)j Fp(Y)m(etter-)423 1947 y(Drinfel'd)c(Hopf)h(algebra)257 1997 y(Hopf)21 b(Galois)e(extension,)i Fa(se)n(e)j Fp(Exten-)423 2046 y(sion,)14 b(Hopf)f(Galois)257 2096 y(Hopf)h(mo)q(dule,)e(130)257 2188 y(Ideal,)i(88,)e(89)340 2237 y(left,)i(32,)e(92,)h(101,)g(126,)g (127,)f(136)340 2287 y(one-dimensional,)g(90)340 2337 y(prime,)h(28,)g(29)340 2387 y(righ)o(t,)g(101)340 2437 y(t)o(w)o(o-sided,)c(30{34,)f(43,)g(80,)g(85,)g(90{)423 2487 y(92,)j(101,)e(113,)h(114,)g(126,)g(130,)1212 262 y(137)1046 311 y(Idemp)q(oten)o(t,)f(57,)f(63,)g(96,)g(121,)g(126,)g (127,)1212 361 y(130)1129 411 y(cen)o(tral,)14 b(126,)f(130)1129 461 y(cen)o(trally)k(primitiv)o(e,)d(28{30,)h(33{)1212 511 y(35,)e(91,)g(113,)g(132)1129 560 y(coin)o(v)n(arian)o(t,)f(90,)h (94,)g(97)1129 610 y(in)o(v)n(arian)o(t,)f(93,)h(94,)g(97)1129 660 y(primitiv)o(e,)c(37,)h(43,)g(46,)g(49,)g(52,)g(55,)1212 710 y(66,)j(88,)h(90,)f(92,)g(93,)g(95,)h(114,)1212 760 y(126,)f(128{130,)e(135,)i(136)1129 809 y(purely)h(unstable,)g Fa(30)1129 859 y Fp(stable,)g Fa(30)1046 909 y Fp(Idemp)q(oten)o(ts,)k (orthogonal,)e(128,)h(131,)1212 959 y(132,)c(135{137)1046 1009 y(In)o(tegral,)18 b(20,)f(24,)h(68,)f(69,)h(81{83,)e(89,)1212 1059 y(126,)d(128,)f(135)1129 1108 y(c)o(haracter,)j(81)1129 1158 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y(induced,)h(32,)f(33,)g(35)1129 2428 y(left,)g(73,)g(137)1129 2478 y(nonab)q(elian,)g(24,)g(26,)f(31,)h(37,)g(49)942 2628 y(148)p eop %%Page: 149 149 149 148 bop 382 262 a Fp(tensor)15 b(pro)q(duct,)f(25)340 311 y(one-dimensional,)e(93,)g(95,)h(96)340 361 y(purely)g(unstable,)g Fa(30)p Fp(,)g(32{34,)e(88{)423 411 y(90,)i(95,)g(114,)g(133)340 461 y(restricted,)j(33)340 511 y(righ)o(t,)d(11,)g(73)340 560 y(simple,)8 b(24,)g(28{30,)f(32,)i(34{36,)e(89,)423 610 y(92,)15 b(95,)f(113,)g(129,)g(131,)g(132,)423 660 y(135{137,)e(139)340 710 y(stable,)i Fa(30)p Fp(,)g(33,)f(88,)g(90,)g (95,)f(113)340 760 y(trivial,)c(11,)h(26,)g(46,)g(59,)f(60,)h(62,)g (64,)423 809 y(108{110,)j(132)340 859 y(t)o(wisted,)i(126)257 949 y(Nic)o(hols-Zo)q(eller)c(theorem,)e(19,)g(20,)h(111,)423 999 y(112,)k(125,)g(128,)f(130)257 1049 y(Normal)g(basis,)i(84,)f(86) 257 1098 y(Normal)e(basis)i(theorem,)f(70,)g(111{113,)423 1148 y(125,)h(126,)g(130)257 1238 y(Orbit,)d(28{30,)f(32,)g(34{36,)f (89{95,)h(101{)423 1288 y(103,)k(120,)g(126)340 1337 y(purely)g(unstable,)g Fa(30)p Fp(,)g(90{92,)e(133)340 1387 y(stable,)j Fa(30)p Fp(,)g(31,)f(91)257 1437 y(Orthogonalit)o(y)g (relation,)g(24)257 1527 y(P)o(olynomial)340 1577 y(minim)o(um)n(,)e (34,)h(131)340 1626 y(separable,)i(34)257 1676 y(P)o(on)o(try)o(agin)f (dualit)o(y)m(,)f(22)257 1726 y(Prime)d(n)o(um)o(b)q(er,)g(22,)f(23,)h (28,)f(37,)h(45,)g(49,)423 1776 y(55,)19 b(58,)f(88,)h(100,)f(109,)g (111,)423 1826 y(121,)8 b(124,)g(125,)g(133{135,)g(137{)423 1875 y(139)257 1925 y(Pro)q(duct,)15 b(semidirect,)e Fa(se)n(e)j Fp(Semidirect)423 1975 y(pro)q(duct)257 2065 y(Quasisymmetry)m(,)c(11{14,)g(73)257 2115 y(Quaternion)e(group,)f Fa(se)n(e)j Fp(Group,)c(quater-)423 2164 y(nion)257 2254 y(Radford)19 b(bipro)q(duct,)g(17,)f(19,)g(20,)h(28,)423 2304 y(35,)g(66{69,)f(77,)g(83,)g(84,)h(88,)423 2354 y(100,)8 b(111{116,)f(121,)i(132,)f(133)257 2403 y(Radford)h(pro)r (jection)g(theorem,)g(113,)f(132)257 2453 y(Represen)o(tation,)13 b(23,)e Fa(se)n(e)i(also)h Fp(Action,)423 2503 y(Mo)q(dule)1129 262 y(adjoin)o(t,)e(129)1129 311 y(coregular,)i(126)1129 361 y(irreducible,)c(31,)e(32,)g(34,)g(131,)g(135{)1212 411 y(137)1129 461 y(one-dimensional,)j(131)1129 511 y(regular,)j(20)1129 560 y(restricted,)i(34)1129 610 y(trivial,)c(136,)h(137)1046 660 y(Represen)o(tativ)o(e,)i(105)1046 710 y(Represen)o(tativ)o(es,)j(System)e(of,)g(23,)f(27,)1212 760 y(59,)e(101,)g(103)1046 809 y(Ring,)f(\014nite,)i(37,)f(42,)g(52,)g (54,)f(66,)h(67)1046 859 y(Ro)q(ot)g(of)h(unit)o(y)m(,)e(34,)h(45,)g (105)1129 909 y(fourth,)j(37,)f(46,)h(48,)f(49,)g(60,)h(108,)1212 959 y(121)1129 1009 y(primitiv)o(e,)7 b(22,)h(23,)g(49,)h(55,)f(88,)g (100,)1212 1059 y(105,)g(114,)g(115,)g(127,)g(129,)g(132)1046 1150 y(Sc)o(h)o(ur's)14 b(lemma,)d(95,)i(132)1046 1200 y(Second)i(construction,)f(66,)f(71,)g(76,)f(77,)1212 1249 y(83)1046 1299 y(Semidirect)d(pro)q(duct,)h(83,)e(124,)g(133,)g (134)1046 1349 y(Set,)14 b(p)q(oin)o(ted,)g(24,)e(26,)h(31)1046 1399 y(Sk)o(olem-No)q(ether)g(theorem,)g(31)1046 1449 y(Smash)18 b(pro)q(duct,)i(17,)f(18,)g(30{32,)e(35,)1212 1499 y(92)1129 1548 y(left,)c(18,)g(91)1046 1598 y(Structure)i (theorem,)c(45,)h(97,)g(107{110,)1212 1648 y(118)1129 1698 y(for)i(\014nite)g(ab)q(elian)f(groups,)g(123)1046 1748 y(Sw)o(eedler)21 b(notation,)e Fa(se)n(e)k Fp(Heyneman-)1212 1797 y(Sw)o(eedler)15 b(sigma)d(notation)1046 1847 y(Sylo)o(w)h (subgroup,)h(133)1046 1897 y(Symmetric)g(group,)h Fa(se)n(e)k Fp(Group,)c(sym-)1212 1947 y(metric)1046 2038 y(W)m(edderburn)d (structure)i(theorem,)c(126)1046 2130 y(Y)m(etter-Drinfel'd)k(algebra,) f(22)1046 2179 y(Y)m(etter-Drinfel'd)c(bialgebra,)f(15,)g(18,)h(22,) 1212 2229 y(23,)k(40,)g(51,)g(108,)f(109)1046 2279 y(Y)m (etter-Drinfel'd)g(category)m(,)f(11,)g(18,)g(22)1046 2329 y(Y)m(etter-Drinfel'd)j(coalgebra,)f(22)1046 2379 y(Y)m(etter-Drinfel'd)c(condition,)f(17,)g(21,)h(22,)1212 2428 y(39)1046 2478 y(Y)m(etter-Drinfel'd)14 b(form,)d(13)942 2628 y(149)p eop %%Page: 150 150 150 149 bop 257 262 a Fp(Y)m(etter-Drinfel'd)9 b(Hopf)g(algebra,)g Fa(12)p Fp(,)g(14{)423 311 y(20,)17 b(28,)g(37,)g(38,)f(42,)h(43,)g (45,)423 361 y(47{52,)k(54,)g(55,)g(59{62,)f(64,)423 411 y(66,)14 b(69,)f(71,)g(73,)g(76,)h(97,)f(101,)423 461 y(103,)g(107{112,)f(115,)g(132)340 511 y(co)q(comm)o(utativ)o(e,)7 b(12,)h(37,)g(100,)g(109)340 560 y(comm)o(utativ)o(e,)j(88,)i(109,)f (133)340 610 y(isomorphism,)f(107,)h(108)340 660 y(morphism,)i(51,)h (53,)h(55,)f(61,)h(108,)423 710 y(109)340 760 y(of)e(dimension)e(4,)h (121)340 809 y(of)h(dimension)e Fo(p)605 794 y Fn(2)623 809 y Fp(,)i(110,)e(114,)h(116)340 859 y(quotien)o(t,)h(101)340 909 y(subalgebra,)g(97,)f(101)340 959 y(trivial,)f(12,)h(23,)g(47,)g (58,)g(97)257 1009 y(Y)m(etter-Drinfel'd)h(mo)q(dule,)e Fa(11)p Fp(,)i(14,)f(22)340 1059 y(dual,)g(13)257 1150 y(Zh)o(u's)h(theorem,)g(125)942 2628 y(150)p eop %%Page: 151 151 151 150 bop 257 262 a Fr(Sym)n(b)r(ol)24 b(index)257 421 y Fo(A)288 406 y Fn(op)12 b(cop)386 421 y Fp(,)i(14)257 471 y Fo(A)288 477 y Fn(+)316 471 y Fp(,)f(48)257 521 y Fo(A)288 527 y Fl(\000)317 521 y Fp(,)g(48)257 570 y Fo(A)288 576 y Fj(G)316 570 y Fp(\()p Fo(\013;)7 b(\014)r(;)g(q)q Fp(\),)13 b(45)257 620 y Fo(A)288 626 y Fj(p)308 620 y Fp(\()p Fo(\013;)7 b(\014)r(;)g(q)q Fp(\),)13 b(45)257 670 y Fo(B)288 676 y Fn(+)316 670 y Fp(,)h(121)257 720 y Fo(B)288 726 y Fl(\000)317 720 y Fp(,)f(121)257 770 y Fo(B)288 776 y Fj(p)308 770 y Fp(\()p Fo(a;)7 b(b;)g(q)q Fp(\),)12 b(114)257 819 y Fo(C)287 825 y Fj(e)305 819 y Fp(,)h(30)257 869 y Fo(E)r Fp(,)h(28)257 919 y Fo(G)p Fp(\()p Fo(A)p Fp(\),)g(16)257 969 y Fo(G)290 975 y Fj(C)318 969 y Fp(\()p Fo(A)p Fp(\),)g(16)257 1019 y Fo(G)290 1025 y Fj(I)309 1019 y Fp(\()p Fo(A)p Fp(\),)g(16)257 1068 y Fo(G)290 1074 y Fn(op)327 1068 y Fp(,)f(83)257 1118 y Fo(H)f Fm(\000)e Fp(Mo)q(d,)j(18)257 1168 y Fo(H)295 1153 y Fj(i)309 1168 y Fp(\()p Fo(G;)7 b(M)e Fp(\),)13 b(24,)f(25)257 1218 y Fo(K)s Fp([)p Fo(G)p Fp(],)h(21)257 1268 y Fo(K)295 1253 y Fl(\002)324 1268 y Fp(,)g(10)257 1318 y Fo(K)295 1302 y Fc(Z)318 1306 y Ff(p)334 1318 y Fp(,)h(30)257 1367 y Fo(M)5 b Fp(,)14 b(120)257 1417 y Fo(N)5 b Fp(,)14 b(120)257 1467 y Fo(T)6 b Fp(,)14 b(83)257 1517 y Fo(U)5 b Fp(\()p Fo(R)p Fp(\),)14 b(31,)f(42,)f(66)257 1567 y Fo(U)290 1552 y Fn(+)318 1567 y Fp(,)h(101)257 1616 y Fo(V)291 1601 y Fl(\003)310 1616 y Fp(,)g(13)257 1666 y Fo(W)296 1672 y Fj(e)314 1666 y Fp(,)h(32)257 1716 y Fo(Z)288 1701 y Fj(i)302 1716 y Fp(\()p Fo(G;)7 b(M)e Fp(\),)13 b(24,)g(25)257 1766 y Fo(Z)285 1772 y Fj(C)314 1766 y Fp(\()p Fo(G)p Fp(\),)g(52)257 1816 y Fm(C)279 1822 y Fj(\037;\021)330 1816 y Fp(\()p Fo(R)p Fp(\),)h(54)257 1866 y(Ch\()p Fo(H)s Fp(\),)g(23)257 1915 y(\003)286 1921 y Fj(H)318 1915 y Fp(,)f(24)257 1965 y(\012,)h(115)257 2015 y Fk(Z)288 2021 y Fj(n)308 2015 y Fp(,)f(10)257 2065 y Fk(Z)288 2050 y Fl(\002)288 2075 y Fj(p)313 2065 y Fp(,)g(102)257 2115 y Fm(O)q Fp(,)h(35)257 2164 y Fo(\037)283 2170 y Fj(V)312 2164 y Fp(,)g(23)257 2214 y Fo(\016)275 2220 y Fj(V)305 2214 y Fp(,)f(11)257 2264 y Fo(\016)275 2270 y Fj(i)290 2264 y Fp(,)g(26)257 2314 y(\001)292 2320 y Fj(H)323 2314 y Fp(,)h(11)257 2364 y Fo(\017)274 2370 y Fj(H)306 2364 y Fp(,)f(11)257 2413 y Fm(\021)p Fp(,)h(80,)f(85)257 2463 y Fo(\021)278 2469 y Fj(V)307 2463 y Fp(,)g(33)1046 421 y Fo(\021)1067 427 y Fj(e)1085 421 y Fp(,)g(32)1046 471 y Fo(\015)r(:V)d Fp(,)j(36)1046 521 y Fo(\015)1067 527 y Fj(s)1085 521 y Fp(,)h(38)1050 560 y(^)1046 570 y Fk(Z)1077 576 y Fj(p)1093 570 y Fp(,)f(22)1045 620 y(^)-20 b Fo(\023)p Fp(,)13 b(60)1048 670 y(^)-23 b Fo(\031)q Fp(,)14 b(60)1046 720 y Fo(\023)p Fp(,)f(46)1046 770 y Fo(\024)p Fp(\()p Fo(V)d Fp(\),)j(29)1046 819 y Fo(\024)1070 804 y Fl(\003)1089 819 y Fp(\()p Fo(V)d Fp(\),)j(29)1046 869 y Fo(\025)1070 875 y Fj(H)1102 869 y Fp(,)g(32)1046 919 y Fm(h\001)p Fo(;)7 b Fm(\001i)1121 925 y Fl(\003)1139 919 y Fp(,)13 b(24)1046 969 y Fm(h\001)p Fo(;)7 b Fm(\001i)1121 975 y Fj(A)1147 969 y Fp(,)13 b(71)1046 1019 y Fm(h\001)p Fo(;)7 b Fm(\001i)1121 1025 y Fj(B)1148 1019 y Fp(,)14 b(82)1046 1068 y Fo(\026)1071 1074 y Fj(H)1103 1068 y Fp(,)f(11)p 1046 1095 26 2 v 1046 1118 a Fo(\037)p Fp(,)g(23)1046 1168 y Fo(\036)p Fp(,)g(22)1046 1218 y Fo(\031)q Fp(,)h(103)1046 1268 y Fo( )q Fp(,)g(22)1046 1318 y Fo( )1073 1324 y Fj(\015)1094 1318 y Fp(,)g(35)1046 1367 y Fm(!)p Fp(,)f(11)1046 1417 y Fo(\032)1067 1423 y Fj(A)1095 1417 y Fp(,)g(20)1046 1467 y Fo(\032)1067 1473 y Fj(ut)1102 1467 y Fp(,)g(86)1046 1517 y Fo(S)1071 1523 y Fj(H)1103 1517 y Fp(,)g(11)1046 1567 y Fo(\033)1071 1549 y Fl(\006)1070 1578 y Fj(i)1099 1567 y Fp(,)h(48)1046 1616 y Fo(\033)1070 1622 y Fj(u)1091 1616 y Fp(\()p Fo(s;)7 b(t)p Fp(\),)14 b(38)1046 1666 y Fo(\033)1070 1672 y Fj(V)r(;W)1139 1666 y Fp(,)f(11)1046 1716 y Fo(\034)1064 1722 y Fj(u)1086 1716 y Fp(,)g(70)1046 1766 y Fo(\030)1064 1772 y Fj(j)1082 1766 y Fp(,)g(130)1046 1816 y Fo(\020)s Fp(,)h(22)1046 1866 y Fo(b)1064 1848 y Fn(+)1064 1878 y Fj(ij)r(k)1111 1866 y Fp(,)g(121)1046 1924 y Fo(b)1064 1906 y Fl(\000)1064 1936 y Fj(ij)r(k)1111 1924 y Fp(,)g(121)1046 1973 y Fo(b)1064 1979 y Fj(ij)r(k)1111 1973 y Fp(,)g(114)1046 2023 y Fo(b)1064 2029 y Fj(uv)1103 2023 y Fp(\()p Fo(s)p Fp(\),)g(67)1046 2073 y Fo(d)1068 2079 y Fj(k)1088 2073 y Fp(,)f(114)1046 2123 y Fo(e)1065 2129 y Fj(i)1079 2123 y Fp(,)h(30)1046 2173 y Fo(f)1070 2158 y Fl(\003)1090 2173 y Fp(,)f(10)1046 2222 y Fo(f)1070 2207 y Fj(i)1085 2222 y Fp(,)g(26)1046 2272 y Fo(f)1066 2278 y Fj(A)1093 2272 y Fp(,)h(51)1046 2322 y Fo(f)1066 2328 y Fj(i)1080 2322 y Fp(,)g(26)1046 2372 y Fo(g)1066 2378 y Fj(A)1093 2372 y Fp(,)f(82)1046 2422 y Fo(q)1065 2428 y Fn(+)1092 2422 y Fp(,)h(48)1046 2472 y Fo(q)1065 2478 y Fl(\000)1093 2472 y Fp(,)f(48)942 2628 y(151)p eop %%Page: 152 152 152 151 bop 257 262 a Fo(s)10 b Fm([)f Fo(t)p Fp(,)k(25)257 311 y Fo(u)281 317 y Fj(e)299 311 y Fp(,)g(31)257 361 y Fo(y)277 367 y Fj(v)q(w)322 361 y Fp(\()p Fo(s)p Fp(\),)h(83)257 411 y Fo(z)278 396 y Fl(0)276 421 y Fj(uv)q(w)341 411 y Fp(\()p Fo(s;)7 b(t)p Fp(\),)13 b(84)257 461 y Fo(z)276 467 y Fj(s)294 461 y Fp(,)h(38)257 511 y Fo(z)276 517 y Fj(uv)q(w)341 511 y Fp(\()p Fo(s;)7 b(t)p Fp(\),)13 b(77)257 560 y Fm(Y)s(D)q(H)q Fp(\()p Fo(H)s Fp(\),)h(55)942 2628 y(152)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF