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FA(b)s(e)g(a)f(\014nite)h(Cartan)g(matrix.)41 b(The)33 b(problem)e(of)g(actually)g(\014nding)g(all)f(the)i (collections)-180 745 y Fx(g)-133 760 y Ft(i)-73 745 y Fu(2)g Fx(\000)14 b FA(\()p Fx(s)p FA(\),)36 b Fx(\037)345 760 y Ft(i)405 745 y Fu(2)507 716 y Fy([)503 745 y Fx(\000)14 b FA(\()p Fx(s)p FA(\),)36 b(1)c Fu(\024)g Fx(i)g Fu(\024)h Fx(\022)s FA(,)j(suc)m(h)g(that)f(\(1.1\))g(and)g(\(1.2\))f(hold)h(has) g(b)s(een)h(discussed)h(in)d([AS2].)51 b(It)36 b(can)-180 885 y(b)s(e)i(stated)g(as)g(the)g(problem)f(of)g(\014nding)g(all)f(the) i(solutions)f(of)g(a)g(system)i(of)e(algebraic)f(equations)i(o)m(v)m (er)h Fy(Z)p Fx(=)p FA(\()p Fx(p)p FA(\))-180 1024 y(and)33 b(it)e(is)h(in)g(principle)f(solv)-5 b(able.)42 b(Note)33 b(that)g(in)e(particular)1675 1241 y Fx(\022)g Fu(\024)d FA(2)p Fx(s)1961 1174 y(p)22 b Fu(\000)g FA(1)p 1961 1218 220 4 v 1961 1310 a Fx(p)g Fu(\000)g FA(2)2190 1241 y Fx(;)-180 1470 y FA(see)34 b([AS2,)f(Prop.)43 b(8.3].)-80 1726 y(\(iv\).)f(The)33 b(question)g(of)e(\014nding)h(all)e(the)i(p)s (ossible)g(linking)e(elemen)m(ts)i(attac)m(hed)h(to)f(a)f(\014xed)i (collection)e Fx(g)3869 1741 y Ft(i)3896 1726 y FA(,)i Fx(\037)4017 1741 y Ft(i)4045 1726 y FA(,)-180 1865 y(1)f Fu(\024)g Fx(i)h Fu(\024)f Fx(\022)s FA(,)k(\()p Fx(a)385 1880 y Ft(ij)446 1865 y FA(\))484 1880 y Fs(1)p Fr(\024)p Ft(i;j)t Fr(\024)p Ft(\022)744 1865 y FA(,)g(is)e(also)g(of)h(com)m (binatorial)d(nature,)k(see)g(Section)f(5,)h(and)f(also)f([Di)o(].)51 b(Once)36 b(these)g(t)m(w)m(o)-180 2005 y(problems)27 b(are)g(solv)m(ed)h(e\013ectiv)m(ely)-8 b(,)30 b(the)d(isomorphism)f (classes)i(of)f(the)h(Hopf)f(algebras)g Fm(u)p FA(\()p Fu(D)s FA(\))g(can)g(b)s(e)h(determined)-180 2144 y(using)k([AS2,)h (Prop.)44 b(6.3],)32 b([AS3,)h(Lemma)e(1.2].)-80 2458 y(\(v\).)50 b(As)36 b(a)e(consequence)k(of)d(Theorem)g(1.8)f(one)h (obtains)g(the)g(complete)f(classi\014cation)g(of)g(all)f(\014nite)h (dimen-)-180 2597 y(sional)i(p)s(oin)m(ted)i(Hopf)g(algebras)f(with)g (group)h(of)f(group-lik)m(es)g Fx(\000)14 b FA(\(1\))37 b(=)f Fy(Z)p Fx(=)p FA(\()p Fx(p)p FA(\),)h Fx(p)f Fu(6)p FA(=)h(5)p Fx(;)17 b FA(7.)59 b(It)38 b(is)f(the)h(list)f(giv)m(en)-180 2737 y(in)30 b([AS2,)i(Theorem)f(1.3])f(plus)h(the)h(F)-8 b(rob)s(enius-Lusztig)29 b(k)m(ernels)j(as)f(describ)s(ed)h(in)e ([AS1].)43 b(Indeed,)33 b(replacing)d(in)-180 2876 y(the)37 b(pro)s(of)e(of)h(Theorem)h(1.8)f([AS2,)h(Cor.)55 b(1.2])37 b(b)m(y)g([AS2,)g(Th.)56 b(1.3])36 b(w)m(e)h(get)g(the)g (classi\014cation)d(for)i(all)f(primes)-180 3016 y Fx(p)g Fu(6)p FA(=)g(5)i(or)f(7,)i(in)f(view)g(of)f(Theorem)i(6.8)e(and)h ([AS3,)i(Lemma)c(4.2].)56 b(The)38 b(only)f(cases)h(not)f(co)m(v)m (ered)i(are)e Fx(p)e FA(=)g(5,)-180 3155 y(t)m(yp)s(e)d Fx(B)112 3170 y Fs(2)182 3155 y FA(and)f Fx(p)c FA(=)h(7,)j(t)m(yp)s(e) g Fx(G)951 3170 y Fs(2)991 3155 y FA(.)43 b(This)31 b(result)f(w)m(as)i (indep)s(enden)m(tly)f(obtained)g(b)m(y)g(Musson)i([Ms)q(])d(using)h (di\013eren)m(t)-180 3295 y(metho)s(ds)h(starting)g(from)f(our)i (previous)g(article)e([AS2].)-80 3550 y(\(vi\).)42 b(Up)29 b(to)h(no)m(w,)h(the)f(determination)e(of)h(all)e(\014nite)j (dimensional)d(p)s(oin)m(ted)i(Hopf)h(algebras)f Fx(A)h FA(with)f Fx(G)p FA(\()p Fx(A)p FA(\))e Fu(')-180 3690 y FA(\000,)h(for)f(a)g(\014xed)h(group)g(\000,)g(w)m(as)g(kno)m(wn)h (only)d(for)h(\000)h(=)f Fy(Z)p Fx(=)p FA(\(2\))d([N)q(].)41 b(Other)28 b(classi\014cation)e(results)h(of)g(p)s(oin)m(ted)g(Hopf) -180 3829 y(algebras)37 b(are)h(kno)m(wn)h(for)f(some)f(\014xed)i (dimension)e Fx(d)p FA(:)53 b Fx(d)37 b FA(=)f Fx(p)2179 3793 y Fs(2)2256 3829 y FA(is)i(easy)h(and)f(follo)m(ws)e(from)h([N],)i ([NZ];)i Fx(d)36 b FA(=)h Fx(p)4033 3793 y Fs(3)-180 3969 y FA(w)m(as)e(done)f(in)f([AS1)q(],)h(and)g(b)m(y)h(di\013eren)m (t)f(metho)s(ds)g(in)f([CD],)h([SvO)q(];)h Fx(d)29 b FA(=)h Fx(p)2667 3933 y Fs(4)2740 3969 y FA(in)k([AS3])g(\(and)g(do)s (es)g(not)g(seem)g(to)-180 4108 y(b)s(e)j(p)s(ossible)f(via)g(the)h (other)f(metho)s(ds\);)j Fx(d)34 b FA(=)g(16)i(in)g([CDR],)i Fx(d)c FA(=)g(32)i(in)g([G)s(~)-51 b(n1)o(];)38 b(results)f(on)g(the)g (case)g(when)h(\000)-180 4248 y(has)33 b(exp)s(onen)m(t)h(2)e(can)h(b)s (e)g(found)g(in)e([AD].)-80 4503 y(\(vii\).)39 b(The)24 b(classi\014cation)e(of)h(all)e Fw(c)-5 b(or)g(adic)g(al)5 b(ly)26 b(gr)-5 b(ade)g(d)22 b FA(p)s(oin)m(ted)h(Hopf)h(algebras)e(of) h(dimension)f Fx(p)3462 4467 y Fs(5)3525 4503 y FA(w)m(as)i(obtained) -180 4643 y(in)34 b([G)s(~)-51 b(n2)n(].)50 b(It)35 b(is)f(not)g (di\016cult)g(to)g(deduce)j(the)e(classi\014cation)e(of)h(all)e(p)s (oin)m(ted)j(Hopf)f(algebras)g(of)g(dimension)g Fx(p)4033 4607 y Fs(5)-180 4782 y FA(using)e(Theorem)h(1.8)f(and)h(results)g(in)f ([AS3].)-80 5038 y(\(viii\).)86 b(The)49 b(Hopf)f(algebras)f Fm(u)p FA(\()p Fu(D)s FA(\))f(can)i(b)s(e)g(de\014ned)h(for)f(an)m(y)g (Cartan)g(datum)f(of)g(\014nite)h(t)m(yp)s(e)g Fu(D)j FA(of)c(an)-180 5177 y(arbitrary)42 b(\014nite)g(ab)s(elian)e(group.)74 b(P)m(art)42 b(\(a\))g(of)h(Theorem)f(1.8)g(is)g(a)h(sp)s(ecial)e(case) j(of)e(the)h(general)f(Theorem)-180 5317 y(5.17.)h(F)-8 b(or)31 b(suitable)f(c)m(hoices)j(of)e Fu(D)s FA(,)h(the)g(F)-8 b(rob)s(enius-Lusztig)30 b(k)m(ernels)j(and)f(their)f(parab)s(olic)f (subalgebras)i(are)f(of)p eop %%Page: 4 4 4 3 bop -180 0 a Fn(4)884 b(NICOL)1001 -19 y(\023)991 0 y(AS)24 b(ANDR)n(USKIEWITSCH)f(AND)h(HANS-J)2419 -19 y(\177)2409 0 y(UR)n(GEN)f(SCHNEIDER)-180 203 y FA(the)38 b(form)f Fm(u)p FA(\()p Fu(D)s FA(\).)59 b(See)39 b(example)e(5.12.)60 b(Otherwise)38 b(Theorem)h(5.17)e(pro)m(vides)i(man)m(y)f(new)h (examples)f(of)f(\014nite)-180 342 y(dimensional)30 b(Hopf)i(algebras)g (arising)f(from)h(exotic)g(linking)f(data.)-80 600 y(\(ix\).)40 b(The)24 b(results)h(of)e(this)h(pap)s(er)g(hea)m(vily)f(dep)s(end)i (on)f(our)g(pap)s(er)g([AS2])g(and)g(on)g(previous)g(w)m(ork)h(on)f (quan)m(tum)-180 739 y(groups)33 b([L1],)f([L2],)h([Ro1)o(],)g([Ro2],)f ([dCP)q(],)h([Mu)q(].)-80 870 y FC(Con)m(v)m(en)m(tions.)78 b 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y FA(,)g(and)g(in)e([AS4])h(for)g(t)m(yp)s(e)-180 5202 y Fx(A)-107 5217 y Ft(n)-27 5202 y FA(for)c(an)m(y)h Fx(n)g FA(\(up)f(to)h(some)f(exceptional)g(cases)i(concerning)f(the)g (orders)g(of)f(the)h(ro)s(ots)f(of)h(unit)m(y\).)p eop %%Page: 6 6 6 5 bop -180 0 a Fn(6)884 b(NICOL)1001 -19 y(\023)991 0 y(AS)24 b(ANDR)n(USKIEWITSCH)f(AND)h(HANS-J)2419 -19 y(\177)2409 0 y(UR)n(GEN)f(SCHNEIDER)1268 203 y FA(3.)49 b Fz(Braided)38 b(Hopf)g(algebras)-180 377 y FA(3.1.)49 b FC(Bipro)s(ducts.)f FA(Let)32 b Fx(R)g FA(b)s(e)g(a)g(braided)f(Hopf) h(algebra)e(in)2176 341 y Ft(H)2176 403 y(H)2244 377 y Fu(Y)8 b(D)r FA(;)32 b(this)g(means)f(that)h Fx(R)h FA(is)e(an)h(algebra)e(and)i(a)-180 494 y(coalgebra)e(in)365 457 y Ft(H)365 519 y(H)432 494 y Fu(Y)8 b(D)34 b FA(and)e(that)f(the)h (com)m(ultiplication)27 b(\001)1982 509 y Ft(R)2067 494 y FA(:)h Fx(R)h Fu(!)e Fx(R)21 b Fu(\012)g Fx(R)32 b FA(is)f(an)g(algebra)g(map)f(when)j(in)e Fx(R)21 b Fu(\012)f Fx(R)-180 610 y FA(the)41 b(m)m(ultiplication)36 b 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FA(\001\()p Fx(r)s FA(#)p Fx(h)p FA(\))28 b(=)g Fx(r)2227 1127 y Fs(\(1\))2321 1168 y FA(#\()p Fx(r)2487 1127 y Fs(\(2\))2581 1168 y FA(\))2619 1184 y Fs(\()p Fr(\000)p Fs(1\))2768 1168 y Fx(h)2824 1184 y Fs(\(1\))2941 1168 y Fu(\012)22 b FA(\()p Fx(r)3125 1127 y Fs(\(2\))3219 1168 y FA(\))3257 1184 y Fs(\(0\))3351 1168 y FA(#)p Fx(h)3488 1184 y Fs(\(2\))3583 1168 y Fx(:)-80 1376 y FA(The)51 b(maps)f Fx(\031)62 b FA(:)d Fx(A)f Fu(!)g Fx(H)g FA(and)51 b Fx(\023)58 b FA(:)h Fx(H)66 b Fu(!)57 b Fx(A)p FA(,)f Fx(\031)t FA(\()p Fx(r)s FA(#)p Fx(h)p FA(\))i(=)g Fx(\017)p FA(\()p Fx(r)s FA(\))p Fx(h)p FA(,)d Fx(\023)p FA(\()p Fx(h)p FA(\))k(=)f(1#)p Fx(h)p FA(,)d(are)c(Hopf)f(algebra)-180 1492 y(homomorphisms;)61 b(w)m(e)54 b(ha)m(v)m(e)g Fx(R)63 b FA(=)g Fu(f)p Fx(a)f Fu(2)h Fx(A)f FA(:)g(\(id)33 b Fu(\012)p Fx(\031)t FA(\)\001\()p Fx(a)p FA(\))63 b(=)f Fx(a)36 b Fu(\012)h FA(1)p Fu(g)p FA(.)104 b(Con)m(v)m(ersely)-8 b(,)60 b(let)52 b Fx(A)p FA(,)58 b Fx(H)j FA(b)s(e)-180 1608 y(Hopf)44 b(algebras)g(pro)m(vided)g(with)g(Hopf)g(algebra)f (homomorphisms)f Fx(\031)51 b FA(:)d Fx(A)f Fu(!)g Fx(H)52 b FA(and)44 b Fx(\023)k FA(:)f Fx(H)55 b Fu(!)47 b Fx(A)p FA(.)79 b(Then)-180 1724 y Fx(R)29 b FA(=)e Fu(f)p Fx(a)h Fu(2)g Fx(A)g FA(:)g(\(id)k Fu(\012)p Fx(\031)t FA(\)\001\()p Fx(a)p FA(\))c(=)g Fx(a)14 b Fu(\012)g FA(1)p Fu(g)28 b FA(is)f(a)i(braided)f(Hopf)g(algebra)f(in)2554 1688 y Ft(H)2554 1750 y(H)2621 1724 y Fu(Y)8 b(D)r FA(.)42 b(The)30 b(action)d Fx(:)h FA(of)g Fx(H)36 b FA(on)28 b Fx(R)i FA(is)d(the)-180 1841 y(restriction)k(of)h(the)g(adjoin)m(t)g (action)f(\(comp)s(osed)h(with)g Fx(\023)p FA(\))g(and)h(the)f (coaction)f(is)h(\()p Fx(\031)25 b Fu(\012)d FA(id)16 b(\)\001;)32 b Fx(R)i FA(is)d(a)h(subalgebra)-180 1957 y(of)d Fx(A)h FA(and)f(the)h(com)m(ultiplication)25 b(is)k(\001)1273 1972 y Ft(R)1331 1957 y FA(\()p Fx(r)s FA(\))f(=)f Fx(r)1629 1972 y Fs(\(1\))1723 1957 y Fx(\023\031)t Fu(S)7 b FA(\()p Fx(r)1965 1972 y Fs(\(2\))2061 1957 y FA(\))16 b Fu(\012)g Fx(r)2252 1972 y Fs(\(3\))2346 1957 y FA(.)43 b(These)31 b(constructions)f(are)g(in)m(v)m(erse)h(to)e(eac)m(h)-180 2073 y(other.)44 b(W)-8 b(e)33 b(shall)e(mostly)g(omit)g Fx(\023)i FA(in)f(what)h(follo)m(ws.)-80 2189 y(Let)f Fx(#)d FA(:)e Fx(A)h Fu(!)f Fx(R)34 b FA(b)s(e)f(the)g(map)f(giv)m(en)g (b)m(y)i Fx(#)p FA(\()p Fx(a)p FA(\))28 b(=)g Fx(a)1845 2205 y Fs(\(1\))1939 2189 y Fx(\031)t Fu(S)7 b FA(\()p Fx(a)2154 2205 y Fs(\(2\))2249 2189 y FA(\).)44 b(Then)-180 2397 y(\(3.1\))1399 b Fx(#)p FA(\()p Fx(ab)p FA(\))28 b(=)g Fx(a)1828 2412 y Fs(\(1\))1922 2397 y Fx(#)p FA(\()p Fx(b)p FA(\))p Fx(\031)t Fu(S)7 b FA(\()p Fx(a)2311 2412 y Fs(\(2\))2407 2397 y FA(\))p Fx(;)-180 2604 y FA(for)38 b(all)e Fx(a;)17 b(b)39 b Fu(2)f Fx(A)g FA(and)h Fx(#)p FA(\()p Fx(h)p FA(\))f(=)g Fx(")p FA(\()p Fx(h)p FA(\))g(for)g(all)e Fx(h)i Fu(2)g Fx(H)8 b FA(;)41 b(therefore,)g(for)d(all)e Fx(a)i Fu(2)g Fx(A)p FA(,)j Fx(h)d Fu(2)g Fx(H)8 b FA(,)39 b(w)m(e)h(ha)m(v)m(e)g Fx(#)p FA(\()p Fx(ah)p FA(\))e(=)-180 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FA(is)f(just)-180 4637 y(\(3.3\))1143 b(ad)1283 4652 y Ft(c)1317 4637 y Fx(x)p FA(\()p Fx(y)t FA(\))28 b(=)f Fx(\026)p FA(\(id)32 b Fu(\000)p Fx(c)p FA(\)\()p Fx(x)23 b Fu(\012)g Fx(y)t FA(\))k(=:)g([)p Fx(x;)17 b(y)t FA(])2667 4652 y Ft(c)2701 4637 y Fx(:)-180 4845 y FA(The)24 b(elemen)m(t)g([)p Fx(x;)17 b(y)t FA(])568 4860 y Ft(c)625 4845 y FA(de\014ned)25 b(b)m(y)g(the)f(second)g (equalit)m(y)f(for)g(an)m(y)i Fx(x)e FA(and)h Fx(y)t FA(,)h(regardless)e(of)g(whether)i Fx(x)f FA(is)f(primitiv)m(e,)-180 4961 y(will)30 b(b)s(e)j(called)e(a)i(braided)f(comm)m(utator.)-80 5077 y(When)h Fx(A)28 b FA(=)f Fx(R)q FA(#)p Fx(H)8 b FA(,)33 b(then)g(for)f(all)f Fx(b;)17 b(d)27 b Fu(2)h Fx(R)q FA(,)-180 5285 y(\(3.4\))1302 b(ad)1443 5300 y Fs(\()p Ft(b)p Fs(#1\))1626 5285 y FA(\()p Fx(d)p FA(#1\))27 b(=)h(\(ad)2171 5300 y Ft(c)2206 5285 y Fx(b)p FA(\()p Fx(d)p FA(\)\)#1)p Fx(:)p eop %%Page: 7 7 7 6 bop 446 0 a Fn(FINITE)33 b(QUANTUM)g(GR)n(OUPS)f(O)n(VER)g(ABELIAN) g(GR)n(OUPS)g(OF)i(PRIME)f(EXPONENT)588 b(7)-180 203 y FA(3.2.)49 b FC(Nic)m(hols)36 b(algebras.)49 b FA(Let)34 b Fx(H)41 b FA(b)s(e)33 b(a)g(Hopf)g(algebra)f(and)i(let)e Fx(R)e FA(=)f Fu(\010)2609 218 y Ft(n)p Fr(2)p Fi(N)2752 203 y Fx(R)q FA(\()p Fx(n)p FA(\))k(b)s(e)h(a)f Fw(gr)-5 b(ade)g(d)32 b FA(braided)h(Hopf)-180 319 y(algebra)j(in)285 283 y Ft(H)285 344 y(H)352 319 y Fu(Y)8 b(D)s FA(.)58 b(W)-8 b(e)37 b(sa)m(y)i(that)e Fx(R)i FA(is)d(a)i Fw(Nichols)g(algebr) -5 b(a)37 b FA(if)f(2.1,)j(2.2)e(and)g(2.3)g(hold,)h Fw(cf.)58 b FA([N],)39 b([AS2],)g([A)m(G].)-180 435 y(A)h(Nic)m(hols)f (algebra)f Fx(R)j FA(is)e(uniquely)h(determined)g(b)m(y)g(the)h(Y)-8 b(etter-Drinfeld)38 b(mo)s(dule)g Fu(P)8 b FA(\()p Fx(R)q FA(\);)44 b(giv)m(en)39 b(a)h(Y)-8 b(etter-)-180 552 y(Drinfeld)36 b(mo)s(dule)g Fx(V)21 b FA(,)39 b(there)f(exists)g(a)f (unique)h(\(up)g(to)f(isomorphism\))e(Nic)m(hols)i(algebra)f Fx(R)i FA(with)g Fu(P)8 b FA(\()p Fx(R)q FA(\))36 b Fu(')g Fx(V)22 b FA(.)-180 668 y(It)46 b(will)e(b)s(e)i(denoted)i Fm(B)p FA(\()p Fx(V)21 b FA(\).)84 b(In)47 b(fact,)i(the)e(k)m(ernel)g (of)e(the)i(canonical)e(map)g Fx($)53 b FA(:)e Fx(T)14 b FA(\()p Fx(V)21 b FA(\))51 b Fu(!)g Fm(B)p FA(\()p Fx(V)21 b FA(\))46 b(can)h(b)s(e)-180 784 y(describ)s(ed)38 b(in)d(sev)m(eral)j(di\013eren)m(t)f(w)m(a)m(ys.)57 b(F)-8 b(or)36 b(instance,)i(Ker)17 b Fx($)37 b FA(=)d Fu(\010)2443 799 y Ft(n)p Fr(\025)p Fs(0)2581 784 y FA(Ker)16 b FC(S)2835 799 y Ft(n)2919 784 y FA(where)38 b FC(S)3286 799 y Ft(n)3369 784 y FA(is)f(the)g("quan)m(tum)-180 900 y(an)m(tisymmetrizer")d (de\014ned)j(from)d(the)i(braiding)e Fx(c)p FA(;)j(so)f(that)f Fm(B)p FA(\()p Fx(V)22 b FA(\))35 b(is)h(a)f("quan)m(tum)h(sh)m(u\017e) h(algebra")d(and)i(as)-180 1017 y(algebra)c(and)h(coalgebra)f(only)g (dep)s(ends)j(on)e(the)g(braiding)f Fx(c)c FA(:)h Fx(V)44 b Fu(\012)22 b Fx(V)50 b Fu(!)28 b Fx(V)44 b Fu(\012)23 b Fx(V)f FA(.)45 b(See)34 b([N],)f([W)-8 b(o],)33 b([L3],)g([Ro1],)-180 1133 y([Ro2)o(],)g([Sc)m(h)q(].)-80 1249 y(Let)43 b Fx(H)54 b FA(=)46 b Fy(|)-9 b FA(\000)37 b(where)45 b(\000)e(is)g(a)g(\014nite) g(ab)s(elian)e(group.)76 b(Let)43 b Fx(V)65 b FA(b)s(e)44 b(a)f(\014nite)g(dimensional)e(Y)-8 b(etter-Drinfeld)-180 1374 y(mo)s(dule)32 b(o)m(v)m(er)j(\000.)47 b(Then)35 b(there)f(exist)g(a)g(basis)f Fx(x)1626 1389 y Fs(1)1666 1374 y Fx(;)17 b(:)g(:)g(:)f(;)h(x)1940 1389 y Ft(\022)2013 1374 y FA(of)33 b Fx(V)55 b FA(and)34 b(elemen)m(ts)g Fx(g)2875 1389 y Fs(1)2914 1374 y Fx(;)17 b(:)g(:)g(:)f(;)h(g)3180 1389 y Ft(\022)3248 1374 y Fu(2)30 b FA(\000,)k Fx(\037)3527 1389 y Fs(1)3567 1374 y Fx(;)17 b(:)g(:)g(:)f(;)h(\037)3847 1389 y Ft(\022)3915 1374 y Fu(2)4014 1349 y Fv(b)4011 1374 y FA(\000)-180 1490 y(suc)m(h)34 b(that)-180 1667 y(\(3.5\))1259 b Fx(x)1335 1682 y Ft(j)1400 1667 y Fu(2)28 b Fx(V)1572 1626 y Ft(\037)1616 1636 y Fq(j)1551 1691 y Ft(g)1585 1701 y Fq(j)1653 1667 y Fx(;)212 b FA(for)32 b(all)e(1)e Fu(\024)g Fx(j)33 b Fu(\024)c Fx(\022)s(:)-180 1855 y FA(In)37 b(what)g(follo)m(ws)e(w)m(e)j(shall)d(only)h(consider)h (Y)-8 b(etter-Drinfeld)34 b(mo)s(dules)i Fx(V)58 b FA(suc)m(h)38 b(that)f Fx(\037)3185 1870 y Ft(i)3213 1855 y FA(\()p Fx(g)3298 1870 y Ft(i)3326 1855 y FA(\))d Fu(6)p FA(=)g(1,)k(1)c Fu(\024)h Fx(i)g Fu(\024)g Fx(\022)s FA(.)-180 1971 y(The)f(braiding)c Fx(c)j FA(is)f(giv)m(en)h(with)f(resp)s(ect)i(to)e(the)h(basis)f Fx(x)1970 1986 y Ft(i)2021 1971 y Fu(\012)23 b Fx(x)2176 1986 y Ft(j)2245 1971 y FA(b)m(y)34 b Fx(c)p FA(\()p Fx(x)2516 1986 y Ft(i)2566 1971 y Fu(\012)23 b Fx(x)2721 1986 y Ft(j)2758 1971 y FA(\))28 b(=)f Fx(b)2968 1986 y Ft(ij)3046 1971 y Fx(x)3101 1986 y Ft(j)3160 1971 y Fu(\012)22 b Fx(x)3314 1986 y Ft(i)3343 1971 y FA(,)33 b(where)1356 2148 y(\()p Fx(b)1435 2163 y Ft(ij)1496 2148 y FA(\))1534 2163 y Fs(1)p Fr(\024)p Ft(i;j)t Fr(\024)p Ft(\022)1821 2148 y FA(=)28 b(\()p Fx(\037)2024 2163 y Ft(j)2060 2148 y FA(\()p Fx(g)2145 2163 y Ft(i)2173 2148 y FA(\)\))2249 2163 y Fs(1)p Fr(\024)p Ft(i;j)t Fr(\024)p Ft(\022)2509 2148 y Fx(:)-180 2347 y Fw(R)-5 b(emark)45 b FA(3.6)p Fw(.)c FA(Let)33 b Fx(V)21 b FA(,)33 b(resp.)957 2322 y Fv(e)945 2347 y Fx(V)22 b FA(,)33 b(b)s(e)f(a)h(\014nite)f(dimensional)e(Y)-8 b(etter-Drinfeld)31 b(mo)s(dule)g(o)m(v)m(er)i(\000,)g(resp.)3683 2322 y Fv(e)3680 2347 y FA(\000,)g(with)f(a)-180 2487 y(basis)k Fx(x)118 2502 y Fs(1)158 2487 y Fx(;)17 b(:)g(:)g(:)f(;)h(x)432 2502 y Ft(\022)507 2487 y FA(suc)m(h)38 b(that)e Fx(x)1001 2502 y Ft(i)1064 2487 y Fu(2)f Fx(V)1243 2451 y Ft(\037)1287 2461 y Fq(i)1222 2512 y Ft(g)1256 2522 y Fq(i)1317 2487 y FA(,)j(resp.)56 b(with)36 b(a)g(basis)j Fv(e)-58 b Fx(x)2247 2502 y Fs(1)2287 2487 y Fx(;)17 b(:)g(:)g(:)f(;)j Fv(e)-57 b Fx(x)2561 2502 y Ft(\022)2636 2487 y FA(suc)m(h)38 b(that)h Fv(e)-58 b Fx(x)3130 2502 y Ft(i)3193 2487 y Fu(2)34 b Fx(V)3378 2440 y Fk(e)-45 b Ft(\037)3416 2450 y Fq(i)3350 2517 y Fk(e)-39 b Ft(g)3384 2527 y Fq(i)3446 2487 y FA(.)55 b(Assume)37 b(that)-180 2626 y Fx(\037)-119 2641 y Ft(i)-91 2626 y FA(\()p Fx(g)-6 2641 y Ft(j)30 2626 y FA(\))c(=)42 b Fv(e)-64 b Fx(\037)271 2641 y Ft(i)300 2626 y FA(\()p Fv(e)-55 b Fx(g)385 2641 y Ft(j)421 2626 y FA(\))36 b(for)f(all)e(1)g Fu(\024)h Fx(i;)17 b(j)39 b Fu(\024)33 b Fx(\022)s FA(.)54 b(Then)36 b(there)h(exists)f(a)g (unique)g(algebra)f(and)h(coalgebra)e(isomorphism)-180 2766 y Fm(B)p FA(\()p Fx(V)22 b FA(\))27 b Fu(!)g Fm(B)p FA(\()355 2741 y Fv(e)343 2766 y Fx(V)22 b FA(\))32 b(suc)m(h)i(that)f Fx(x)979 2781 y Ft(i)1035 2766 y Fu(7!)d Fv(e)-58 b Fx(x)1217 2781 y Ft(i)1279 2766 y FA(for)32 b(all)e(1)e Fu(\024)g Fx(i)g Fu(\024)g Fx(\022)s FA(.)-180 2936 y FC(De\014nition)42 b(3.7.)i FA(W)-8 b(e)38 b(shall)e(sa)m(y)i(that)g(a)f(braiding)f(giv)m (en)h(b)m(y)i(a)e(matrix)f FC(b)g FA(=)g(\()p Fx(b)2936 2951 y Ft(ij)2997 2936 y FA(\))3035 2951 y Fs(1)p Fr(\024)p Ft(i;j)t Fr(\024)p Ft(\022)3332 2936 y FA(whose)j(en)m(tries)f(are)-180 3075 y(ro)s(ots)32 b(of)g(unit)m(y)h(is)f Fw(of)j(Cartan)f(typ)-5 b(e)33 b FA(if)e(for)h(all)f Fx(i;)17 b(j)6 b FA(,)32 b Fx(b)1777 3090 y Ft(ii)1858 3075 y Fu(6)p FA(=)27 b(1)33 b(and)f(there)i(exists)f Fx(a)2802 3090 y Ft(ij)2890 3075 y Fu(2)c Fy(Z)g FA(suc)m(h)34 b(that)1696 3275 y Fx(b)1737 3290 y Ft(ij)1798 3275 y Fx(b)1839 3290 y Ft(j)t(i)1928 3275 y FA(=)27 b Fx(b)2072 3221 y Ft(a)2109 3231 y Fq(ij)2072 3300 y Ft(ii)2169 3275 y Fx(:)-180 3475 y FA(The)34 b(in)m(tegers)e Fx(a)432 3490 y Ft(ij)526 3475 y FA(are)g(uniquely)h(determined)f(b)m (y)i(the)f(follo)m(wing)d(rules:)43 3644 y Fu(\017)42 b FA(If)32 b Fx(i)c FA(=)g Fx(j)38 b FA(w)m(e)c(tak)m(e)f Fx(a)881 3659 y Ft(ii)961 3644 y FA(=)28 b(2;)43 3784 y Fu(\017)42 b FA(if)31 b Fx(i)d Fu(6)p FA(=)f Fx(j)6 b FA(,)33 b(w)m(e)h(select)f(the)g(unique)g Fx(a)1440 3799 y Ft(ij)1533 3784 y FA(suc)m(h)h(that)e Fu(\000)17 b FA(ord)g Fx(b)2257 3799 y Ft(ii)2337 3784 y Fx(<)28 b(a)2492 3799 y Ft(ij)2580 3784 y Fu(\024)h FA(0.)-80 3930 y(Then)44 b(\()p Fx(a)274 3945 y Ft(ij)335 3930 y FA(\))f(is)f(a)h(generalized)f(Cartan)i(matrix)d([K].)75 b(W)-8 b(e)44 b(shall)d(sa)m(y)j(a)f(Y)-8 b(etter-Drinfeld)42 b(mo)s(dule)f Fx(V)65 b FA(is)42 b(of)-180 4047 y(Cartan)29 b(t)m(yp)s(e,)j(resp.)43 b(\014nite)29 b(Cartan)h(t)m(yp)s(e,)h(if)d (its)h(corresp)s(onding)g(braiding)f(is)h(of)g(Cartan)h(t)m(yp)s(e,)h (resp.)43 b(the)30 b(same)-180 4163 y(plus)i(the)h(matrix)f(\()p Fx(a)601 4178 y Ft(ij)661 4163 y FA(\))h(is)f(of)g(\014nite)g(t)m(yp)s (e.)-180 4377 y(3.3.)49 b FC(The)35 b(t)m(wisting)e(functor.)49 b FA(Let)31 b Fx(H)38 b FA(b)s(e)30 b(a)h(Hopf)f(algebra)g(and)g(let)g Fx(F)45 b FA(b)s(e)30 b(an)h(in)m(v)m(ertible)f(elemen)m(t)g(in)g Fx(H)c Fu(\012)18 b Fx(H)-180 4493 y FA(suc)m(h)34 b(that)-180 4670 y(\(3.8\))519 b Fx(F)603 4685 y Fs(12)678 4670 y FA(\(\001)22 b Fu(\012)h FA(id)16 b(\))p Fx(F)41 b FA(=)27 b Fx(F)1325 4685 y Fs(23)1400 4670 y FA(\(id)33 b Fu(\012)p FA(\001\))p Fx(F)s(;)212 b FA(\()p Fx(")22 b Fu(\012)g FA(id)16 b(\)\()p Fx(F)e FA(\))27 b(=)h(1)f(=)h(\(id)k Fu(\012)p Fx(")p FA(\)\()p Fx(F)14 b FA(\))p Fx(:)-180 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FA(,)30 b(let)g Fx(A)e FA(=)f Fx(R)q FA(#)p Fx(H)38 b FA(b)s(e)31 b(its)f(b)s(osonization)e(and)j(consider)f(the)-180 5317 y(Hopf)k(algebra)e Fx(A)475 5332 y Ft(F)534 5317 y FA(.)47 b(It)33 b(follo)m(ws)g(from)f(the)i(de\014nitions)f(that)h Fx(\031)f FA(:)d Fx(A)2340 5332 y Ft(F)2428 5317 y Fu(!)g Fx(H)2639 5332 y Ft(F)2731 5317 y FA(and)j Fx(\023)d FA(:)g Fx(H)3123 5332 y Ft(F)3211 5317 y Fu(!)f Fx(A)3413 5332 y Ft(F)3506 5317 y FA(are)k(also)g(Hopf)p eop %%Page: 8 8 8 7 bop -180 0 a Fn(8)884 b(NICOL)1001 -19 y(\023)991 0 y(AS)24 b(ANDR)n(USKIEWITSCH)f(AND)h(HANS-J)2419 -19 y(\177)2409 0 y(UR)n(GEN)f(SCHNEIDER)-180 203 y FA(algebra)31 b(homomorphisms.)42 b(Hence)1084 372 y Fx(R)1158 387 y Ft(F)1244 372 y FA(:=)28 b Fu(f)p Fx(a)g Fu(2)g Fx(A)1671 387 y Ft(F)1757 372 y FA(:)g(\(id)k Fu(\012)p Fx(\031)t FA(\)\001)2219 387 y Ft(F)2278 372 y FA(\()p Fx(a)p FA(\))c(=)g Fx(a)22 b Fu(\012)h FA(1)p Fu(g)-180 549 y FA(is)31 b(a)g(braided)g (Hopf)g(algebra)f(in)h(the)h(category)1596 504 y Ft(H)1654 515 y Fq(F)1596 576 y Ft(H)1654 587 y Fq(F)1709 549 y Fu(Y)8 b(D)s FA(.)43 b(W)-8 b(e)32 b(consider)f(the)h(corresp)s(onding) f(map)g Fx(#)3544 564 y Ft(F)3634 549 y FA(and)h(de\014ne)-180 665 y Fx( )g FA(:)27 b Fx(R)i Fu(!)e Fx(R)273 680 y Ft(F)365 665 y FA(b)m(y)-180 835 y(\(3.9\))1391 b Fx( )t FA(\()p Fx(r)s FA(\))27 b(=)g Fx(#)1789 850 y Ft(F)1848 835 y FA(\()p Fx(r)s FA(\))p Fx(;)212 b(r)30 b Fu(2)e Fx(R)q(:)-180 1004 y FA(The)35 b(map)f Fx( )39 b FA(w)m(as)c(de\014ned)h(in)e([AS2])g (in)g(the)h(case)g Fx(H)k FA(=)31 b Fy(|)-9 b FA(\000)28 b(is)34 b(the)h(group)g(algebra)e(of)h(a)g(\014nite)g(ab)s(elian)f (group.)-180 1120 y(The)f(follo)m(wing)c(Lemma)h(generalizes)i([AS2,)g (Lemma)f(2.3];)h(part)g(\(iii\),)d(new)k(ev)m(en)g(for)e Fx(H)36 b FA(=)27 b Fy(|)-8 b FA(\000,)25 b(will)j(b)s(e)j(needed)-180 1236 y(in)h(the)h(sequel.)-180 1402 y FC(Lemma)k(3.10.)42 b Fw(\(i\).)i Fx( )39 b Fw(is)c(an)f(isomorphism)f(of)i Fx(H)8 b Fw(-mo)-5 b(dules.)43 b(\(R)-5 b(e)g(c)g(al)5 b(l)34 b(that)i Fx(H)f FA(=)27 b Fx(H)3067 1417 y Ft(F)3161 1402 y Fw(as)34 b(algebr)-5 b(as\).)-80 1595 y(\(ii\).)44 b(If)34 b Fx(r)m(;)17 b(s)28 b Fu(2)g Fx(R)36 b Fw(then)-180 1787 y FA(\(3.11\))1338 b Fx( )t FA(\()p Fx(r)s(s)p FA(\))28 b(=)f Fx(F)1852 1746 y Fs(1)1891 1787 y Fx(: )t FA(\()p Fx(r)s FA(\))17 b Fx(F)2202 1746 y Fs(2)2241 1787 y Fx(: )t FA(\()p Fx(s)p FA(\))p Fx(:)-80 1979 y Fw(\(iii\).)44 b(If)34 b Fx(r)c Fu(2)e Fx(R)36 b Fw(then)-180 2172 y FA(\(3.12\))1119 b(\001)1270 2187 y Ft(R)1323 2198 y Fq(F)1379 2172 y Fx( )t FA(\()p Fx(r)s FA(\))27 b(=)h Fx(F)1777 2131 y Fs(1)1816 2172 y Fx(: )t FA(\()p Fx(r)1995 2131 y Fs(\(1\))2089 2172 y FA(\))22 b Fu(\012)g Fx(F)2325 2131 y Fs(2)2365 2172 y Fx(: )t FA(\()p Fx(r)2544 2131 y Fs(\(2\))2638 2172 y FA(\))p Fx(:)-80 2364 y Fw(\(iv\).)66 b(If)41 b Fx(R)i Fw(is)f(a)g(gr)-5 b(ade)g(d)41 b(br)-5 b(aide)g(d)42 b(Hopf)g(algebr)-5 b(a,)43 b(then)f Fx(R)2167 2379 y Ft(F)2268 2364 y Fw(also)f(is)h(and)f Fx( )47 b Fw(is)41 b(a)h(gr)-5 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b(then)-180 4099 y(\(3.13\))483 b Fx( )t FA(\()p Fx(r)705 4058 y Fs(\(1\))799 4099 y FA(\))22 b Fu(\012)g Fx( )t FA(\()p Fx(r)1110 4058 y Fs(\(2\))1204 4099 y FA(\))28 b(=)g Fx(#)1431 4114 y Ft(F)1490 4099 y FA(\()p Fx(r)1572 4114 y Fs(\(1\))1666 4099 y Fx(\031)t Fu(S)7 b FA(\()p Fx(r)1874 4114 y Fs(\(2\))1969 4099 y FA(\)\))22 b Fu(\012)g Fx(#)2223 4114 y Ft(F)2283 4099 y FA(\()p Fx(r)2365 4114 y Fs(\(3\))2459 4099 y FA(\))27 b(=)h Fx( )t FA(\()p Fx(r)2777 4114 y Fs(\(1\))2871 4099 y FA(\))22 b Fu(\012)h Fx( )t FA(\()p Fx(r)3180 4114 y Fs(\(2\))3274 4099 y FA(\))p Fx(:)-180 4291 y FA(Using)32 b(that)h Fx(#)363 4306 y Ft(F)454 4291 y FA(is)f(a)h(coalgebra)e(map,)h(\(3.2\))g(and)h (\(3.13\),)f(w)m(e)h(conclude)g(that)494 4623 y(\001)575 4638 y Ft(R)628 4649 y Fq(F)684 4623 y Fx( )t FA(\()p Fx(r)s FA(\))27 b(=)h(\001)1086 4638 y Ft(R)1139 4649 y Fq(F)1194 4623 y Fx(#)1251 4638 y Ft(F)1310 4623 y FA(\()p Fx(r)s FA(\))f(=)h Fx(#)1621 4638 y Ft(F)1680 4623 y FA(\()p Fx(r)1762 4638 y Fs(\(1)p Ft(;F)10 b Fs(\))1930 4623 y FA(\))22 b Fu(\012)h Fx(#)2147 4638 y Ft(F)2206 4623 y FA(\()p Fx(r)2288 4638 y Fs(\(2)p Ft(;F)10 b Fs(\))2456 4623 y FA(\))901 4792 y(=)28 b Fx(#)1062 4807 y Ft(F)1121 4792 y FA(\()p Fx(F)1236 4750 y Fs(1)1275 4792 y Fx(r)1319 4807 y Fs(\(1\))1413 4792 y Fx(G)1490 4750 y Fs(1)1529 4792 y FA(\))22 b Fu(\012)h Fx(#)1746 4807 y Ft(F)1805 4792 y FA(\()p Fx(F)1920 4750 y Fs(2)1959 4792 y Fx(r)2003 4807 y Fs(\(2\))2098 4792 y Fx(G)2175 4750 y Fs(2)2214 4792 y FA(\))28 b(=)f Fx(#)2440 4807 y Ft(F)2499 4792 y FA(\()p Fx(F)2614 4750 y Fs(1)2653 4792 y Fx(r)2697 4807 y Fs(\(1\))2792 4792 y FA(\))22 b Fu(\012)g Fx(#)3008 4807 y Ft(F)3067 4792 y FA(\()p Fx(F)3182 4750 y Fs(2)3221 4792 y Fx(r)3265 4807 y Fs(\(2\))3360 4792 y FA(\))901 4960 y(=)28 b Fx(F)1082 4919 y Fs(1)1121 4960 y Fx(:#)1205 4975 y Ft(F)1264 4960 y FA(\()p Fx(r)1346 4976 y Fs(\(1\))1440 4960 y FA(\))22 b Fu(\012)h Fx(F)1677 4919 y Fs(2)1716 4960 y Fx(:#)1800 4975 y Ft(F)1859 4960 y FA(\()p Fx(r)1941 4976 y Fs(\(2\))2035 4960 y FA(\))28 b(=)f Fx(F)2281 4919 y Fs(1)2321 4960 y Fx(: )t FA(\()p Fx(r)2500 4919 y Fs(\(1\))2594 4960 y FA(\))22 b Fu(\012)g Fx(F)2830 4919 y Fs(2)2869 4960 y Fx(: )t FA(\()p Fx(r)3048 4919 y Fs(\(2\))3142 4960 y FA(\);)-80 5152 y(The)41 b(pro)s(of)f(of)h (\(iv\))f(has)h(no)g(di\013erence)g(with)g(the)g(pro)s(of)f(of)g(the)h (analogous)f(statemen)m(t)h(in)f([AS2,)j(Lemma)-180 5292 y(2.3].)3995 b Fj(\003)p eop %%Page: 9 9 9 8 bop 446 0 a Fn(FINITE)33 b(QUANTUM)g(GR)n(OUPS)f(O)n(VER)g(ABELIAN) g(GR)n(OUPS)g(OF)i(PRIME)f(EXPONENT)588 b(9)-80 217 y FA(W)-8 b(e)33 b(no)m(w)h(consider)f(the)h(sp)s(ecial)e(case)i(when)g Fx(H)i FA(=)28 b Fy(|)-8 b FA(\000,)27 b(\000)33 b(a)g(\014nite)g(ab)s (elian)e(group.)45 b(Let)33 b Fx(!)f FA(:)3426 192 y Fv(b)3423 217 y FA(\000)23 b Fu(\002)3609 192 y Fv(b)3607 217 y FA(\000)28 b Fu(!)g Fy(|)3886 181 y Fr(\002)3972 217 y FA(b)s(e)-180 333 y(a)k(2-co)s(cycle,)h(i.e.)44 b Fx(!)t FA(\()p Fx(\034)6 b(;)17 b FA(1\))26 b(=)i Fx(!)t FA(\(1)p Fx(;)17 b(\034)11 b FA(\))28 b(=)f(1)33 b(and)g Fx(!)t FA(\()p Fx(\034)6 b(;)17 b(\020)8 b FA(\))p Fx(!)t FA(\()p Fx(\034)j(\020)d(;)17 b(\021)t FA(\))24 b(=)k Fx(!)t FA(\()p Fx(\034)6 b(;)17 b(\020)8 b(\021)t FA(\))p Fx(!)t FA(\()p Fx(\020)g(;)17 b(\021)t FA(\).)40 b(The)33 b(co)s(cycle)h Fx(!)i FA(allo)m(ws)31 b(to)-180 464 y(de\014ne)j(a)e (map)g(\011)27 b(:)561 439 y Fv(b)558 464 y FA(\000)22 b Fu(\002)h FA(\000)28 b Fu(!)f FA(\000)32 b(b)m(y)-180 666 y(\(3.14\))869 b Fu(h)p Fx(\034)6 b(;)17 b FA(\011\()p Fx(\037;)g(g)t FA(\))p Fu(i)25 b FA(=)j Fx(!)t FA(\()p Fx(\034)6 b(;)17 b(\037)p FA(\))p Fx(!)t FA(\()p Fx(\037;)g(\034)11 b FA(\))2139 625 y Fr(\000)p Fs(1)2232 666 y Fu(h)p Fx(\034)6 b(;)17 b(g)t Fu(i)p Fx(;)210 b(\034)39 b Fu(2)2868 641 y Fv(b)2865 666 y FA(\000)p Fx(:)-180 877 y FA(W)-8 b(e)43 b(iden)m(tify)e Fx(H)50 b FA(with)42 b(the)h(Hopf)f(algebra)f Fy(|)1565 824 y Fk(b)1563 841 y Fs(\000)1647 877 y FA(of)h(functions)g (on)h(the)f(group)2810 852 y Fv(b)2807 877 y FA(\000;)47 b(w)m(e)d(denote)f(b)m(y)g Fx(\016)3608 892 y Ft(\034)3695 877 y Fu(2)i Fx(H)k FA(the)-180 1002 y(function)32 b(giv)m(en)h(b)m(y)g Fx(\016)635 1017 y Ft(\034)679 1002 y FA(\()p Fx(\020)8 b FA(\))26 b(=)i Fx(\016)979 1017 y Ft(\034)t(;\020)1073 1002 y FA(,)33 b Fx(\034)6 b(;)17 b(\020)35 b Fu(2)1400 977 y Fv(b)1397 1002 y FA(\000.)43 b(Then)34 b Fx(\016)1826 1017 y Ft(\034)1897 1002 y FA(=)2034 963 y Fs(1)p 2010 979 84 4 v 2010 1036 a Fr(j)p Fs(\000)p Fr(j)2120 927 y Fv(P)2225 1031 y Ft(g)r Fr(2)p Fs(\000)2357 1002 y Fu(h)p Fx(\034)6 b(;)17 b(g)2539 966 y Fr(\000)p Fs(1)2632 1002 y Fu(i)p Fx(g)t(:)31 b FA(Let)i Fx(F)41 b Fu(2)29 b Fx(H)g Fu(\012)23 b Fx(H)40 b FA(b)s(e)33 b(giv)m(en)f(b)m(y)1443 1243 y Fx(F)41 b FA(=)1670 1148 y Fv(X)1651 1379 y Ft(\034)t(;\020)5 b Fr(2)1791 1362 y Fk(b)1789 1379 y Fs(\000)1849 1243 y Fx(!)t FA(\()p Fx(\034)h(;)17 b(\020)8 b FA(\))p Fx(\016)2176 1258 y Ft(\034)2239 1243 y Fu(\012)23 b Fx(\016)2382 1258 y Ft(\020)2422 1243 y Fx(:)-180 1569 y FA(Then)38 b Fx(F)51 b FA(satis\014es)38 b(\(3.8\);)h(note)f(that)f Fx(H)43 b FA(=)35 b Fx(H)1578 1584 y Ft(F)1637 1569 y FA(.)57 b(Let)37 b(no)m(w)h Fx(R)g FA(b)s(e)g(a)f(braided)g(Hopf)g (algebra)f(in)3508 1533 y Fs(\000)3508 1594 y(\000)3556 1569 y Fu(Y)8 b(D)s FA(;)39 b(w)m(e)g(can)-180 1685 y(consider)d(the)g (Hopf)f(algebras)g Fx(A)e FA(=)g Fx(R)q FA(#)p Fy(|)-8 b FA(\000)29 b(and)36 b Fx(A)1787 1700 y Ft(F)1845 1685 y FA(,)h(the)f(braided)f(Hopf)h(algebra)e Fx(R)3095 1700 y Ft(F)3187 1685 y Fu(2)3286 1649 y Fs(\000)3286 1710 y(\000)3334 1685 y Fu(Y)8 b(D)38 b FA(and)e(the)g(map)-180 1801 y Fx( )c FA(:)27 b Fx(R)i Fu(!)e Fx(R)273 1816 y Ft(F)332 1801 y FA(.)44 b(W)-8 b(e)33 b(ha)m(v)m(e)710 2011 y Fx( )t FA(\()p Fx(r)s FA(\))27 b(=)1031 1916 y Fv(X)1038 2147 y Ft(\034)8 b Fr(2)1126 2130 y Fk(b)1124 2147 y Fs(\000)1191 2011 y Fx(!)t FA(\()p Fx(\037;)17 b(\034)11 b FA(\))1490 1970 y Fr(\000)p Fs(1)1584 2011 y Fx(r)s FA(#)p Fx(\016)1755 2026 y Ft(\034)1799 2011 y Fx(;)114 b(r)30 b Fu(2)e Fx(R)2183 1970 y Ft(\037)2231 2011 y FA(;)212 b Fx( )t FA(\()p Fx(R)2650 1970 y Ft(\037)2649 2036 y(g)2698 2011 y FA(\))28 b(=)f Fx(R)2942 1964 y Ft( )2941 2042 y Fs(\011\()p Ft(\037;g)r Fs(\))3155 2011 y Fx(:)-180 2321 y FA(See)33 b([AS2)q(,)f(Lemma)f(2.3].)44 b(Note)32 b(that)h(\(3.11\))e(is)i(no)m(w)g Fx( )t FA(\()p Fx(r)s(s)p FA(\))27 b(=)h Fx(!)t FA(\()p Fx(\037;)17 b(\034)11 b FA(\))p Fx( )t FA(\()p Fx(r)s FA(\))p Fx( )t FA(\()p Fx(s)p FA(\),)31 b Fx(r)f Fu(2)e Fx(R)3203 2285 y Ft(\037)3251 2321 y FA(,)33 b Fx(s)28 b Fu(2)g Fx(R)3554 2285 y Ft(\034)3597 2321 y FA(.)-180 2499 y FC(Lemma)37 b(3.15.)42 b Fw(If)34 b Fx(r)d Fu(2)d Fx(P)14 b FA(\()p Fx(R)q FA(\))988 2463 y Ft(\037)988 2524 y(g)1070 2499 y Fw(and)34 b Fx(s)28 b Fu(2)g Fx(R)1502 2463 y Ft(\034)1580 2499 y Fw(then)-180 2716 y FA(\(3.16\))1202 b Fx( )21 b FA(\([)p Fx(r)m(;)c(s)p FA(])1579 2731 y Ft(c)1614 2716 y FA(\))27 b(=)h Fx(!)t FA(\()p Fx(\037;)17 b(\034)11 b FA(\)[)p Fx( )t FA(\()p Fx(r)s FA(\))p Fx(;)17 b( )t FA(\()p Fx(s)p FA(\)])2559 2731 y Ft(c)2592 2716 y Fx(:)-180 2950 y Fw(Pr)-5 b(o)g(of.)41 b FA(W)-8 b(e)33 b(ha)m(v)m(e)818 3167 y Fx( )21 b FA(\()o([)p Fx(r)m(;)c(s)p FA(])1124 3182 y Ft(c)1159 3167 y FA(\))28 b(=)f Fx( )21 b FA(\()p Fx(r)s(s)h Fu(\000)g Fx(\034)11 b FA(\()p Fx(g)t FA(\))p Fx(sr)s FA(\))1225 3335 y(=)27 b Fx(!)t FA(\()p Fx(\037;)17 b(\034)11 b FA(\))p Fx( )t FA(\()p Fx(r)s FA(\))p Fx( )t FA(\()p Fx(s)p FA(\))21 b Fu(\000)i Fx(!)t FA(\()p Fx(\034)6 b(;)17 b(\037)p FA(\))p Fx(\034)11 b FA(\()p Fx(g)t FA(\))p Fx( )t FA(\()p Fx(s)p FA(\))p Fx( )t FA(\()p Fx(r)s FA(\))1225 3504 y(=)27 b Fx(!)t FA(\()p Fx(\037;)17 b(\034)11 b FA(\))17 b(\()o Fx( )t FA(\()p Fx(r)s FA(\))p Fx( )t FA(\()p Fx(s)p FA(\))22 b Fu(\000)h(h)p Fx(\034)6 b(;)17 b FA(\011\()p Fx(\037;)g(g)t FA(\))p Fu(i)p Fx( )t FA(\()p Fx(s)p FA(\))p Fx( )t FA(\()p Fx(r)s FA(\)\))1225 3672 y(=)27 b Fx(!)t FA(\()p Fx(\037;)17 b(\034)11 b FA(\)[)p Fx( )t FA(\()p Fx(r)s FA(\)\))p Fx(;)17 b( )t FA(\()p Fx(s)p FA(\)])2142 3687 y Ft(c)2176 3672 y Fx(;)-180 3889 y FA(where)34 b(w)m(e)f(used)h(\(3.14\))o(.)3250 b Fj(\003)-180 4123 y Fw(R)-5 b(emark)42 b FA(3.17)p Fw(.)c FA(It)29 b(is)g(p)s(ossible)f(to)h(sho)m(w)h(that)f(\()p Fx( )19 b Fu(\012)c Fx( )t FA(\))p Fx(c)p FA(\()p Fx(r)k Fu(\012)c Fx(s)p FA(\))28 b(=)f Fx(F)s(:c)2455 4138 y Ft(F)2514 4123 y FA(\()p Fx( )t FA(\()p Fx(r)s FA(\))15 b Fu(\012)g Fx( )t FA(\()p Fx(s)p FA(\)\),)30 b(for)f(all)e Fx(r)j Fu(2)e Fx(R)3654 4087 y Ft(\037)3653 4148 y(g)3702 4123 y FA(,)i Fx(s)e Fu(2)g Fx(R)4002 4087 y Ft(\034)4045 4123 y FA(.)-80 4259 y(F)-8 b(rom)31 b(the)i(previous)g(considerations) f(and)h(Lemma)e(3.10)h(w)m(e)h(immediately)d(get)-180 4437 y FC(Prop)s(osition)45 b(3.18.)h Fw(L)-5 b(et)42 b Fx(R)h Fw(b)-5 b(e)42 b(an)g(algebr)-5 b(a)41 b(in)1741 4401 y Fs(\000)1741 4462 y(\000)1789 4437 y Fu(Y)8 b(D)s Fw(,)44 b FA(\()p Fx(x)2111 4452 y Ft(i)2139 4437 y FA(\))2177 4452 y Ft(i)p Fr(2)p Ft(I)2330 4437 y Fw(a)e(family)g(of)g(elements)f (of)h Fx(R)q Fw(,)i Fx(x)3582 4452 y Ft(i)3652 4437 y Fu(2)e Fx(R)3835 4401 y Ft(\037)3879 4411 y Fq(i)3834 4461 y Ft(g)3868 4471 y Fq(i)3951 4437 y Fw(for)-180 4580 y(some)34 b Fx(g)116 4595 y Ft(i)172 4580 y Fu(2)28 b FA(\000)p Fw(,)34 b Fx(\037)452 4595 y Ft(i)508 4580 y Fu(2)605 4554 y Fv(b)602 4580 y FA(\000)p Fw(.)45 b(Then:)-80 4719 y(\(i\).)f Fx( )t FA(\()p Fy(|)-9 b Fu(h)p Fx(x)364 4734 y Ft(i)387 4719 y Fu(i)p FA(\))27 b(=)g Fy(|)-8 b Fu(h)p Fx( )t FA(\()o Fx(x)854 4734 y Ft(i)877 4719 y FA(\))p Fu(i)p Fw(,)35 b Fx( )t FA(\()p Fu(h)p Fx(x)1218 4734 y Ft(i)1246 4719 y Fu(i)p FA(\))27 b(=)h Fu(h)p Fx( )t FA(\()p Fx(x)1653 4734 y Ft(i)1681 4719 y FA(\))p Fu(i)p Fw(.)-80 4859 y(\(ii\).)43 b(If)30 b Fx(R)j Fw(has)d(a)i(pr)-5 b(esentation)30 b(by)i(gener)-5 b(ators)30 b Fx(x)1788 4874 y Ft(i)1848 4859 y Fw(and)h(r)-5 b(elations)30 b Fx(t)2462 4874 y Ft(j)2499 4859 y Fw(,)i(wher)-5 b(e)31 b(also)f(the)i Fx(t)3222 4874 y Ft(j)3258 4859 y Fw('s)f(ar)-5 b(e)32 b(homo)-5 b(gene)g(ous)-180 4998 y(then)35 b Fx(R)111 5013 y Ft(F)204 4998 y Fw(has)g(a)f(pr)-5 b(esentation)35 b(by)g(gener)-5 b(ators)34 b Fx( )t FA(\()p Fx(x)1778 5013 y Ft(i)1806 4998 y FA(\))h Fw(and)g(r)-5 b(elations)34 b Fx( )t FA(\()p Fx(t)2606 5013 y Ft(j)2642 4998 y FA(\))p Fw(.)-80 5138 y(\(iii\).)44 b(If)34 b Fx(x)321 5153 y Ft(i)384 5138 y Fw(is)h(c)-5 b(entr)g(al)35 b(and)f Fx(!)t FA(\()p Fx(\037)1160 5153 y Ft(i)1188 5138 y Fx(;)17 b(\034)11 b FA(\))27 b(=)h Fx(!)t FA(\()p Fx(\034)6 b(;)17 b(\037)1710 5153 y Ft(i)1737 5138 y FA(\))35 b Fw(for)g(al)5 b(l)34 b Fx(\034)46 b Fw(such)35 b(that)g Fx(R)2689 5101 y Ft(\034)2760 5138 y Fu(6)p FA(=)28 b(0)p Fw(,)34 b(then)h Fx( )t FA(\()p Fx(x)3354 5153 y Ft(i)3383 5138 y FA(\))f Fw(is)h(c)-5 b(entr)g(al.)121 b Fj(\003)p eop %%Page: 10 10 10 9 bop -180 0 a Fn(10)846 b(NICOL)1001 -19 y(\023)991 0 y(AS)24 b(ANDR)n(USKIEWITSCH)f(AND)h(HANS-J)2419 -19 y(\177)2409 0 y(UR)n(GEN)f(SCHNEIDER)729 203 y FA(4.)48 b Fz(R)m(oot)38 b(vectors)f(and)h(Quantum)h(Serre)d(rela)-7 b(tions)-180 377 y FA(4.1.)49 b FC(Ro)s(ot)36 b(v)m(ectors.)49 b FA(In)33 b(this)f(Section,)g(w)m(e)i(assume)f(the)g(follo)m(wing)d (situation:)-80 502 y(W)-8 b(e)25 b(\014x)h(a)e(\014nite)h(ab)s(elian)e (group)i(\000,)i(a)d(\014nite)h(Cartan)g(matrix)f(\()p Fx(a)2278 517 y Ft(ij)2338 502 y FA(\))2376 517 y Fs(1)p Fr(\024)p Ft(i;j)t Fr(\024)p Ft(\022)2661 502 y FA(and)h Fx(g)2890 517 y Fs(1)2930 502 y Fx(;)17 b(:)g(:)g(:)e(;)i(g)3195 517 y Ft(\022)3261 502 y Fu(2)29 b FA(\000,)d Fx(\037)3531 517 y Fs(1)3571 502 y Fx(;)17 b(:)g(:)g(:)e(;)i(\037)3850 517 y Ft(\022)3917 502 y Fu(2)4014 477 y Fv(b)4011 502 y FA(\000)-180 618 y(suc)m(h)37 b(that)f(\(1.1\))f(and)g(\(1.2\))g (hold.)52 b(Let)36 b Fx(d)1415 633 y Fs(1)1454 618 y Fx(;)17 b(:)g(:)g(:)f(;)h(d)1724 633 y Ft(\022)1796 618 y Fu(2)33 b(f)p FA(1)p Fx(;)17 b FA(2)p Fx(;)g FA(3)p Fu(g)34 b FA(suc)m(h)j(that)e Fx(d)2752 633 y Ft(i)2780 618 y Fx(a)2831 633 y Ft(ij)2925 618 y FA(=)e Fx(d)3085 633 y Ft(j)3121 618 y Fx(a)3172 633 y Ft(j)t(i)3268 618 y FA(for)i(all)f Fx(i)p FA(,)j Fx(j)6 b FA(.)52 b(W)-8 b(e)36 b(set)-180 735 y Fx(q)-137 750 y Ft(i)-78 735 y FA(=)30 b Fx(\037)89 750 y Ft(i)118 735 y FA(\()p Fx(g)203 750 y Ft(i)230 735 y FA(\),)35 b Fx(N)408 750 y Ft(i)470 735 y FA(the)g(order)f(of)g Fx(q)1052 750 y Ft(i)1081 735 y FA(.)48 b(W)-8 b(e)35 b(assume,)g(for)f(all)e Fx(i)i FA(and)h Fx(j)6 b FA(,)34 b(that)g(the)h(order)f(of)g Fx(\037)3159 750 y Ft(i)3187 735 y FA(\()p Fx(g)3272 750 y Ft(j)3308 735 y FA(\))h(is)e(o)s(dd,)i(and)f(that)-180 851 y Fx(N)-102 866 y Ft(i)-41 851 y FA(is)e(not)g(divisible)f(b)m(y)i (3)g(if)e Fx(i)i FA(b)s(elongs)f(to)g(a)h(connected)h(comp)s(onen)m(t)e (of)g(t)m(yp)s(e)i Fx(G)2897 866 y Fs(2)2936 851 y FA(.)-80 1002 y(Let)42 b Fu(X)56 b FA(b)s(e)43 b(the)f(set)h(of)e(connected)j (comp)s(onen)m(ts)e(of)g(the)g(Dynkin)g(diagram)e(corresp)s(onding)i (to)f(\()p Fx(a)3739 1017 y Ft(ij)3800 1002 y FA(\).)72 b(W)-8 b(e)-180 1118 y(assume)45 b(that)f(for)g(eac)m(h)h Fx(I)56 b Fu(2)49 b(X)15 b FA(,)47 b(there)e(exist)g Fx(c)1704 1133 y Ft(I)1744 1118 y Fx(;)17 b(d)1839 1133 y Ft(I)1922 1118 y FA(suc)m(h)46 b(that)e Fx(I)56 b FA(=)48 b Fu(f)p Fx(j)54 b FA(:)48 b Fx(c)2861 1133 y Ft(I)2948 1118 y Fu(\024)h Fx(j)54 b Fu(\024)48 b Fx(d)3344 1133 y Ft(I)3384 1118 y Fu(g)p FA(;)i(that)44 b(is,)j(after)-180 1234 y(reordering)31 b(the)g(Cartan)g(matrix)f(is)h(a)f(matrix)g(of)h (blo)s(c)m(ks)g(corresp)s(onding)g(to)g(the)g(connected)i(comp)s(onen)m (ts.)44 b(Let)-180 1350 y Fx(I)57 b Fu(2)49 b(X)60 b FA(and)45 b Fx(i)k Fu(\030)g Fx(j)i FA(in)44 b Fx(I)8 b FA(;)52 b(then)45 b Fx(N)1235 1365 y Ft(i)1312 1350 y FA(=)k Fx(N)1515 1365 y Ft(j)1552 1350 y FA(,)f(hence)e Fx(N)1988 1365 y Ft(I)2077 1350 y FA(:=)j Fx(N)2307 1365 y Ft(i)2380 1350 y FA(is)c(w)m(ell)f(de\014ned.)82 b(Let)46 b(\010)3370 1365 y Ft(I)3410 1350 y FA(,)i(resp.)82 b(\010)3837 1309 y Fs(+)3837 1377 y Ft(I)3897 1350 y FA(,)48 b(b)s(e)-180 1467 y(the)41 b(ro)s(ot)e(system,)k(resp.)68 b(the)41 b(subset)h(of)e(p)s(ositiv)m(e)f(ro)s(ots,)j(corresp)s(onding)e(to)g (the)h(Cartan)f(matrix)f(\()p Fx(a)3783 1482 y Ft(ij)3844 1467 y FA(\))3882 1482 y Ft(i;j)t Fr(2)p Ft(I)4045 1467 y FA(;)-180 1584 y(then)k(\010)j(=)289 1509 y Fv(S)372 1613 y Ft(I)5 b Fr(2X)535 1584 y FA(\010)605 1599 y Ft(I)646 1584 y FA(,)45 b(resp.)75 b(\010)1063 1548 y Fs(+)1167 1584 y FA(=)1288 1509 y Fv(S)1371 1613 y Ft(I)5 b Fr(2X)1535 1584 y FA(\010)1605 1543 y Fs(+)1605 1611 y Ft(I)1707 1584 y FA(is)42 b(the)h(ro)s(ot)f(system,)k(resp.)75 b(the)44 b(subset)g(of)e(p)s(ositiv)m(e)g(ro)s(ots,)-180 1700 y(corresp)s(onding)32 b(to)h(the)g(Cartan)f(matrix)f(\()p Fx(a)1470 1715 y Ft(ij)1531 1700 y FA(\))1569 1715 y Fs(1)p Fr(\024)p Ft(i;j)t Fr(\024)p Ft(\022)1829 1700 y Fx(:)i FA(Let)f Fx(\013)2125 1715 y Fs(1)2165 1700 y Fx(;)17 b(:)g(:)g(:)f(;)h(\013)2446 1715 y Ft(\022)2517 1700 y FA(b)s(e)33 b(the)g(set)g(of)f(simple)f(ro)s(ots.)-80 1851 y(Let)47 b Fu(W)207 1866 y Ft(I)294 1851 y FA(b)s(e)h(the)f(W)-8 b(eyl)47 b(group)g(corresp)s(onding)g(to)g(the)h(Cartan)f(matrix)e(\()p Fx(a)2899 1866 y Ft(ij)2960 1851 y FA(\))2998 1866 y Ft(i;j)t Fr(2)p Ft(I)3161 1851 y FA(;)54 b(w)m(e)49 b(iden)m(tify)d(it) g(with)-180 1967 y(a)h(subgroup)h(of)f(the)h(W)-8 b(eyl)48 b(group)f Fu(W)56 b FA(corresp)s(onding)48 b(to)f(the)h(Cartan)f (matrix)f(\()p Fx(a)3095 1982 y Ft(ij)3156 1967 y FA(\).)88 b(W)-8 b(e)48 b(\014x)g(a)f(reduced)-180 2083 y(decomp)s(osition)37 b(of)i(the)h(longest)e(elemen)m(t)h Fx(!)1527 2098 y Fs(0)p Ft(;I)1661 2083 y FA(of)g Fu(W)1877 2098 y Ft(I)1956 2083 y FA(in)f(terms)h(of)g(simple)f(re\015ections.)64 b(Then)40 b(w)m(e)g(obtain)e(a)-180 2200 y(reduced)44 b(decomp)s(osition)e(of)g(the)h(longest)g(elemen)m(t)g Fx(!)1920 2215 y Fs(0)2004 2200 y FA(=)i Fx(s)2171 2215 y Ft(i)2195 2224 y Fl(1)2251 2200 y Fx(:)17 b(:)g(:)f(s)2428 2215 y Ft(i)2452 2226 y Fq(P)2550 2200 y FA(of)42 b Fu(W)52 b FA(from)42 b(the)h(expression)h(of)f Fx(!)3903 2215 y Fs(0)3985 2200 y FA(as)-180 2316 y(pro)s(duct)34 b(of)g(the)g Fx(!)530 2331 y Fs(0)p Ft(;I)624 2316 y FA('s)h(in)e(some)h(\014xed)h (order)f(of)f(the)h(comp)s(onen)m(ts,)h(sa)m(y)g(the)f(order)g(arising) e(from)h(the)h(order)g(of)-180 2432 y(the)f(v)m(ertices.)45 b(Therefore)33 b Fx(\014)875 2447 y Ft(j)940 2432 y FA(:=)27 b Fx(s)1116 2447 y Ft(i)1140 2456 y Fl(1)1196 2432 y Fx(:)17 b(:)g(:)f(s)1373 2447 y Ft(i)1397 2457 y Fq(j)s Fh(\000)p Fl(1)1512 2432 y FA(\()p Fx(\013)1612 2447 y Ft(i)1636 2457 y Fq(j)1673 2432 y FA(\))32 b(is)g(a)g(n)m(umeration)g (of)g(\010)2621 2396 y Fs(+)2680 2432 y FA(.)-80 2686 y(W)-8 b(e)29 b(\014x)h(a)f(\014nite)g(dimensional)e(Y)-8 b(etter-Drinfeld)28 b(mo)s(dule)g Fx(V)51 b FA(o)m(v)m(er)30 b(\000)f(with)g(a)g(basis)h Fx(x)3086 2701 y Fs(1)3125 2686 y Fx(;)17 b(:)g(:)g(:)f(;)h(x)3399 2701 y Ft(\022)3468 2686 y FA(with)29 b Fx(x)3742 2701 y Ft(i)3798 2686 y Fu(2)f Fx(V)3971 2650 y Ft(\037)4015 2660 y Fq(i)3949 2711 y Ft(g)3983 2721 y Fq(i)4045 2686 y FA(,)-180 2803 y(1)f Fu(\024)i Fx(i)f Fu(\024)g Fx(\022)s FA(.)-80 3057 y(Ma)5 b(jor)37 b(examples)g(of)f(mo)s(dules)g(of)h(Cartan)g(t)m(yp)s (e)h(are)f(the)g(F)-8 b(rob)s(enius-Lusztig)36 b(k)m(ernels.)58 b(Let)37 b Fx(N)46 b(>)35 b FA(1)i(b)s(e)g(an)-180 3173 y(o)s(dd)c(natural)f(n)m(um)m(b)s(er)i(and)f(let)g Fx(q)f Fu(2)e Fy(|)18 b FA(b)s(e)34 b(a)f(primitiv)m(e)e Fx(N)10 b FA(-th)33 b(ro)s(ot)f(of)h(1,)g(not)h(divisible)d(b)m(y)j(3)f(in)g (case)h(\()p Fx(a)3799 3188 y Ft(ij)3859 3173 y FA(\))g(has)-180 3307 y(a)i(comp)s(onen)m(t)h(of)f(t)m(yp)s(e)i Fx(G)824 3322 y Fs(2)863 3307 y FA(.)55 b(Let)37 b Fy(G)62 b FA(=)35 b Fy(Z)p Fx(=)p FA(\()p Fx(N)10 b FA(\))1629 3271 y Ft(\022)1700 3307 y FA(=)34 b Fu(h)p Fx(e)1894 3322 y Fs(1)1933 3307 y Fu(i)25 b(\010)g(\001)17 b(\001)g(\001)23 b(\010)j(h)p Fx(e)2427 3322 y Ft(\022)2466 3307 y Fu(i)p FA(;)38 b(let)e Fx(\021)2763 3322 y Ft(j)2834 3307 y Fu(2)2946 3281 y Fv(b)2935 3307 y Fy(G)64 b FA(b)s(e)37 b(the)g(unique)g(c)m(haracter) -180 3425 y(suc)m(h)k(that)e Fu(h)p Fx(\021)t FA(\()p Fx(j)6 b FA(\))p Fx(;)17 b(e)p FA(\()p Fx(i)p FA(\))p Fu(i)37 b FA(=)i Fx(q)914 3389 y Ft(d)950 3399 y Fq(i)977 3389 y Ft(a)1014 3399 y Fq(ij)1073 3425 y FA(.)63 b(Let)39 b Fy(V)46 b FA(b)s(e)40 b(a)f(Y)-8 b(etter-Drinfeld)37 b(mo)s(dule)h(o)m(v)m(er)i Fy(G)66 b FA(with)39 b(a)g(basis)g Fx(X)3694 3440 y Fs(1)3733 3425 y Fx(;)17 b(:)g(:)g(:)f(;)h(X)4033 3440 y Ft(\022)-180 3542 y FA(suc)m(h)34 b(that)1288 3686 y Fx(X)1369 3701 y Ft(i)1425 3686 y Fu(2)28 b Fy(V)1585 3645 y Ft(e)1618 3655 y Fq(i)1585 3711 y Ft(\021)1621 3721 y Fq(i)1657 3686 y Fx(;)212 b FA(for)32 b(all)f(1)c Fu(\024)h Fx(i)g Fu(\024)g Fx(\022)s(:)-180 3857 y FA(W)-8 b(e)26 b(denote)h(b)m(y)g Fm(c)f FA(the)g(braiding)f(of)g Fy(V)q FA(.)47 b(Lusztig)26 b(de\014ned)i(ro)s(ot)d(v)m(ectors)i Fx(X)2541 3872 y Ft(\013)2619 3857 y Fu(2)h Fm(B)p FA(\()p Fy(V)q FA(\),)33 b Fx(\013)28 b Fu(2)g FA(\010)3257 3821 y Fs(+)3343 3857 y FA([L2].)41 b(One)26 b(can)h(see)-180 3973 y(from)k([L3])h(that,)g(up)h(to)f(a)g(non-zero)g(scalar,)g(eac)m (h)h(ro)s(ot)f(v)m(ector)h(can)g(b)s(e)f(written)g(as)h(an)f(iterated)g (braided)g(com-)-180 4089 y(m)m(utator)e(in)g(some)h(sequence)i Fx(X)1033 4104 y Ft(`)1062 4113 y Fl(1)1101 4089 y Fx(;)17 b(:)g(:)g(:)f(;)h(X)1401 4104 y Ft(`)1430 4112 y Fq(a)1502 4089 y FA(of)30 b(simple)g(ro)s(ot)g(v)m(ectors)i(suc)m(h)h(as)e([[)p Fx(X)2922 4104 y Ft(`)2951 4113 y Fl(1)2989 4089 y Fx(;)17 b FA([)p Fx(X)3141 4104 y Ft(`)3170 4113 y Fl(2)3209 4089 y Fx(;)g(X)3334 4104 y Ft(`)3363 4113 y Fl(3)3401 4089 y FA(])3428 4104 y Fg(c)3459 4089 y FA(])3486 4104 y Fg(c)3517 4089 y Fx(;)g FA([)p Fx(X)3669 4104 y Ft(`)3698 4113 y Fl(4)3736 4089 y Fx(;)g(X)3861 4104 y Ft(`)3890 4113 y Fl(5)3929 4089 y FA(])3956 4104 y Fg(c)3987 4089 y FA(])4014 4104 y Fg(c)4045 4089 y FA(.)-180 4206 y(This)33 b(can)g(also)e(b)s(e)i(seen)h(in)e(the)h(situation)e(in)h([Ri)n(].)-80 4460 y(W)-8 b(e)36 b(no)m(w)g(\014x)g(for)g(eac)m(h)g Fx(\013)e Fu(2)f FA(\010)1078 4424 y Fs(+)1174 4460 y FA(suc)m(h)k(a)e(represen)m(tation)i(of)e Fx(X)2317 4475 y Ft(\013)2402 4460 y FA(as)h(an)f(iterated)h(braided)f(comm)m(utator.) 52 b(In)-180 4576 y(the)30 b(general)f(case)h(of)f(our)g Fx(V)22 b FA(,)30 b(w)m(e)g(de\014ne)h(ro)s(ot)d(v)m(ectors)j Fx(x)1942 4591 y Ft(\013)2022 4576 y FA(in)d(the)i(tensor)g(algebra)e Fx(T)14 b FA(\()p Fx(V)22 b FA(\),)30 b Fx(\013)e Fu(2)g FA(\010)3462 4540 y Fs(+)3522 4576 y FA(,)i(as)f(the)h(same)-180 4692 y(formal)36 b(iteration)g(of)h(braided)h(comm)m(utators)f(in)g (the)h(elemen)m(ts)g Fx(x)2352 4707 y Fs(1)2392 4692 y Fx(;)17 b(:)g(:)g(:)f(;)h(x)2666 4707 y Ft(\022)2743 4692 y FA(instead)38 b(of)f Fx(X)3282 4707 y Fs(1)3322 4692 y Fx(;)17 b(:)g(:)g(:)f(;)h(X)3622 4707 y Ft(\022)3698 4692 y FA(but)38 b(with)-180 4808 y(resp)s(ect)h(to)f(the)h(braiding)d Fx(c)i FA(giv)m(en)g(b)m(y)h(the)g(general)e(matrix)g(\()p Fx(\037)2266 4823 y Ft(j)2302 4808 y FA(\()p Fx(g)2387 4823 y Ft(i)2415 4808 y FA(\)\).)60 b(Note)38 b(that)g(eac)m(h)h Fx(x)3316 4823 y Ft(\013)3404 4808 y FA(is)f(homogeneous)-180 4925 y(and)33 b(has)g(the)g(same)f(degree)i(as)e Fx(X)1100 4940 y Ft(\013)1150 4925 y FA(,)g(where)i(w)m(e)g(mean)e(the)h(degree)g (in)f(the)h(sense)i(of)d([L3].)43 b(Also,)-180 5110 y(\(4.1\))1639 b Fx(x)1715 5125 y Ft(\013)1793 5110 y Fu(2)28 b Fx(T)14 b FA(\()p Fx(V)21 b FA(\))2112 5069 y Ft(\037)2156 5077 y Fq(\013)2112 5134 y Ft(g)2146 5142 y Fq(\013)2204 5110 y Fx(;)-180 5317 y FA(where)34 b Fx(g)149 5332 y Ft(\013)226 5317 y FA(=)27 b Fx(g)380 5273 y Ft(b)410 5282 y Fl(1)376 5341 y Fs(1)465 5317 y Fx(:)17 b(:)g(:)f(g)647 5271 y Ft(b)677 5283 y Fq(\022)643 5345 y Ft(\022)716 5317 y FA(,)33 b Fx(\037)837 5332 y Ft(\013)914 5317 y FA(=)27 b Fx(\037)1078 5273 y Ft(b)1108 5282 y Fl(1)1078 5341 y Fs(1)1164 5317 y Fx(:)17 b(:)g(:)f(\037)1356 5271 y Ft(b)1386 5283 y Fq(\022)1356 5345 y Ft(\022)1426 5317 y FA(,)32 b(where)i Fx(\013)29 b FA(=)e Fx(b)2002 5332 y Fs(1)2042 5317 y Fx(\013)2104 5332 y Fs(1)2166 5317 y FA(+)22 b Fu(\001)17 b(\001)g(\001)j FA(+)i Fx(b)2541 5332 y Ft(\022)2580 5317 y Fx(\013)2642 5332 y Ft(\022)2682 5317 y FA(.)p eop %%Page: 11 11 11 10 bop 446 0 a Fn(FINITE)33 b(QUANTUM)g(GR)n(OUPS)f(O)n(VER)g (ABELIAN)g(GR)n(OUPS)g(OF)i(PRIME)f(EXPONENT)549 b(11)-180 203 y FC(Theorem)37 b(4.2.)42 b Fw(The)34 b(Nichols)h(algebr)-5 b(a)34 b Fm(B)p FA(\()p Fy(V)q FA(\))40 b Fw(is)35 b(pr)-5 b(esente)g(d)34 b(by)h(gener)-5 b(ators)34 b Fx(X)2870 218 y Ft(i)2898 203 y Fw(,)h FA(1)28 b Fu(\024)g Fx(i)g Fu(\024)g Fx(\022)s Fw(,)35 b(and)f(r)-5 b(elations)1057 414 y FA(ad)1176 429 y Ft(c)1211 414 y FA(\()p Fx(X)1330 429 y Ft(i)1358 414 y FA(\))1396 373 y Fs(1)p Fr(\000)p Ft(a)1523 383 y Fq(ij)1582 414 y FA(\()p Fx(X)1701 429 y Ft(j)1738 414 y FA(\))28 b(=)f(0)p Fx(;)216 b Fw(for)34 b(al)5 b(l)35 b Fx(i)28 b Fu(6)p FA(=)f Fx(j;)-2905 b FA(\(4.3\))1620 583 y Fx(X)1709 542 y Ft(N)1701 607 y(\013)1804 583 y FA(=)27 b(0)p Fx(;)216 b Fw(for)34 b(al)5 b(l)35 b Fx(\013)28 b Fu(2)g FA(\010)2748 542 y Fs(+)2808 583 y Fx(:)-3015 b FA(\(4.4\))-180 794 y Fw(Mor)-5 b(e)g(over,)34 b(the)h(fol)5 b(lowing)34 b(elements)g(c)-5 b(onstitute)35 b(a)g(b)-5 b(asis)34 b(of)h Fm(B)p FA(\()p Fy(V)p FA(\))p Fw(:)711 1005 y Fx(X)800 961 y Ft(h)841 970 y Fl(1)792 1033 y Ft(\014)832 1042 y Fl(1)879 1005 y Fx(X)968 961 y Ft(h)1009 970 y Fl(2)960 1033 y Ft(\014)1000 1042 y Fl(2)1064 1005 y Fx(:)17 b(:)g(:)f(X)1284 960 y Ft(h)1325 971 y Fq(P)1276 1033 y Ft(\014)1316 1044 y Fq(P)1380 1005 y Fx(;)216 b Fw(for)35 b(al)5 b(l)34 b FA(0)28 b Fu(\024)g Fx(h)2156 1020 y Ft(j)2220 1005 y Fu(\024)g Fx(N)33 b Fu(\000)23 b FA(1)p Fx(;)116 b FA(1)27 b Fu(\024)h Fx(j)34 b Fu(\024)28 b Fx(P)s(:)-180 1232 y Fw(Pr)-5 b(o)g(of.)41 b FA(It)d(follo)m(ws)f(from)g(results)i(of)f(Lusztig)g ([L1],)h([L2],)h(Rosso)f([Ro1)o(],)h([Ro2)o(])e(and)h(M)s(\177)-51 b(uller)36 b([Mu)q(])i(that)g Fm(B)p FA(\()p Fy(V)q FA(\))-180 1372 y(is)k(the)h(p)s(ositiv)m(e)f(part)g(of)g(the)h(so-called)f(F)-8 b(rob)s(enius-Lusztig)41 b(k)m(ernel)i(corresp)s(onding)g(to)f(the)h (Cartan)f(matrix)-180 1511 y(\()p Fx(a)-91 1526 y Ft(ij)-30 1511 y FA(\).)50 b(See)35 b([AS2,)h(Th.)50 b(3.1])35 b(for)f(details.)49 b(The)35 b(presen)m(tation)h(b)m(y)f(generators)g (and)g(relations)f(follo)m(ws)f(from)g(the)-180 1651 y(considerations)i(in)g(the)i(last)e(paragraph)g(of)g(p.)53 b(15)36 b(and)f(the)i(\014rst)f(paragraph)f(of)g(p.)54 b(16)35 b(in)g([AJS])h(referring)f(to)-180 1790 y([dCP)q(,)e Fu(x)p FA(19,)f(Corollary)f(in)h(p.)43 b(120].)g(The)34 b(statemen)m(t)f(ab)s(out)f(the)h(basis)g(is)f([L1)o(,)h(L2].)983 b Fj(\003)-180 2026 y FA(4.2.)49 b FC(Nic)m(hols)c(algebras)k(of)f (Cartan)g(t)m(yp)s(e.)h FA(W)-8 b(e)42 b(can)g(no)m(w)g(pro)m(v)m(e)h (the)f(\014rst)g(main)e(result)i(of)f(the)h(presen)m(t)-180 2143 y(pap)s(er,)29 b(describing)e Fm(B)p FA(\()p Fx(V)22 b FA(\))27 b(b)m(y)i(generators)f(and)g(relations)e(when)j Fx(V)49 b FA(is)28 b(of)f(\014nite)g(Cartan)h(t)m(yp)s(e,)i(impro)m (ving)c([AS2,)-180 2259 y(Th.)44 b(1.1)32 b(\(i\)].)43 b(As)33 b(in)f Fw(lo)-5 b(c.)44 b(cit.)p FA(,)33 b(w)m(e)g(use)h(rep)s (eatedly)f(Remark)f(3.6.)-180 2434 y FC(Theorem)37 b(4.5.)42 b Fw(The)34 b(Nichols)h(algebr)-5 b(a)34 b Fm(B)p FA(\()p Fx(V)21 b FA(\))35 b Fw(is)g(pr)-5 b(esente)g(d)34 b(by)h(gener)-5 b(ators)34 b Fx(x)2851 2449 y Ft(i)2880 2434 y Fw(,)g FA(1)28 b Fu(\024)g Fx(i)g Fu(\024)g Fx(\022)s Fw(,)35 b(and)f(r)-5 b(elations)932 2645 y FA(ad)1051 2660 y Ft(c)1085 2645 y FA(\()p Fx(x)1178 2660 y Ft(i)1207 2645 y FA(\))1245 2604 y Fs(1)p Fr(\000)p Ft(a)1372 2614 y Fq(ij)1431 2645 y FA(\()p Fx(x)1524 2660 y Ft(j)1561 2645 y FA(\))28 b(=)g(0)p Fx(;)215 b Fw(for)35 b(al)5 b(l)34 b Fx(i)28 b Fu(6)p FA(=)g Fx(j;)-2729 b FA(\(4.6\))1448 2814 y Fx(x)1503 2773 y Ft(N)1559 2784 y Fq(I)1503 2839 y Ft(\013)1627 2814 y FA(=)28 b(0)p Fx(;)215 b Fw(for)35 b(al)5 b(l)34 b Fx(\013)29 b Fu(2)f FA(\010)2572 2773 y Fs(+)2572 2841 y Ft(I)2631 2814 y Fx(;)17 b(I)36 b Fu(2)28 b(X)15 b Fx(:)-3141 b FA(\(4.7\))-180 3025 y Fw(Mor)-5 b(e)g(over,)34 b(the)h(fol)5 b(lowing)34 b(elements)g(c)-5 b(onstitute)35 b(a)g(b)-5 b(asis)34 b(of)h Fm(B)p FA(\()p Fx(V)21 b FA(\))p Fw(:)519 3236 y Fx(x)574 3193 y Ft(h)615 3202 y Fl(1)574 3264 y Ft(\014)614 3273 y Fl(1)654 3236 y Fx(x)709 3193 y Ft(h)750 3202 y Fl(2)709 3264 y Ft(\014)749 3273 y Fl(2)806 3236 y Fx(:)c(:)g(:)f(x)992 3191 y Ft(h)1033 3202 y Fq(P)992 3264 y Ft(\014)1032 3275 y Fq(P)1088 3236 y Fx(;)216 b Fw(for)35 b(al)5 b(l)34 b FA(0)28 b Fu(\024)g Fx(h)1864 3251 y Ft(j)1928 3236 y Fu(\024)g Fx(N)2111 3251 y Ft(I)2174 3236 y Fu(\000)22 b FA(1)p Fx(;)52 b Fw(if)f Fx(\014)2567 3251 y Ft(j)2631 3236 y Fu(2)28 b Fx(I)8 b(;)116 b FA(1)28 b Fu(\024)g Fx(j)34 b Fu(\024)28 b Fx(P)s(:)-180 3463 y Fw(Pr)-5 b(o)g(of.)41 b FA(\(a\))h(Let)h(us)h(\014rst)f(assume)g(that)g(the)g(braiding)e(is)h (symmetric,)j(that)d(is)h Fx(\037)2973 3478 y Ft(i)3001 3463 y FA(\()p Fx(g)3086 3478 y Ft(j)3122 3463 y FA(\))i(=)g Fx(\037)3387 3478 y Ft(j)3424 3463 y FA(\()p Fx(g)3509 3478 y Ft(i)3536 3463 y FA(\))e(for)f(all)f Fx(i;)17 b(j)6 b FA(.)-180 3603 y(By)32 b([AS2)q(,)g(Lemma)e(4.2])i(w)m(e)g(can) g(assume)h(moreo)m(v)m(er)f(that)g(the)g(Cartan)g(matrix)e(\()p Fx(a)2955 3618 y Ft(ij)3016 3603 y FA(\))h(is)h(connected.)45 b(F)-8 b(rom)30 b(our)-180 3742 y(assumptions)23 b(on)g(the)h(orders)g (of)f(the)h Fx(\037)1256 3757 y Ft(i)1284 3742 y FA(\()p Fx(g)1369 3757 y Ft(j)1405 3742 y FA(\))f(w)m(e)i(then)f(conclude)g (that)f(the)g(braiding)f(has)i(the)f(form)g Fx(\037)3548 3757 y Ft(j)3584 3742 y FA(\()p Fx(g)3669 3757 y Ft(i)3697 3742 y FA(\))28 b(=)f Fx(q)3913 3706 y Ft(d)3949 3716 y Fq(i)3976 3706 y Ft(a)4013 3716 y Fq(ij)-180 3882 y FA(for)j(all)e Fx(i;)17 b(j)36 b FA(where)31 b Fx(q)j FA(is)29 b(a)h(ro)s(ot)f(of)h(unit)m(y)g(of)g(order)g Fx(N)38 b FA(=)28 b Fx(\037)1992 3897 y Ft(i)2020 3882 y FA(\()p Fx(g)2105 3897 y Ft(i)2133 3882 y FA(\).)43 b(See)31 b([AS2,)g(Lemma)d(4.3].)43 b(Hence)31 b(the)g(Theorem)-180 4021 y(follo)m(ws)g(directly)h(from)g(Theorem)g(4.2)h(and)f(Remark)g (3.6.)-80 4161 y(\(b\))c(In)g(the)g(case)h(of)e(an)h(arbitrary)f (braiding)f(w)m(e)j(kno)m(w)g(from)e(Lemma)f(4.1)i(of)f([AS2)q(])h (that)f(there)i(exists)g(a)e(\014nite)-180 4300 y(ab)s(elian)k(group)h Fy(G)60 b FA(satisfying:)43 4476 y Fu(\017)42 b FA(The)33 b(braiding)d Fx(c)i FA(of)g Fx(V)53 b FA(can)33 b(b)s(e)f(realized)f (from)g(a)h(Y)-8 b(etter-Drinfeld)30 b(mo)s(dule)h(structure)i(o)m(v)m (er)g Fy(G)60 b FA(that)32 b(w)m(e)135 4615 y(con)m(tin)m(ue)h (denoting)f(b)m(y)h Fx(V)22 b FA(,)32 b Fw(cf.)43 b FA(Remark)33 b(3.6.)43 4755 y Fu(\017)42 b FA(There)h(exists)f(a)f(co)s(cycle)i Fx(!)j FA(:)1329 4729 y Fv(b)1317 4755 y Fy(G)56 b Fu(\002)1540 4729 y Fv(b)1529 4755 y Fy(G)71 b Fu(!)43 b Fy(|)1854 4718 y Fr(\002)1949 4755 y FA(with)e(corresp)s(onding)h Fx(F)57 b Fu(2)43 b Fy(|)-8 b(G)49 b Fu(\012)29 b Fy(|)-8 b(G)63 b FA(suc)m(h)43 b(that)e(the)135 4894 y(braiding)34 b(of)j Fx(V)696 4909 y Ft(F)791 4894 y FA(is)f(symmetric.)54 b(Let)37 b Fx( )i FA(:)34 b Fm(B)p FA(\()p Fx(V)22 b FA(\))34 b Fu(!)h Fm(B)p FA(\()p Fx(V)2354 4909 y Ft(F)2412 4894 y FA(\))i(b)s(e)f(the)h(isomorphism)e(ha)m(ving)h(the)h(same)135 5034 y(meaning)31 b(as)i(in)e(\(3.9\).)43 5173 y Fu(\017)42 b FA(The)31 b(braiding)e(of)g Fx(V)881 5188 y Ft(F)970 5173 y FA(is)h(giv)m(en)h(in)e(the)i(basis)g Fx( )t FA(\()p Fx(x)1994 5188 y Ft(i)2022 5173 y FA(\))18 b Fu(\012)g Fx( )t FA(\()p Fx(x)2333 5188 y Ft(j)2370 5173 y FA(\))30 b(b)m(y)h(a)f(matrix)f(\()p Fx(b)3044 5137 y Ft(F)3044 5198 y(ij)3105 5173 y FA(\))i(suc)m(h)g(that)g Fx(b)3642 5137 y Ft(F)3642 5198 y(ii)3728 5173 y FA(=)d Fx(\037)3893 5188 y Ft(i)3921 5173 y FA(\()p Fx(g)4006 5188 y Ft(i)4034 5173 y FA(\))135 5313 y(and)k(the)h(order)g(of)f(\()p Fx(b)937 5276 y Ft(F)937 5337 y(ij)998 5313 y FA(\))g(is)h(again)e(o)s (dd)h(for)g(all)f Fx(i)h FA(and)h Fx(j)6 b FA(.)p eop %%Page: 12 12 12 11 bop -180 0 a Fn(12)846 b(NICOL)1001 -19 y(\023)991 0 y(AS)24 b(ANDR)n(USKIEWITSCH)f(AND)h(HANS-J)2419 -19 y(\177)2409 0 y(UR)n(GEN)f(SCHNEIDER)-180 203 y FA(If)k Fx($)j FA(:)e Fx(T)14 b FA(\()p Fx(V)21 b FA(\))28 b Fu(!)f Fm(B)p FA(\()p Fx(V)21 b FA(\),)28 b Fx($)836 218 y Ft(F)922 203 y FA(:)g Fx(T)14 b FA(\()p Fx(V)1143 218 y Ft(F)1201 203 y FA(\))28 b Fu(!)f Fm(B)p FA(\()p Fx(V)1577 218 y Ft(F)1636 203 y FA(\))g(denote)g(the)h(canonical)d (maps,)j(then)g(w)m(e)g(ha)m(v)m(e)g(a)f(comm)m(utativ)m(e)-180 342 y(diagram)1507 481 y Fx(T)14 b FA(\()p Fx(V)21 b FA(\))1893 424 y Ft($)1799 481 y Fu(\000)-30 b(\000)-21 b(\000)-30 b(!)80 b Fm(B)p FA(\()p Fx(V)22 b FA(\))1538 658 y Ft( )1586 551 y Fv(?)1586 611 y(?)1586 671 y(y)2217 551 y(?)2217 611 y(?)2217 671 y(y)2284 658 y Ft( )1488 856 y Fx(T)14 b FA(\()p Fx(V)1654 871 y Ft(F)1713 856 y FA(\))1869 797 y Ft($)1928 808 y Fq(F)1799 856 y Fu(\000)-30 b(\000)-21 b(\000)-30 b(!)48 b Fm(B)p FA(\()p Fx(V)2280 871 y Ft(F)2339 856 y FA(\))p Fx(:)-180 1008 y FA(Clearly)-8 b(,)40 b Fx( )t FA(\(Ker)16 b Fx($)s FA(\))38 b(=)h(Ker)16 b Fx($)993 1023 y Ft(F)1052 1008 y FA(;)42 b(if)c(\()p Fx(r)1299 1023 y Ft(j)1335 1008 y FA(\))1373 1023 y Ft(j)t Fr(2)p Ft(J)1541 1008 y FA(is)h(a)f(set)i(of)f(generators)g(of)g(the)h (ideal)d(Ker)17 b Fx($)41 b FA(with)e Fx(r)3589 1023 y Ft(j)3664 1008 y Fu(2)g Fx(T)14 b FA(\()p Fx(V)22 b FA(\))3995 953 y Ft(\021)4030 963 y Fq(j)3995 1035 y Ft(h)4036 1045 y Fq(j)-180 1147 y FA(then)40 b(b)m(y)h(Prop)s(osition)d (3.18)h(\()p Fx( )t FA(\()p Fx(r)1124 1162 y Ft(j)1161 1147 y FA(\)\))1237 1162 y Ft(j)t Fr(2)p Ft(J)1405 1147 y FA(is)g(a)h(set)g(of)f(generators)i(of)e(the)h(ideal)e(Ker)17 b Fx($)3144 1162 y Ft(F)3202 1147 y FA(.)65 b(By)41 b(the)f(symmetric) -180 1287 y(case)c(\(a\),)g(w)m(e)g(kno)m(w)h(the)f(generators)f(of)g (Ker)17 b Fx(')1618 1302 y Ft(F)1676 1287 y FA(.)52 b(Let)35 b(us)h(denote)g Fx(X)2458 1302 y Ft(i)2519 1287 y FA(:=)c Fx( )t FA(\()p Fx(x)2814 1302 y Ft(i)2843 1287 y FA(\).)51 b(Then)37 b(b)m(y)f(Lemma)e(3.15)h(and)-180 1426 y(\(3.11\))o(,)44 b(w)m(e)f(ha)m(v)m(e)f Fx( )21 b FA(\(ad)767 1441 y Ft(c)802 1426 y FA(\()p Fx(x)895 1441 y Ft(i)923 1426 y FA(\))961 1390 y Fs(1)p Fr(\000)p Ft(a)1088 1400 y Fq(ij)1148 1426 y FA(\()p Fx(x)1241 1441 y Ft(j)1278 1426 y FA(\)\))42 b(=)h Fx(u)1571 1441 y Ft(ij)1631 1426 y FA(ad)1750 1441 y Ft(c)1785 1426 y FA(\()p Fx(X)1904 1441 y Ft(i)1932 1426 y FA(\))1970 1390 y Fs(1)p Fr(\000)p Ft(a)2097 1400 y Fq(ij)2157 1426 y FA(\()p Fx(X)2276 1441 y Ft(j)2312 1426 y FA(\))e(and)h Fx( )2673 1345 y Fv(\000)2719 1426 y Fx(x)2774 1390 y Ft(N)2830 1401 y Fq(I)2774 1451 y Ft(\013)2870 1345 y Fv(\001)2958 1426 y FA(=)h Fx(u)3133 1441 y Ft(\013)3182 1426 y Fx(X)3271 1390 y Ft(N)3327 1401 y Fq(I)3263 1451 y Ft(\013)3366 1426 y FA(,)h Fx(\013)g Fu(2)f FA(\010)3722 1385 y Fs(+)3722 1453 y Ft(I)3823 1426 y FA(where)-180 1565 y Fx(u)-124 1580 y Ft(ij)-64 1565 y Fx(;)17 b(u)36 1580 y Ft(\013)124 1565 y FA(are)39 b(non-zero)g(scalars.)64 b(This)39 b(implies)e(the)j(\014rst)f(claim)e (of)i(the)g(Theorem.)64 b(The)40 b(second)h(follo)m(ws)d(in)g(a)-180 1705 y(similar)29 b(w)m(a)m(y)-8 b(.)3671 b Fj(\003)-80 1899 y FA(Let)109 1873 y Fv(b)93 1899 y Fm(B)p FA(\()p Fx(V)21 b FA(\))31 b(b)s(e)g(the)g(braided)f(Hopf)h(algebra)e(in)1701 1863 y Fs(\000)1701 1925 y(\000)1750 1899 y Fu(Y)8 b(D)33 b FA(generated)e(b)m(y)h Fx(x)2566 1914 y Fs(1)2606 1899 y Fx(;)17 b(:)g(:)g(:)f(;)h(x)2880 1914 y Ft(\022)2949 1899 y FA(with)31 b(relations)e(\(4.6\),)i(where)-180 2031 y(the)j Fx(x)44 2046 y Ft(i)73 2031 y FA('s)g(are)f(primitiv)m(e.) 44 b(Let)34 b Fu(K)q FA(\()p Fx(V)22 b FA(\))33 b(b)s(e)h(the)g (subalgebra)f(of)2154 2005 y Fv(b)2138 2031 y Fm(B)p FA(\()p Fx(V)22 b FA(\))33 b(generated)h(b)m(y)h Fx(x)3051 1995 y Ft(N)3107 2006 y Fq(I)3051 2055 y Ft(\013)3147 2031 y FA(,)e Fx(\013)d Fu(2)g FA(\010)3465 1990 y Fs(+)3465 2058 y Ft(I)3525 2031 y FA(,)j Fx(I)38 b Fu(2)29 b(X)15 b FA(;)34 b(it)e(is)-180 2164 y(a)g(Y)-8 b(etter-Drinfeld)31 b(submo)s(dule)h(of)1198 2138 y Fv(b)1182 2164 y Fm(B)p FA(\()p Fx(V)22 b FA(\).)-180 2332 y FC(Theorem)37 b(4.8.)42 b Fu(K)q FA(\()p Fx(V)22 b FA(\))35 b Fw(is)f(a)h(br)-5 b(aide)g(d)34 b(Hopf)h(sub)-5 b(algebr)g(a)34 b(in)2120 2296 y Fs(\000)2120 2357 y(\000)2168 2332 y Fu(Y)8 b(D)38 b Fw(of)2489 2306 y Fv(b)2472 2332 y Fm(B)p FA(\()p Fx(V)22 b FA(\))p Fw(.)-180 2540 y(Pr)-5 b(o)g(of.)41 b FA(\(a\).)j(As)33 b(in)f(the)i(pro)s(of)e(of)g(Theorem)h(4.5)g(w)m(e)g(\014rst)h(assume)f (that)g(the)g(braiding)e(is)h(symmetric.)44 b(If)33 b Fx(i)28 b Fu(6)p FA(=)g Fx(j)6 b FA(,)-180 2680 y(then)34 b Fx(\037)104 2695 y Ft(j)140 2680 y FA(\()p Fx(g)225 2695 y Ft(i)253 2680 y FA(\))p Fx(\037)352 2695 y Ft(i)380 2680 y FA(\()p Fx(g)465 2695 y Ft(j)502 2680 y FA(\))28 b(=)h(1)k(and)g(hence)i(the)e(corresp)s(onding)g(Serre)h(relation)d (\(4.6\))i(sa)m(ys)i(that)e Fx(x)3329 2695 y Ft(i)3357 2680 y Fx(x)3412 2695 y Ft(j)3478 2680 y FA(=)28 b Fx(x)3637 2695 y Ft(j)3674 2680 y Fx(x)3729 2695 y Ft(i)3758 2680 y FA(.)45 b(Th)m(us,)-180 2819 y(w)m(e)35 b(can)e(easily)g(reduce)i(to) e(the)h(connected)i(case.)47 b(In)34 b(suc)m(h)h(case,)g Fx(\037)2357 2834 y Ft(j)2394 2819 y FA(\()p Fx(g)2479 2834 y Ft(i)2506 2819 y FA(\))30 b(=)f Fx(q)2726 2783 y Ft(d)2762 2793 y Fq(i)2789 2783 y Ft(a)2826 2793 y Fq(ij)2918 2819 y FA(as)34 b(b)s(efore)g(and)g(the)g(Theorem)-180 2959 y(is)e(sho)m(wn)i(in)e([dCP)q(].)-80 3098 y(\(b\).)53 b(In)36 b(the)g(general)g(case,)h(w)m(e)g(c)m(hange)g(the)f(group)g(as) g(in)f(the)h(pro)s(of)f(of)h(Theorem)g(4.5.)53 b(The)36 b(isomorphism)-180 3237 y Fx( )50 b FA(:)45 b Fx(T)14 b FA(\()p Fx(V)21 b FA(\))46 b Fu(!)f Fx(T)14 b FA(\()p Fx(V)587 3252 y Ft(F)645 3237 y FA(\))43 b(resp)s(ects)i(the)e(Serre)h (relations)e(up)h(to)g(non-zero)g(scalars)g(b)m(y)h(Lemma)d(3.15.)75 b(Also,)45 b(it)-180 3377 y(maps)33 b(sub)s(coalgebras)g(stable)f (under)i(the)g(action)e(of)g(the)i(group)e(to)h(sub)s(coalgebras)g(b)m (y)h(Lemma)d(3.10)i(\(iii\).)42 b(W)-8 b(e)-180 3516 y(conclude)33 b(from)e(\(a\))i(that)f Fu(K)q FA(\()p Fx(V)22 b FA(\))32 b(is)g(a)h(sub)s(coalgebra)f(of)1974 3490 y Fv(b)1958 3516 y Fm(B)p FA(\()p Fx(V)21 b FA(\).)1767 b Fj(\003)517 3782 y FA(5.)49 b Fz(Linking)38 b(d)n(a)-7 b(tum)38 b(and)g(glueing)g(of)g(connected)f(components)-180 3956 y FA(5.1.)49 b FC(Linking)58 b(datum.)49 b FA(In)i(this)g (Section,)56 b(w)m(e)d(\014x)e(a)g(\014nite)h(ab)s(elian)d(group)i (\000,)56 b(a)51 b(\014nite)g(Cartan)g(matrix)-180 4081 y(\()p Fx(a)-91 4096 y Ft(ij)-30 4081 y FA(\))8 4096 y Fs(1)p Fr(\024)p Ft(i;j)t Fr(\024)p Ft(\022)314 4081 y FA(and)46 b Fx(g)564 4096 y Fs(1)604 4081 y Fx(;)17 b(:)g(:)g(:)e(;)i(g)869 4096 y Ft(\022)959 4081 y Fu(2)51 b FA(\000,)f Fx(\037)1275 4096 y Fs(1)1315 4081 y Fx(;)17 b(:)g(:)g(:)f(;)h(\037)1595 4096 y Ft(\022)1684 4081 y Fu(2)1805 4056 y Fv(b)1802 4081 y FA(\000)46 b(suc)m(h)i(that)e (\(1.1\))g(and)g(\(1.2\))g(hold.)84 b(W)-8 b(e)47 b(preserv)m(e)h(the) -180 4197 y(con)m(v)m(en)m(tions)34 b(and)f(h)m(yp)s(otheses)i(from)c (Section)i(4.)-180 4365 y FC(De\014nition)j(5.1.)42 b FA(W)-8 b(e)33 b(sa)m(y)g(that)g(t)m(w)m(o)g(v)m(ertices)h Fx(i)f FA(and)f Fx(j)39 b Fw(ar)-5 b(e)34 b(linkable)e FA(\(or)g(that)g Fx(i)h Fw(is)i(linkable)e(to)g Fx(j)6 b FA(\))33 b(if)70 4560 y Fx(i)28 b Fu(6\030)g Fx(j;)-483 b FA(\(5.2\))70 4729 y Fx(g)117 4744 y Ft(i)145 4729 y Fx(g)192 4744 y Ft(j)256 4729 y Fu(6)p FA(=)28 b(1)k(and)-778 b(\(5.3\))70 4898 y Fx(\037)131 4913 y Ft(i)160 4898 y Fx(\037)221 4913 y Ft(j)285 4898 y FA(=)27 b(1)p Fx(:)-644 b FA(\(5.4\))-80 5093 y(If)32 b Fx(i)h FA(is)f(link)-5 b(able)30 b(to)j Fx(j)6 b FA(,)32 b(then)h Fx(\037)1049 5108 y Ft(i)1078 5093 y FA(\()p Fx(g)1163 5108 y Ft(j)1199 5093 y FA(\))p Fx(\037)1298 5108 y Ft(j)1334 5093 y FA(\()p Fx(g)1419 5108 y Ft(i)1447 5093 y FA(\))28 b(=)f(1)33 b(b)m(y)i(\(5.2\))o(;)e(it)e(follo)m(ws)h(then)h(from)e(\(5.4\))h(that) -180 5289 y(\(5.5\))1550 b Fx(\037)1632 5304 y Ft(j)1669 5289 y FA(\()p Fx(g)1754 5304 y Ft(j)1790 5289 y FA(\))28 b(=)f Fx(\037)2020 5304 y Ft(i)2049 5289 y FA(\()p Fx(g)2134 5304 y Ft(i)2161 5289 y FA(\))2199 5248 y Fr(\000)p Fs(1)2294 5289 y Fx(:)p eop %%Page: 13 13 13 12 bop 446 0 a Fn(FINITE)33 b(QUANTUM)g(GR)n(OUPS)f(O)n(VER)g (ABELIAN)g(GR)n(OUPS)g(OF)i(PRIME)f(EXPONENT)549 b(13)-180 203 y FC(Lemma)50 b(5.6.)d Fw(Assume)e(that)g Fx(i)h Fw(and)e Fx(k)s Fw(,)j(r)-5 b(esp.)75 b Fx(j)51 b Fw(and)44 b Fx(`)p Fw(,)k(ar)-5 b(e)44 b(linkable.)74 b(Then)44 b Fx(a)3042 218 y Ft(ij)3149 203 y FA(=)j Fx(a)3323 218 y Ft(k)r(`)3394 203 y Fw(,)h Fx(a)3523 218 y Ft(j)t(i)3630 203 y FA(=)e Fx(a)3803 218 y Ft(`k)3875 203 y Fw(.)75 b(In)-180 342 y(p)-5 b(articular,)35 b(a)f(vertex)h Fx(i)g Fw(c)-5 b(an)35 b(not)f(b)-5 b(e)35 b(linkable)f(to)h(two)g(di\013er)-5 b(ent)34 b(vertic)-5 b(es)34 b Fx(j)41 b Fw(and)34 b Fx(h)p Fw(.)-180 534 y(Pr)-5 b(o)g(of.)41 b FA(If)33 b Fx(a)267 549 y Ft(i`)352 534 y Fu(6)p FA(=)27 b(0)32 b(then)i Fx(a)810 549 y Ft(ij)898 534 y FA(=)28 b Fx(a)1053 549 y Ft(j)t(i)1141 534 y FA(=)g(0)k(\(otherwise)h Fx(j)h Fu(\030)28 b Fx(`)p FA(\))k(and)h Fx(a)2329 549 y Ft(k)r(`)2429 534 y FA(=)27 b Fx(a)2583 549 y Ft(`k)2682 534 y FA(=)h(0)k (\(otherwise)h Fx(i)28 b Fu(\030)h Fx(k)s FA(\).)43 b(If)33 b Fx(a)3817 549 y Ft(j)t(k)3920 534 y Fu(6)p FA(=)27 b(0)-180 674 y(then)k Fx(a)91 689 y Ft(ij)179 674 y FA(=)d Fx(a)334 689 y Ft(j)t(i)422 674 y FA(=)f(0)j(\(otherwise)h Fx(i)c Fu(\030)i Fx(k)s FA(\))h(and)g Fx(a)1600 689 y Ft(k)r(`)1699 674 y FA(=)d Fx(a)1853 689 y Ft(`k)1953 674 y FA(=)g(0)j(\(otherwise)g Fx(j)k Fu(\030)28 b Fx(`)p FA(\).)43 b(Assume)30 b(that)g Fx(a)3550 689 y Ft(i`)3635 674 y FA(=)e(0)f(=)h Fx(a)3970 689 y Ft(j)t(k)4045 674 y FA(.)-180 813 y(Then)-145 995 y Fx(\037)-84 1010 y Ft(i)-56 995 y FA(\()p Fx(g)29 1010 y Ft(i)57 995 y FA(\))95 954 y Ft(a)132 964 y Fq(ij)219 995 y FA(=)f Fx(\037)383 1010 y Ft(i)411 995 y FA(\()p Fx(g)496 1010 y Ft(j)533 995 y FA(\))p Fx(\037)632 1010 y Ft(j)668 995 y FA(\()p Fx(g)753 1010 y Ft(i)781 995 y FA(\))h(=)f Fx(\037)1011 954 y Fr(\000)p Fs(1)1011 1023 y Ft(k)1106 995 y FA(\()p Fx(g)1191 1010 y Ft(j)1227 995 y FA(\))p Fx(\037)1326 954 y Fr(\000)p Fs(1)1326 1023 y Ft(`)1420 995 y FA(\()p Fx(g)1505 1010 y Ft(i)1533 995 y FA(\))h(=)f Fx(\037)1763 1010 y Ft(j)1800 995 y FA(\()p Fx(g)1885 1010 y Ft(k)1927 995 y FA(\))p Fx(\037)2026 1010 y Ft(i)2054 995 y FA(\()p Fx(g)2139 1010 y Ft(`)2172 995 y FA(\))h(=)f Fx(\037)2402 954 y Fr(\000)p Fs(1)2402 1023 y Ft(`)2497 995 y FA(\()p Fx(g)2582 1010 y Ft(k)2624 995 y FA(\))p Fx(\037)2723 954 y Fr(\000)p Fs(1)2723 1023 y Ft(k)2817 995 y FA(\()p Fx(g)2902 1010 y Ft(`)2935 995 y FA(\))g(=)h Fx(\037)3165 1010 y Ft(k)3208 995 y FA(\()p Fx(g)3293 1010 y Ft(k)3335 995 y FA(\))3373 954 y Fr(\000)p Ft(a)3465 966 y Fq(k)q(`)3561 995 y FA(=)g Fx(\037)3726 1010 y Ft(i)3754 995 y FA(\()p Fx(g)3839 1010 y Ft(i)3867 995 y FA(\))3905 954 y Ft(a)3942 966 y Fq(k)q(`)4010 995 y Fx(:)-180 1178 y FA(Then)i Fx(N)149 1193 y Ft(i)206 1178 y FA(divides)f Fx(a)582 1193 y Ft(ij)658 1178 y Fu(\000)15 b Fx(a)801 1193 y Ft(k)r(`)902 1178 y FA(and)29 b(analogously)-8 b(,)28 b Fx(N)1711 1193 y Ft(k)1783 1178 y FA(divides)h Fx(a)2159 1193 y Ft(ij)2235 1178 y Fu(\000)15 b Fx(a)2378 1193 y Ft(k)r(`)2450 1178 y FA(.)42 b(So)29 b(that)g Fx(a)2910 1193 y Ft(ij)2998 1178 y FA(=)f Fx(a)3153 1193 y Ft(k)r(`)3254 1178 y FA(b)m(y)i(the)f(assumptions)-180 1317 y(on)36 b(the)g(order)g(of)f Fx(N)580 1332 y Ft(i)644 1317 y FA(and)h Fx(N)915 1332 y Ft(k)958 1317 y FA(;)h(b)m(y)g(symmetry)-8 b(,)36 b Fx(a)1690 1332 y Ft(j)t(i)1784 1317 y FA(=)d Fx(a)1944 1332 y Ft(`k)2016 1317 y FA(.)53 b(Assume)36 b(that)g(a)f(v)m(ertex)j Fx(i)e FA(is)f(link)-5 b(able)34 b(to)h Fx(j)42 b FA(and)36 b Fx(h)p FA(.)-180 1457 y(Then)e(2)27 b(=)h Fx(a)306 1472 y Ft(ii)386 1457 y FA(=)f Fx(a)540 1472 y Ft(j)t(h)618 1457 y FA(,)32 b(so)h Fx(j)h FA(=)27 b Fx(h)p FA(.)2937 b Fj(\003)-80 1648 y FA(A)32 b Fw(linking)i(datum)h (of)g(\014nite)f(Cartan)h(typ)-5 b(e)35 b(for)f FA(\000)f(is)f(a)g (collection)689 1831 y Fu(D)f FA(=)c Fu(D)20 b FA(\()o(\000)p Fx(;)34 b FA(\()p Fx(a)1245 1846 y Ft(ij)1305 1831 y FA(\))1343 1846 y Fs(1)p Fr(\024)p Ft(i;j)t Fr(\024)p Ft(\022)1603 1831 y Fx(;)g FA(\()p Fx(g)1749 1846 y Ft(i)1777 1831 y FA(\))1815 1846 y Fs(1)p Fr(\024)p Ft(i)p Fr(\024)p Ft(\022)2023 1831 y Fx(;)f FA(\()p Fx(\037)2182 1846 y Ft(j)2219 1831 y FA(\))2257 1846 y Fs(1)p Fr(\024)p Ft(j)t Fr(\024)p Ft(\022)2473 1831 y Fx(;)g FA(\()p Fx(\025)2628 1846 y Ft(ij)2689 1831 y FA(\))2727 1846 y Fs(1)p Fr(\024)p Ft(i)g Fu(\010)17 b(\001)g(\001)g(\001)e(\010) 31 b Fx(<)f(Y)2733 2977 y Ft(s)2799 2962 y Fx(>)p FA(;)35 b(let)e Fx(M)3173 2977 y Ft(h)3252 2962 y FA(denote)i(the)f(order)g(of) -180 3102 y Fx(Y)-123 3117 y Ft(h)-79 3102 y FA(,)i(1)31 b Fu(\024)h Fx(h)g Fu(\024)g Fx(s)p FA(.)50 b(Let)35 b Fu(D)f FA(=)d Fu(D)20 b FA(\()o(\000)p Fx(;)34 b FA(\()p Fx(a)1233 3117 y Ft(ij)1293 3102 y FA(\))1331 3117 y Fs(1)p Fr(\024)p Ft(i;j)t Fr(\024)p Ft(\022)1591 3102 y Fx(;)g FA(\()p Fx(g)1737 3117 y Ft(i)1765 3102 y FA(\))1803 3117 y Fs(1)p Fr(\024)p Ft(i)p Fr(\024)p Ft(\022)2011 3102 y Fx(;)f FA(\()p Fx(\037)2170 3117 y Ft(j)2206 3102 y FA(\))2244 3117 y Fs(1)p Fr(\024)p Ft(j)t Fr(\024)p Ft(\022)2461 3102 y Fx(;)g FA(\()p Fx(\025)2616 3117 y Ft(ij)2677 3102 y FA(\))2715 3117 y Fs(1)p Fr(\024)p Ft(i)g Fu(\010)17 b(\001)g(\001)g(\001)e(\010)33 b Fx(<)f(Z)3617 852 y Fk(e)3613 869 y Ft(\022)3684 838 y Fx(>)p FA(,)k(where)-180 977 y(the)h(order)g(of)f Fx(Z)433 992 y Ft(i)497 977 y FA(is)g(the)h(least)g(common)e(m)m(ultiple)f(of)i(ord)17 b Fx(g)2104 992 y Ft(i)2168 977 y FA(and)37 b(ord)16 b Fx(\037)2580 992 y Ft(i)2608 977 y FA(,)38 b(1)c Fu(\024)h Fx(i)g Fu(\024)3053 951 y Fv(e)3048 977 y Fx(\022)s FA(.)56 b(Let)36 b Fx(\021)3405 992 y Ft(j)3479 977 y FA(b)s(e)g(the)h(unique) -180 1116 y(c)m(haracter)f(of)g(\007)f(suc)m(h)i(that)e Fx(\021)959 1131 y Ft(j)996 1116 y FA(\()p Fx(Z)1101 1131 y Ft(i)1129 1116 y FA(\))e(=)f Fx(\037)1369 1131 y Ft(j)1406 1116 y FA(\()p Fx(g)1491 1131 y Ft(i)1519 1116 y FA(\),)k(1)d Fu(\024)g Fx(i)g Fu(\024)1993 1090 y Fv(e)1989 1116 y Fx(\022)s FA(,)j(1)d Fu(\024)g Fx(j)39 b Fu(\024)2486 1090 y Fv(e)2481 1116 y Fx(\022)s FA(.)53 b(This)36 b(is)f(w)m(ell)g(de\014ned)i(b)s(ecause)g(ord)16 b Fx(g)4044 1131 y Ft(i)-180 1256 y FA(divides)32 b(ord)17 b Fx(Z)373 1271 y Ft(i)433 1256 y FA(for)32 b(all)f Fx(i)p FA(.)43 1427 y Fu(\017)42 b(B)33 b FA(:=)d Fm(u)p FA(\()p Fu(D)533 1442 y Fs(1)572 1427 y FA(\),)k(with)f Fu(D)971 1442 y Fs(1)1041 1427 y FA(=)c Fu(D)1243 1316 y Fv(\020)1302 1427 y FA(\000)p Fx(;)k FA(\()p Fx(a)1512 1442 y Ft(ij)1573 1427 y FA(\))1615 1441 y Fk(e)1611 1458 y Ft(\022)r()28 b FA(7)p Fw(.)-80 1303 y(\(a\))49 b(Assume)h Fx(i)56 b Fu(\030)h Fx(j)f Fw(and)49 b(let)h Fx(I)58 b Fw(b)-5 b(e)50 b(the)g(c)-5 b(onne)g(cte)g(d)49 b(c)-5 b(omp)g(onent)49 b(c)-5 b(ontaining)49 b Fx(i;)17 b(j)6 b Fw(.)90 b(If)50 b(the)g(typ)-5 b(e)51 b(of)e Fx(I)58 b Fw(is)-180 1443 y Fx(B)-106 1458 y Ft(n)-59 1443 y Fx(;)17 b(C)55 1458 y Ft(n)136 1443 y Fw(or)35 b Fx(F)325 1458 y Fs(4)365 1443 y Fw(,)48 b(assume)d(that)h Fx(N)1083 1458 y Ft(i)1157 1443 y Fw(is)g(not)f(divisible)g(by)g(5.)77 b(If)45 b(the)h(typ)-5 b(e)46 b(is)g Fx(G)2828 1458 y Fs(2)2867 1443 y Fw(,)i(assume)d(that)h Fx(N)3585 1458 y Ft(i)3659 1443 y Fw(is)g(not)f(di-)-180 1582 y(visible)34 b(by)h(5)g(or)f(7.)45 b(Then)34 b FA(\(ad)987 1597 y Ft(c)1022 1582 y Fx(x)1077 1597 y Ft(i)1105 1582 y FA(\))1143 1546 y Fs(1)p Fr(\000)p Ft(a)1270 1556 y Fq(ij)1330 1582 y Fx(x)1385 1597 y Ft(j)1450 1582 y FA(=)27 b(0)p Fw(.)-80 1722 y(\(b\))34 b(Assume)h Fx(i)28 b Fy(\034)g Fx(j)41 b Fw(and)34 b Fx(q)922 1737 y Ft(i)951 1722 y Fx(q)994 1737 y Ft(j)1058 1722 y FA(=)28 b(1)34 b Fw(or)h Fx(or)s(d)p FA(\()p Fx(q)1597 1737 y Ft(i)1625 1722 y Fx(q)1668 1737 y Ft(j)1704 1722 y FA(\))28 b(=)f Fx(or)s(d)p FA(\()p Fx(q)2099 1737 y Ft(i)2127 1722 y FA(\))p Fw(.)45 b(Then)34 b Fx(x)2549 1737 y Ft(i)2577 1722 y Fx(x)2632 1737 y Ft(j)2692 1722 y Fu(\000)22 b Fx(b)2832 1737 y Ft(ij)2893 1722 y Fx(x)2948 1737 y Ft(j)2985 1722 y Fx(x)3040 1737 y Ft(i)3096 1722 y FA(=)28 b(0)p Fw(.)-180 1968 y(Pr)-5 b(o)g(of.)41 b FA(De\014ne)34 b Fx(z)466 1983 y Fs(1)534 1968 y FA(:=)29 b Fx(x)721 1983 y Ft(i)749 1968 y FA(,)k Fx(z)854 1983 y Fs(2)923 1968 y FA(:=)28 b(\(ad)1211 1983 y Ft(c)1246 1968 y Fx(x)1301 1983 y Ft(i)1330 1968 y FA(\))1368 1932 y Fs(1)p Fr(\000)p Ft(a)1495 1942 y Fq(ij)1554 1968 y Fx(x)1609 1983 y Ft(j)1646 1968 y FA(.)45 b(In)34 b(b)s(oth)f(cases)h(w)m(e)g(ha)m(v)m(e)h(to)d(sho)m(w)j Fx(z)3094 1983 y Fs(2)3162 1968 y FA(=)28 b(0.)45 b(W)-8 b(e)34 b(assume)f(that)-180 2107 y Fx(z)-135 2122 y Fs(2)-66 2107 y FA(is)c(not)g(0.)42 b(Let)29 b Fx(g)535 2122 y Ft(i)591 2107 y Fu(2)f FA(\000)p Fx(;)17 b(\037)851 2122 y Ft(i)906 2107 y Fu(2)1003 2082 y Fv(b)1001 2107 y FA(\000,)29 b(1)f Fu(\024)g Fx(i;)17 b(j)34 b Fu(\024)28 b Fx(\022)s FA(,)i(with)f Fx(b)1921 2122 y Ft(ij)2009 2107 y FA(=)f Fx(\037)2174 2122 y Ft(j)2211 2107 y FA(\()p Fx(g)2296 2122 y Ft(i)2323 2107 y FA(\))i(for)e(all)f Fx(i;)17 b(j)6 b FA(.)43 b(Then)30 b(action)e(and)i(coaction)e(on)-180 2247 y Fx(z)-135 2262 y Fs(1)-95 2247 y Fx(;)17 b(z)-6 2262 y Fs(2)65 2247 y FA(are)33 b(giv)m(en)f(b)m(y)i Fx(\016)t FA(\()p Fx(z)748 2262 y Fs(1)787 2247 y FA(\))28 b(=)f Fx(g)1003 2262 y Ft(i)1053 2247 y Fu(\012)22 b Fx(z)1197 2262 y Fs(1)1237 2247 y FA(,)32 b Fx(\016)t FA(\()p Fx(z)1426 2262 y Fs(2)1466 2247 y FA(\))27 b(=)h Fx(g)1686 2193 y Fs(1)p Fr(\000)p Ft(a)1813 2203 y Fq(ij)1682 2272 y Ft(i)1872 2247 y Fx(g)1919 2262 y Ft(j)1977 2247 y Fu(\012)22 b Fx(z)2121 2262 y Fs(2)2193 2247 y FA(and)32 b Fx(h)22 b Fu(\001)f Fx(z)2554 2262 y Fs(1)2622 2247 y FA(=)27 b Fx(\037)2786 2262 y Ft(i)2815 2247 y FA(\()p Fx(h)p FA(\))p Fx(z)2992 2262 y Fs(1)3032 2247 y FA(,)32 b Fx(h)22 b Fu(\001)f Fx(z)3263 2262 y Fs(2)3330 2247 y FA(=)28 b(\()p Fx(\037)3533 2193 y Fs(1)p Fr(\000)p Ft(a)3660 2203 y Fq(ij)3533 2272 y Ft(i)3720 2247 y Fx(\037)3781 2262 y Ft(j)3817 2247 y FA(\)\()p Fx(h)p FA(\))p Fx(z)4032 2262 y Fs(2)-180 2386 y FA(for)35 b(all)e Fx(h)g Fu(2)g FA(\000.)52 b(The)36 b(elemen)m(ts)g Fx(z)1088 2401 y Fs(1)1128 2386 y FA(,)g Fx(z)1236 2401 y Fs(2)1311 2386 y FA(are)g(linearly)d(indep)s(enden)m(t)k(since)f(they)g(are)g (non-zero)f(and)h(of)f(di\013eren)m(t)-180 2526 y(degree.)70 b(The)42 b(braiding)e(\()p Fx(B)902 2541 y Ft(k)r(l)966 2526 y FA(\))1004 2541 y Fs(1)p Fr(\024)p Ft(k)r(;l)q Fr(\024)p Fs(2)1310 2526 y FA(of)h(the)g(2-dimensional)d(Y)-8 b(etter-Drinfeld)39 b(mo)s(dule)h(with)h(basis)g Fx(z)3797 2541 y Fs(1)3837 2526 y Fx(;)17 b(z)3926 2541 y Fs(2)4006 2526 y FA(is)-180 2665 y(giv)m(en)33 b(b)m(y)295 2892 y Fx(B)369 2907 y Fs(11)471 2892 y FA(=)28 b Fx(\037)636 2907 y Ft(i)664 2892 y FA(\()p Fx(g)749 2907 y Ft(i)777 2892 y FA(\))g(=)f Fx(q)989 2907 y Ft(i)1018 2892 y Fx(;)984 b(B)2103 2907 y Fs(12)2205 2892 y FA(=)28 b(\()p Fx(\037)2408 2837 y Fs(1)p Fr(\000)p Ft(a)2535 2847 y Fq(ij)2408 2917 y Ft(i)2594 2892 y Fx(\037)2655 2907 y Ft(j)2692 2892 y FA(\)\()p Fx(g)2815 2907 y Ft(i)2843 2892 y FA(\))f(=)h Fx(q)3055 2907 y Ft(i)3083 2892 y Fx(b)3124 2850 y Fr(\000)p Fs(1)3124 2917 y Ft(j)t(i)3219 2892 y Fx(;)295 3065 y(B)369 3080 y Fs(21)471 3065 y FA(=)g Fx(\037)636 3080 y Ft(i)664 3065 y FA(\()p Fx(g)753 3011 y Fs(1)p Fr(\000)p Ft(a)880 3021 y Fq(ij)749 3090 y Ft(i)939 3065 y Fx(g)986 3080 y Ft(j)1022 3065 y FA(\))g(=)f Fx(q)1238 3011 y Fs(1)p Fr(\000)p Ft(a)1365 3021 y Fq(ij)1234 3090 y Ft(i)1425 3065 y Fx(b)1466 3080 y Ft(j)t(i)1527 3065 y Fx(;)475 b(B)2103 3080 y Fs(22)2205 3065 y FA(=)28 b(\()p Fx(\037)2408 3011 y Fs(1)p Fr(\000)p Ft(a)2535 3021 y Fq(ij)2408 3090 y Ft(i)2594 3065 y Fx(\037)2655 3080 y Ft(j)2692 3065 y FA(\)\()p Fx(g)2819 3011 y Fs(1)p Fr(\000)p Ft(a)2946 3021 y Fq(ij)2815 3090 y Ft(i)3005 3065 y Fx(g)3052 3080 y Ft(j)3088 3065 y FA(\))f(=)h Fx(q)3304 3011 y Fs(1)p Fr(\000)p Ft(a)3431 3021 y Fq(ij)3300 3090 y Ft(i)3490 3065 y Fx(q)3533 3080 y Ft(j)3570 3065 y Fx(:)-180 3296 y FA(Then)41 b Fx(B)156 3311 y Fs(12)231 3296 y Fx(B)305 3311 y Fs(21)419 3296 y FA(=)f Fx(q)582 3242 y Fs(2)p Fr(\000)p Ft(a)709 3252 y Fq(ij)578 3322 y Ft(i)768 3296 y FA(.)65 b(W)-8 b(e)40 b(claim)d(that)i(\()p Fx(B)1632 3311 y Ft(k)r(l)1697 3296 y FA(\))g(is)g(of)g(Cartan)h(t)m(yp)s(e)h (and)e(satis\014es)i(the)f(relativ)m(e)f(primeness)-180 3435 y(condition,)31 b(that)i(is)f(there)h(are)g(in)m(tegers)f Fx(A)1430 3450 y Fs(12)1505 3435 y Fx(;)17 b(A)1622 3450 y Fs(21)1729 3435 y FA(suc)m(h)34 b(that)539 3662 y Fx(q)586 3608 y Fs(2)p Fr(\000)p Ft(a)713 3618 y Fq(ij)582 3687 y Ft(i)800 3662 y FA(=)28 b Fx(q)951 3618 y Ft(A)1004 3627 y Fl(12)947 3687 y Ft(i)1105 3662 y FA(,and)33 b Fx(A)1395 3677 y Fs(12)1502 3662 y FA(is)f(relativ)m(ely)g(prime)f(to)h Fx(N)2499 3677 y Ft(i)2528 3662 y Fx(;)-2735 b FA(\(7.3\))539 3835 y Fx(q)586 3781 y Fs(2)p Fr(\000)p Ft(a)713 3791 y Fq(ij)582 3861 y Ft(i)800 3835 y FA(=)28 b(\()p Fx(q)989 3781 y Fs(1)p Fr(\000)p Ft(a)1116 3791 y Fq(ij)985 3861 y Ft(i)1175 3835 y Fx(q)1218 3850 y Ft(j)1255 3835 y FA(\))1293 3794 y Ft(A)1346 3803 y Fl(21)1447 3835 y FA(,)33 b(and)f Fx(A)1769 3850 y Fs(21)1877 3835 y FA(is)g(relativ)m (ely)f(prime)h(to)g(ord\()p Fx(q)3022 3781 y Fs(1)p Fr(\000)p Ft(a)3149 3791 y Fq(ij)3018 3861 y Ft(i)3208 3835 y Fx(q)3251 3850 y Ft(j)3288 3835 y FA(\))p Fx(:)-3533 b FA(\(7.4\))-180 4062 y(In)33 b(b)s(oth)f(cases)i(2)22 b Fu(\000)h Fx(a)639 4077 y Ft(ij)732 4062 y FA(is)32 b(relativ)m(ely)g(prime)f(to)h Fx(N)1729 4077 y Ft(i)1757 4062 y FA(,)h(b)s(ecause)h(of)e(the)h(h)m (yp)s(othesis)h(on)e Fx(N)3151 4077 y Ft(i)3180 4062 y FA(.)43 b(This)33 b(sho)m(ws)h(\(7.3\))o(.)-80 4201 y(W)-8 b(e)26 b(no)m(w)g(pro)m(v)m(e)h(\(7.4\))e(in)g(case)i(\(a\).)41 b(Then)27 b Fx(N)1585 4216 y Ft(I)1652 4201 y FA(=)h Fx(N)1834 4216 y Ft(i)1890 4201 y FA(=)f Fx(N)2071 4216 y Ft(j)2108 4201 y FA(,)g(and)f(it)f(su\016ces)j(to)d(\014nd)h(an)g(in) m(teger)g Fx(A)3579 4216 y Fs(21)3679 4201 y FA(relativ)m(ely)-180 4341 y(prime)31 b(to)i Fx(N)294 4356 y Ft(i)354 4341 y FA(with)g Fx(q)624 4286 y Fs(2)p Fr(\000)p Ft(a)751 4296 y Fq(ij)620 4366 y Ft(i)838 4341 y FA(=)27 b(\()p Fx(q)1026 4286 y Fs(1)p Fr(\000)p Ft(a)1153 4296 y Fq(ij)1022 4366 y Ft(i)1212 4341 y Fx(q)1255 4356 y Ft(j)1292 4341 y FA(\))1330 4304 y Ft(A)1383 4313 y Fl(21)1452 4341 y FA(.)-80 4480 y(First)45 b(assume)j(that)e Fx(a)799 4495 y Ft(ij)912 4480 y Fu(6)p FA(=)52 b(0.)85 b(Since)47 b Fx(a)1521 4495 y Ft(j)t(i)1634 4480 y Fu(6)p FA(=)52 b(0)46 b(is)h(relativ)m(ely)e(prime)h(to)g Fx(N)2911 4495 y Ft(i)2940 4480 y FA(,)k(it)c(is)g(enough)h(to)g(consider)-180 4619 y(the)g Fx(a)53 4634 y Ft(j)t(i)113 4619 y FA(-th)f(p)s(o)m(w)m (er)h(of)53 b(\(7.4\))o(.)84 b(Since)47 b Fx(q)1341 4565 y Ft(a)1378 4575 y Fq(ij)1337 4645 y Ft(i)1488 4619 y FA(=)j Fx(q)1661 4565 y Ft(a)1698 4575 y Fq(j)s(i)1657 4645 y Ft(j)1804 4619 y FA(b)m(y)d(the)f(Cartan)g(condition)f(for)h(\() p Fx(b)3162 4634 y Ft(ij)3223 4619 y FA(\),)j(w)m(e)e(ha)m(v)m(e)h(to)d (solv)m(e)-180 4759 y(\(2)18 b Fu(\000)g Fx(a)71 4774 y Ft(ij)132 4759 y FA(\))p Fx(a)221 4774 y Ft(j)t(i)309 4759 y Fu(\021)29 b FA(\(\(1)17 b Fu(\000)i Fx(a)704 4774 y Ft(ij)764 4759 y FA(\))p Fx(a)853 4774 y Ft(j)t(i)932 4759 y FA(+)f Fx(a)1077 4774 y Ft(ij)1138 4759 y FA(\))p Fx(A)1249 4774 y Fs(21)1390 4759 y FA(mo)s(d)32 b Fx(N)1688 4774 y Ft(i)1717 4759 y FA(.)42 b(Since)31 b(\()p Fx(a)2128 4774 y Ft(ij)2189 4759 y FA(\))f(is)g(of)g(\014nite)h(Cartan)f(t)m(yp)s (e,)i(the)f(p)s(ossible)f(v)-5 b(alues)-180 4898 y(of)30 b(\(2)19 b Fu(\000)g Fx(a)182 4913 y Ft(ij)243 4898 y FA(\))p Fx(a)332 4913 y Ft(j)t(i)424 4898 y FA(are)31 b(-3,)f(-4,)h(-5,)g(-6,)f(-9)g(\(-4,)h(-6)f(resp.)44 b(-5,)31 b(-9)f(only)g(o)s(ccur)i(if)d(the)j(t)m(yp)s(e)g(is)e Fx(B)3157 4913 y Ft(n)3204 4898 y Fx(;)17 b(C)3318 4913 y Ft(n)3396 4898 y FA(or)30 b Fx(F)3576 4913 y Fs(4)3647 4898 y FA(resp.)44 b Fx(G)3968 4913 y Fs(2)4007 4898 y FA(\);)-180 5038 y(the)35 b(p)s(ossible)f(v)-5 b(alues)35 b(of)f(\(\(1)23 b Fu(\000)h Fx(a)1065 5053 y Ft(ij)1126 5038 y FA(\))p Fx(a)1215 5053 y Ft(j)t(i)1299 5038 y FA(+)f Fx(a)1449 5053 y Ft(ij)1510 5038 y FA(\))35 b(are)f(-3,)h(-5,)f (-7,)h(\(-5,)f(resp.)51 b(-7)34 b(only)g(o)s(ccur)h(if)f(the)h(t)m(yp)s (e)g(is)f Fx(B)3864 5053 y Ft(n)3912 5038 y Fx(;)17 b(C)4026 5053 y Ft(n)-180 5177 y FA(or)31 b Fx(F)1 5192 y Fs(4)72 5177 y FA(resp.)44 b Fx(G)393 5192 y Fs(2)432 5177 y FA(\).)f(Hence)33 b(\(2)19 b Fu(\000)h Fx(a)1083 5192 y Ft(ij)1144 5177 y FA(\))p Fx(a)1233 5192 y Ft(j)t(i)1325 5177 y FA(and)31 b(\(\(1)19 b Fu(\000)h Fx(a)1805 5192 y Ft(ij)1866 5177 y FA(\))p Fx(a)1955 5192 y Ft(j)t(i)2035 5177 y FA(+)f Fx(a)2181 5192 y Ft(ij)2242 5177 y FA(\))31 b(are)h(relativ)m(ely)e(prime)g(to)h Fx(N)3368 5192 y Ft(i)3427 5177 y FA(b)m(y)h(assumption,)-180 5317 y(and)h(the)g(claim)d (follo)m(ws.)42 b(\(Note)33 b(that)f Fx(a)22 b FA(+)g Fx(b)33 b FA(is)f(nev)m(er)j(0\).)p eop %%Page: 23 23 23 22 bop 446 0 a Fn(FINITE)33 b(QUANTUM)g(GR)n(OUPS)f(O)n(VER)g (ABELIAN)g(GR)n(OUPS)g(OF)i(PRIME)f(EXPONENT)549 b(23)-80 203 y FA(If)38 b Fx(a)74 218 y Ft(ij)172 203 y FA(=)g(0,)i(then)f(b)m (y)g(connectedness)j(there)d(is)f(a)g(sequence)j Fx(i)d FA(=)g Fx(i)2481 218 y Fs(1)2520 203 y Fx(;)17 b(i)2597 218 y Fs(2)2637 203 y Fx(;)g(:)g(:)g(:)f(;)h(i)2889 218 y Ft(t)2956 203 y FA(=)37 b Fx(j)45 b FA(of)38 b(elemen)m(ts)h(in)e Fx(I)46 b FA(suc)m(h)-180 342 y(that)32 b Fx(a)82 357 y Ft(i)106 369 y Fq(`)137 357 y Ft(i)161 369 y Fq(`)p Fl(+1)299 342 y Fu(6)p FA(=)c(0)p Fx(;)17 b FA(2)32 b(for)g(all)e Fx(`)p FA(,)j(1)27 b Fu(\024)h Fx(`)g(<)g(t)p FA(.)43 b(Then)34 b(as)f(in)f(the)h(pro)s(of)e(of)h(Lemma)g(6.1)g(\(b\),)562 542 y Fx(q)609 501 y Ft(a)605 566 y(i)678 542 y FA(=)c Fx(q)829 501 y Ft(b)825 566 y(j)863 542 y Fx(;)49 b FA(where)34 b Fx(a)28 b FA(=)f Fx(a)1454 557 y Ft(i)1478 566 y Fl(1)1513 557 y Ft(i)1537 566 y Fl(2)1576 542 y Fx(a)1627 557 y Ft(i)1651 566 y Fl(2)1686 557 y Ft(i)1710 566 y Fl(3)1765 542 y Fx(:)17 b(:)g(:)f(a)1947 557 y Ft(i)1971 566 y Fq(t)p Fh(\000)p Fl(1)2078 557 y Ft(i)2102 565 y Fq(t)2134 542 y Fx(;)49 b FA(and)33 b Fx(b)28 b FA(=)f Fx(a)2623 557 y Ft(i)2647 566 y Fl(2)2682 557 y Ft(i)2706 566 y Fl(1)2745 542 y Fx(a)2796 557 y Ft(i)2820 566 y Fl(2)2855 557 y Ft(i)2879 566 y Fl(3)2934 542 y Fx(:)17 b(:)g(:)f(a)3116 557 y Ft(i)3140 566 y Fq(t)p Fh(\000)p Fl(1)3247 557 y Ft(i)3271 565 y Fq(t)3303 542 y Fx(:)-180 741 y FA(Since)44 b(the)g(p)s(ossible)f(v)-5 b(alues)44 b(of)g Fx(a;)17 b(b)44 b FA(are)g(1,)i(2,)g(-1,)g(-2,)g(the)f Fx(b)p FA(-th)f(p)s(o)m(w)m(er)g(of)51 b(\(7.4\))43 b(leads)h(to)f(the)h (congruence)-180 880 y(2)p Fx(b)28 b Fu(\021)g FA(\()p Fx(b)23 b FA(+)f Fx(a)p FA(\))p Fx(A)405 895 y Fs(21)546 880 y FA(mo)s(d)32 b Fx(N)844 895 y Ft(i)873 880 y FA(,)g(and)h(the)g (claim)d(again)h(follo)m(ws.)-80 1020 y(Assume)45 b(case)h(\(b\),)i(in) c(particular)f Fx(a)1356 1035 y Ft(ij)1465 1020 y FA(=)48 b(0.)80 b(If)45 b Fx(q)1898 1035 y Ft(i)1926 1020 y Fx(q)1969 1035 y Ft(j)2054 1020 y FA(=)j(1,)g(w)m(e)e(get)f(a)f(con)m(tradiction) g(since)h(the)g(algebra)-180 1159 y(generated)33 b(b)m(y)h Fx(z)445 1174 y Fs(1)485 1159 y Fx(;)17 b(z)574 1174 y Fs(2)646 1159 y FA(is)32 b(\014nite-dimensional,)e(hence)k Fx(B)1903 1174 y Fs(22)2005 1159 y Fu(6)p FA(=)28 b(1)k(b)m(y)i([AS1,)f (Lemma)e(3.1].)44 b(If)32 b(ord\()p Fx(q)3479 1174 y Ft(i)3507 1159 y Fx(q)3550 1174 y Ft(j)3587 1159 y FA(\))c(=)g(ord\()p Fx(q)3979 1174 y Ft(i)4007 1159 y FA(\),)-180 1299 y(\(7.4\))k(is)g (solv)-5 b(able)31 b(since)i Fx(N)834 1314 y Ft(i)895 1299 y FA(is)f(o)s(dd.)-80 1438 y(Th)m(us)39 b(w)m(e)h(ha)m(v)m(e)f (sho)m(wn)h(that)e(\()p Fx(B)1183 1453 y Ft(k)r(l)1248 1438 y FA(\))g(is)g(of)f(Cartan)i(t)m(yp)s(e)g(and)f(satis\014es)h(the) g(relativ)m(e)f(primeness)g(condition.)-180 1578 y(Hence)33 b(\()p Fx(B)221 1593 y Ft(k)r(l)286 1578 y FA(\))e(is)h(of)f(\014nite)h (Cartan)f(t)m(yp)s(e)i(b)m(y)g(Lemma)d(7.1.)43 b(In)32 b(b)s(oth)g(cases)h Fx(A)2705 1593 y Fs(12)2808 1578 y FA(=)27 b(2)21 b Fu(\000)g Fx(a)3130 1593 y Ft(ij)3211 1578 y Fu(\000)g Fx(N)3387 1593 y Ft(i)3447 1578 y FA(is)32 b(a)f(solution)f(of)-180 1717 y(\(7.3\),)h(and)f Fu(\000)p Fx(N)421 1732 y Ft(i)478 1717 y Fx(<)d(A)654 1732 y Fs(12)757 1717 y Fu(\024)h FA(0.)43 b(Hence)32 b(the)f(p)s(ossible)f(v)-5 b(alues)31 b(of)f Fx(A)2271 1732 y Fs(12)2376 1717 y FA(are)h(0,)g(-1,)f(-2,)h(-3,)f(and)h(w)m(e)h(see)f(that)g Fx(N)3835 1732 y Ft(i)3891 1717 y Fu(\024)d FA(8.)-180 1857 y(This)33 b(con)m(tradicts)g(our)f(assumption,)g(and)h(w)m(e)g(ha) m(v)m(e)h(sho)m(wn)g(the)f(Serre)g(relation)e Fx(z)2935 1872 y Fs(2)3003 1857 y FA(=)c(0.)812 b Fj(\003)-180 2070 y FC(Lemma)53 b(7.5.)c Fw(L)-5 b(et)48 b Fx(S)57 b FA(=)51 b Fu(\010)958 2085 y Ft(n)p Fr(\025)p Fs(0)1096 2070 y Fx(S)6 b FA(\()p Fx(n)p FA(\))47 b Fw(b)-5 b(e)48 b(a)f(\014nite-dimensional)e(gr)-5 b(ade)g(d)47 b(Hopf)h(algebr)-5 b(a)47 b(in)3423 2033 y Fs(\000)3423 2095 y(\000)3471 2070 y Fu(Y)8 b(D)50 b Fw(such)e(that)-180 2209 y Fx(S)6 b FA(\(0\))52 b(=)g Fy(|)-9 b FA(1)p Fw(.)79 b(Assume)48 b(that)g Fx(V)74 b FA(=)52 b Fx(S)6 b FA(\(1\))48 b Fw(is)g(of)g (Cartan)g(typ)-5 b(e)48 b(with)g(b)-5 b(asis)48 b FA(\()p Fx(x)2872 2224 y Ft(i)2900 2209 y FA(\))2938 2224 y Fs(1)p Fr(\024)p Ft(i;j)t Fr(\024)p Ft(\022)3246 2209 y Fw(as)g(describ)-5 b(e)g(d)47 b(in)h(the)-180 2348 y(b)-5 b(e)g(ginning)33 b(of)i(Se)-5 b(ction)34 b(4.)45 b(Assume)35 b(the)f(Serr)-5 b(e)35 b(r)-5 b(elations)796 2548 y FA(\(ad)953 2563 y Ft(c)988 2548 y Fx(x)1043 2563 y Ft(i)1072 2548 y FA(\))1110 2507 y Fs(1)p Fr(\000)p Ft(a)1237 2517 y Fq(ij)1296 2548 y Fx(x)1351 2563 y Ft(j)1416 2548 y FA(=)27 b(0)35 b Fw(for)g(al)5 b(l)34 b FA(1)28 b Fu(\024)g Fx(i;)17 b(j)33 b Fu(\024)c Fx(\022)s(;)17 b(i)27 b Fu(6)p FA(=)h Fx(j)41 b Fw(and)34 b Fx(i)28 b Fu(\030)g Fx(j:)-180 2747 y Fw(Then)34 b(the)h(r)-5 b(o)g(ot)35 b(ve)-5 b(ctor)35 b(r)-5 b(elations)1190 2946 y Fx(x)1245 2905 y Ft(N)1301 2916 y Fq(I)1245 2971 y Ft(\013)1369 2946 y FA(=)27 b(0)p Fx(;)116 b Fw(for)35 b(al)5 b(l)34 b Fx(\013)29 b Fu(2)f FA(\010)2214 2905 y Fs(+)2214 2973 y Ft(I)2273 2946 y Fx(;)117 b(I)35 b Fu(2)28 b(X)15 b Fx(;)-180 3146 y Fw(hold)34 b(in)h Fx(S)6 b Fw(.)-180 3358 y(Pr)-5 b(o)g(of.)41 b FA(W)-8 b(e)38 b(\014x)g(a)g(connected)h(comp)s(onen)m(t)f Fx(I)44 b Fu(2)36 b(X)15 b FA(.)58 b(Let)38 b Fx(V)2082 3373 y Ft(I)2159 3358 y FA(b)s(e)g(the)g(Y)-8 b(etter-Drinfeld)36 b(submo)s(dule)h(of)g Fx(V)59 b FA(with)-180 3498 y(basis)29 b Fx(x)111 3513 y Ft(i)140 3498 y Fx(;)17 b(i)27 b Fu(2)i Fx(I)8 b FA(,)29 b(and)649 3472 y Fv(b)633 3498 y Fm(B)p FA(\()p Fx(V)816 3513 y Ft(I)855 3498 y FA(\))h(the)f(quotien)m(t)h(of) e Fx(T)14 b FA(\()p Fx(V)1742 3513 y Ft(I)1782 3498 y FA(\))29 b(mo)s(dulo)e(the)j(Serre)g(relations)d(of)i(all)e(elemen)m (ts)j Fx(x)3689 3513 y Ft(i)3717 3498 y Fx(;)17 b(x)3816 3513 y Ft(j)3882 3498 y FA(with)-180 3637 y Fx(i)39 b Fu(6)p FA(=)f Fx(j)45 b FA(in)38 b(I.)h(Let)g Fx(N)571 3652 y Ft(I)649 3637 y FA(=)g Fx(N)10 b FA(.)62 b(The)40 b(map)e(\011)h(:)f Fx(T)14 b FA(\()p Fx(V)1717 3652 y Ft(I)1756 3637 y FA(\))39 b Fu(\032)g Fx(T)14 b FA(\()p Fx(V)21 b FA(\))38 b Fu(!)g Fx(S)45 b FA(factorizes)38 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4159 y Fs(\000)144 4220 y(\000)192 4195 y Fu(Y)8 b(D)35 b FA(with)e(trivial)d(coradical.)43 b(W)-8 b(e)33 b(ha)m(v)m(e)h(to)f(sho)m(w)h(the)f(ro)s(ot)f(v)m(ector)i (relation)d Fx(x)3181 4159 y Ft(N)3237 4170 y Fq(I)3181 4220 y Ft(\013)3305 4195 y FA(=)d(0,)k Fx(\013)d Fu(2)g FA(\010)3773 4154 y Fs(+)3773 4222 y Ft(I)3865 4195 y FA(in)j Fx(S)6 b FA(,)-180 4335 y(or)32 b(equiv)-5 b(alen)m(tly)32 b(that)h Fx(K)39 b FA(is)32 b(one-dimensional,)e(that)j(is)f Fu(P)8 b FA(\()p Fx(K)f FA(\))28 b(=)f(0.)-80 4474 y(Assume)39 b Fu(P)8 b FA(\()p Fx(K)f FA(\))38 b Fu(6)p FA(=)f(0.)61 b(Since)39 b Fu(P)8 b FA(\()p Fx(K)f FA(\))38 b(is)g(in)1584 4438 y Fs(\000)1584 4499 y(\000)1632 4474 y Fu(Y)8 b(D)s FA(,)40 b(there)f(are)f Fx(g)j Fu(2)d FA(\000)p Fx(;)17 b(\037)37 b Fu(2)2779 4449 y Fv(b)2776 4474 y FA(\000)i(with)f Fu(P)8 b FA(\()p Fx(K)f FA(\))3347 4438 y Ft(\037)3347 4499 y(g)3433 4474 y Fu(6)p FA(=)37 b(0.)61 b(By)39 b([AS1,)-180 4617 y(Lemma)c(3.1],)h(w)m(e)h(conclude)g Fx(\037)p FA(\()p 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Ft(b)3376 5228 y Fq(\014)3376 5282 y(i)3346 5342 y Ft(i)3422 5317 y FA(.)66 b(Hence)41 b(for)f(all)p eop %%Page: 24 24 24 23 bop -180 0 a Fn(24)846 b(NICOL)1001 -19 y(\023)991 0 y(AS)24 b(ANDR)n(USKIEWITSCH)f(AND)h(HANS-J)2419 -19 y(\177)2409 0 y(UR)n(GEN)f(SCHNEIDER)-180 203 y Fx(\014)-125 218 y Fs(1)-85 203 y Fx(;)17 b(:)g(:)g(:)e(;)i(\014)188 218 y Ft(m)282 203 y Fu(2)28 b FA(\010)446 162 y Fs(+)446 230 y Ft(I)506 203 y FA(,)528 394 y(\011\()p Fx(x)697 353 y Ft(N)697 418 y(\014)737 427 y Fl(1)776 394 y FA(\))17 b Fu(\001)g(\001)g(\001)d FA(\011\()p Fx(x)1132 353 y Ft(N)1132 418 y(\014)1172 426 y Fq(m)1235 394 y FA(\))28 b Fu(2)g Fx(K)1485 353 y Ft(\037)1478 418 y(g)1533 394 y Fx(;)114 b FA(where)99 b Fx(\037)28 b FA(=)f Fx(\037)2274 353 y Ft(N)2274 418 y(\014)2314 427 y Fl(1)2369 394 y Fu(\001)17 b(\001)g(\001)e Fx(\037)2563 353 y Ft(N)2563 418 y(\014)2603 426 y Fq(m)2666 394 y Fx(;)38 b(g)31 b FA(=)d Fx(g)2964 353 y Ft(N)2960 418 y(\014)3000 427 y Fl(1)3055 394 y Fu(\001)17 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Fq(i)488 5038 y FA(with)h Fx(\037)776 5053 y Ft(i)839 5038 y FA(=)e Fx(\021)1002 4997 y Fr(\000)p Fs(1)998 5063 y Ft(i)1096 5038 y Fx(;)17 b(g)1187 5053 y Ft(i)1249 5038 y FA(=)35 b Fx(h)1416 4997 y Fr(\000)p Fs(1)1416 5063 y Ft(i)1547 5038 y FA(and)i Fx(b)1782 5053 y Ft(ij)1878 5038 y FA(=)d Fx(\037)2049 5053 y Ft(j)2086 5038 y FA(\()p Fx(g)2171 5053 y Ft(i)2199 5038 y FA(\))h(=)f Fx(\021)2430 5053 y Ft(j)2467 5038 y FA(\()p Fx(h)2561 5053 y Ft(i)2589 5038 y FA(\))j(for)f(all)f(1)g Fu(\024)g Fx(i;)17 b(j)41 b Fu(\024)35 b Fx(\022)s FA(.)56 b(Th)m(us)39 b Fx(V)58 b FA(is)36 b(a)-180 5177 y(Y)-8 b(etter-Drinfeld)26 b(mo)s(dule)g(o)m(v)m(er)i(\000)g(with)f(the)h(same)f(braiding)f(as)i Fx(R)q FA(\(1\).)41 b(By)28 b([AS2,)h(Lemma)d(5.5],)j Fx(R)f FA(is)f(generated)-180 5317 y(b)m(y)j Fx(R)q FA(\(1\))f(if)f (and)h(only)g(if)f Fx(S)6 b FA(\(1\))27 b(=)g Fu(P)8 b FA(\()p Fx(S)e FA(\).)43 b(Hence)30 b(b)m(y)g(dualit)m(y)-8 b(,)29 b Fx(S)35 b FA(is)28 b(generated)i(b)m(y)g Fx(S)6 b FA(\(1\),)30 b(since)f Fx(R)q FA(\(1\))f(=)f Fu(P)8 b FA(\()p Fx(R)q FA(\).)43 b(It)p eop %%Page: 25 25 25 24 bop 446 0 a Fn(FINITE)33 b(QUANTUM)g(GR)n(OUPS)f(O)n(VER)g (ABELIAN)g(GR)n(OUPS)g(OF)i(PRIME)f(EXPONENT)549 b(25)-180 203 y FA(is)33 b(easy)i(to)e(see)i(that)e Fx(V)51 b FA(=)29 b Fx(S)6 b FA(\(1\))29 b Fu(\032)h(P)8 b FA(\()p Fx(S)e FA(\).)47 b(Hence)35 b(there)f(are)g(canonical)e(surjections)i(of)f (graded)h(braided)f(Hopf)-180 342 y(algebras)1511 485 y Fx(T)14 b FA(\()p Fx(V)21 b FA(\))28 b Fu(!)f Fx(S)33 b Fu(!)28 b Fm(B)p FA(\()p Fx(V)21 b FA(\))p Fx(:)-180 647 y FA(Here)31 b Fx(T)14 b FA(\()p Fx(V)21 b FA(\))30 b(is)g(the)g(tensor)h(algebra,)f(the)g(elemen)m(ts)h Fx(x)1839 662 y Ft(i)1898 647 y FA(are)f(primitiv)m(e)e(and)i(of)g (degree)h(one,)g(and)f(b)s(oth)g(maps)g(are)-180 787 y(the)38 b(iden)m(tit)m(y)g(on)g Fx(V)21 b FA(.)59 b(The)39 b(k)m(ernel)f Fx(I)46 b FA(of)37 b(the)i(\014rst)f(map)f(is)g(a)h (homogeneous)g(ideal)e(generated)j(b)m(y)f(elemen)m(ts)g(of)-180 926 y(degree)d Fu(\025)30 b FA(2,)k(a)f(coideal)g(and)h(stable)f(under) i(the)f(action)f(and)h(coaction)f(of)g(\000.)48 b(Since)34 b Fm(B)p FA(\()p Fx(V)21 b FA(\))30 b(=)f Fx(T)14 b FA(\()p Fx(V)22 b FA(\))p Fx(=J)9 b FA(,)34 b(where)-180 1066 y Fx(J)42 b FA(is)32 b(the)h(largest)f(ideal)f(with)h(the)h(same)f (prop)s(erties)h(as)g Fx(I)8 b FA(,)32 b(there)h(is)f(a)h(canonical)e (surjection)h Fx(S)i Fu(!)27 b Fm(B)p FA(\()p Fx(V)22 b FA(\).)-80 1205 y(The)33 b Fx(x)175 1220 y Ft(i)237 1205 y FA(satisfy)f(the)i(Serre)f(relations)e(\(4.6\))i(b)m(y)g(Lemma)f (7.2,)g(and)h(then)h(the)f(ro)s(ot)f(v)m(ector)h(relations)f(\(4.7\))g (b)m(y)-180 1345 y(Lemma)d(7.5.)42 b(Therefore)32 b(it)d(follo)m(ws)h (from)f(the)i(description)f(of)g Fm(B)p FA(\()p Fx(V)21 b FA(\))31 b(in)e(Theorem)i(4.5)f(that)g Fx(S)k FA(=)28 b Fm(B)p FA(\()p Fx(V)21 b FA(\).)43 b(This)-180 1484 y(means)33 b Fx(S)6 b FA(\(1\))27 b(=)g Fu(P)8 b FA(\()p Fx(S)e FA(\),)33 b(hence)h(b)m(y)f(dualit)m(y)f(that)g Fx(R)i FA(is)e(generated)i(b)m(y)f Fx(R)q FA(\(1\).)1314 b Fj(\003)-80 1654 y FA(A)24 b(sp)s(ecial)f(case)i(of)e(the)i(last)e (Theorem)h(together)h(with)e(a)h(main)f(result)h(in)f([AS2])h(allo)m (ws)f(to)h(pro)m(v)m(e)h(the)g(follo)m(wing)-180 1815 y FC(Corollary)43 b(7.7.)h Fw(L)-5 b(et)40 b Fx(p)35 b(>)h FA(17)j Fw(b)-5 b(e)39 b(a)g(prime)f(numb)-5 b(er.)58 b(Then)38 b(any)h(\014nite-dimensional)e(p)-5 b(ointe)g(d)39 b(Hopf)g(algebr)-5 b(a)-180 1955 y(with)43 b(c)-5 b(or)g(adic)g(al)43 b Fy(|)-9 b FA(\()p Fy(Z)n Fx(=)o FA(\()p Fx(p)p FA(\)\))831 1918 y Ft(s)905 1955 y Fw(for)44 b(some)f(natur)-5 b(al)43 b(numb)-5 b(er)43 b Fx(s)h Fw(is)f(gener)-5 b(ate)g(d)43 b(by)h(gr)-5 b(oup-like)43 b(and)g(skew-primitive)-180 2094 y(elements.)-180 2287 y(Pr)-5 b(o)g(of.)41 b FA(Let)30 b Fx(A)g FA(b)s(e)g(a)g(\014nite-dimensional)d(p)s(oin)m(ted)i(Hopf)h (algebra)f(with)g(coradical)f Fy(|)-8 b FA(\()p Fy(Z)m Fx(=)p FA(\()p Fx(p)p FA(\)\))3316 2251 y Ft(s)3376 2287 y FA(and)30 b(let)f Fx(R)i FA(b)s(e)f(the)-180 2426 y(diagram)h(of)i Fx(A)p FA(.)45 b(Then)34 b Fx(R)q FA(\(1\))f(is)g(a)f(Y)-8 b(etter-Drinfeld)32 b(mo)s(dule)f(of)i(\014nite)g(Cartan)g(t)m(yp)s(e)h (b)m(y)g([AS2,)g(Corollary)d(1.2].)-180 2566 y(Hence)j(the)f(claim)d (follo)m(ws)h(from)h(Theorem)h(7.6.)2341 b Fj(\003)-80 2735 y FA(Let)32 b(us)i(state)f(explicitly)d(another)j(Corollary)e(of)h (the)h(Theorem.)-180 2897 y FC(Corollary)h(7.8.)40 b Fw(Under)33 b(the)f(hyp)-5 b(othesis)32 b(of)g(The)-5 b(or)g(em)32 b(7.6,)g(if)g(the)h(Dynkin)f(diagr)-5 b(am)31 b(attache)-5 b(d)32 b(to)h(the)g(p)-5 b(ointe)g(d)-180 3036 y(Hopf)35 b(algebr)-5 b(a)34 b(is)g(c)-5 b(onne)g(cte)g(d,)34 b(then)h Fx(A)g Fw(is)f(gener)-5 b(ate)g(d)35 b(by)g(gr)-5 b(oup-like)34 b(and)g(skew-primitive)f(elements.)364 b Fj(\003)-80 3174 y FA(In)31 b(principle,)f(the)i(idea)f(b)s(ehind)g (the)g(pro)s(of)g(of)g(Theorem)g(7.6)g(is)g(as)g(follo)m(ws.)42 b(Let)31 b Fx(A)h FA(b)s(e)f(a)g(\014nite-dimensional)-180 3290 y(p)s(oin)m(ted)48 b(Hopf)g(algebra)f(with)h(coradical)e Fy(|)-8 b FA(\000,)46 b(\000)i(an)m(y)h(\014nite)f(group.)90 b(Let)48 b Fx(R)i FA(b)s(e)e(the)h(diagram)d(of)i Fx(A)p FA(,)k(and)-180 3415 y Fx(S)38 b FA(:=)33 b Fx(R)129 3379 y Fr(\003)204 3415 y FA(the)j(dual)e(braided)i(Hopf)f(algebra.)51 b(Consider)36 b(the)g(diagram)2560 3390 y Fv(e)2542 3415 y Fx(R)h FA(of)e(the)g(b)s(osonization)f Fx(S)6 b FA(#)p Fy(|)-8 b FA(\000.)46 b(Then)-180 3540 y Fu(P)8 b FA(\()p Fx(S)e FA(\))41 b(is)f(naturally)f(em)m(b)s(edded)i(in)f Fu(P)8 b FA(\()1337 3515 y Fv(e)1319 3540 y Fx(R)r FA(\))40 b(\(and)g(this)h(em)m(b)s(edding)f(is)g(in)g(fact)g(an)g (isomorphism\).)65 b(Moreo)m(v)m(er,)-180 3670 y(dim)o(\()p Fu(P)8 b FA(\()p Fx(R)q FA(\)\))28 b Fu(\024)h FA(dim)n(\()p Fu(P)8 b FA(\()p Fx(S)e FA(\)\))28 b Fu(\024)h FA(dim)n(\()p Fu(P)8 b FA(\()1346 3645 y Fv(e)1327 3670 y Fx(R)r FA(\)\),)33 b(and)g(dim)n(\()p Fu(P)8 b FA(\()p Fx(R)q FA(\)\))29 b(=)f(dim)n(\()p Fu(P)8 b FA(\()2662 3645 y Fv(e)2643 3670 y Fx(R)r FA(\)\))32 b(if)g(and)h(only)f(if)g Fx(S)6 b FA(\(1\))27 b(=)h Fu(P)8 b FA(\()p Fx(S)e FA(\))33 b(or)-180 3786 y Fx(R)c FA(=)e Fm(B)p FA(\()p Fu(P)8 b FA(\()p Fx(R)q FA(\)\).)-80 3917 y(Corollary)33 b(7.7)i(can)g(also)g (b)s(e)g(seen)i(as)e(a)g(direct)g(consequence)j(of)d(Section)g(6)g(and) h([AS2]:)49 b(By)35 b([AS2)q(])g Fu(P)8 b FA(\()3876 3892 y Fv(e)3858 3917 y Fx(R)q FA(\))35 b(is)-180 4033 y(of)d(\014nite)g(Cartan)h(t)m(yp)s(e.)44 b(Then)34 b(the)f(result)g (follo)m(ws)e(from)g(Theorem)i(6.8)f(and)h(6.9)f(applied)g(to)g Fx(A)c FA(=)f Fx(S)6 b FA(#)p Fy(|)-8 b FA(\000.)-80 4149 y(The)34 b(next)h(theorem)e(is)g(another)h(application)d(of)i (this)g(principle.)45 b(It)34 b(sho)m(ws)h(that)f(only)f(v)m(ery)i(sp)s (ecial)d(dimen-)-180 4265 y(sions)h(are)f(p)s(ossible)g(for)g (\014nite-dimensional)d(p)s(oin)m(ted)k(Hopf)f(algebras.)-180 4427 y FC(Theorem)46 b(7.9.)h Fw(F)-7 b(or)41 b(any)h(\014nite)g(gr)-5 b(oup)43 b FA(\000)f Fw(of)g(o)-5 b(dd)42 b(or)-5 b(der)42 b(ther)-5 b(e)43 b(is)f(a)g(natur)-5 b(al)42 b(numb)-5 b(er)42 b Fx(n)p FA(\(\000\))h Fw(such)f(that)h(the)-180 4566 y(dimension)33 b(of)i(any)g(\014nite-dimensional)d(p)-5 b(ointe)g(d)34 b(Hopf)h(algebr)-5 b(a)34 b(with)h(c)-5 b(or)g(adic)g(al)34 b Fy(|)-9 b FA(\000)29 b Fw(is)34 b Fu(\024)29 b Fx(n)p FA(\(\000\))p Fw(.)-180 4759 y(Pr)-5 b(o)g(of.)41 b FA(Let)h Fx(A)f FA(b)s(e)h(a)f(\014nite-dimensional)d(p) s(oin)m(ted)j(Hopf)h(algebra)e(with)h(coradical)f Fy(|)-9 b FA(\000)36 b(and)41 b(diagram)e Fx(R)k FA(and)-162 4873 y Fv(e)-180 4898 y Fx(R)c FA(as)f(de\014ned)i(ab)s(o)m(v)m(e.)60 b(Since)39 b Fx(R)g FA(and)1317 4873 y Fv(e)1299 4898 y Fx(R)g FA(are)f(braided)g(Hopf)f(algebras)h(o)m(v)m(er)h(\000)f(of)f (the)i(same)f(dimension,)g(and)-180 5038 y(dim)o(\()p Fu(P)8 b FA(\()p Fx(R)q FA(\)\))49 b Fu(\024)h FA(dim)n(\()p Fu(P)8 b FA(\()797 5013 y Fv(e)778 5038 y Fx(R)r FA(\)\),)48 b(w)m(e)e(can)g(iterate)e(this)h(pro)s(cess)h(and)g(after)f(\014nitely) f(man)m(y)h(steps)i(w)m(e)f(obtain)e(a)-180 5177 y(graded)39 b(braided)g(Hopf)g(algebra)f Fx(T)53 b FA(o)m(v)m(er)41 b(\000)e(with)g(dim)n(\()p Fx(R)q FA(\))g(=)g(dim)o(\()p Fx(T)14 b FA(\))38 b(and)i Fx(T)52 b FA(=)39 b Fm(B)p FA(\()p Fu(P)8 b FA(\()p Fx(T)14 b FA(\)\).)64 b(By)39 b(a)g(result)g(of)-180 5317 y(Gra)s(~)-51 b(na)33 b([G)s(~)-51 b(n3)o(])35 b(using)g([AS2,)h(Theorem)f(3.1])g(whic)m(h)h(follo)m(ws)e (from)g([L3],)h(the)h(n)m(um)m(b)s(er)g(of)e(isomorphism)f(classes)p eop %%Page: 26 26 26 25 bop -180 0 a Fn(26)846 b(NICOL)1001 -19 y(\023)991 0 y(AS)24 b(ANDR)n(USKIEWITSCH)f(AND)h(HANS-J)2419 -19 y(\177)2409 0 y(UR)n(GEN)f(SCHNEIDER)-180 203 y FA(of)29 b(Y)-8 b(etter-Drinfeld)28 b(mo)s(dules)h Fx(V)51 b FA(o)m(v)m(er)31 b(the)f(\014xed)h(group)e(\000)h(with)f(\014nite-dimensional)e Fm(B)p FA(\()p Fx(V)21 b FA(\))30 b(is)f(\014nite.)43 b(Th)m(us)31 b(w)m(e)-180 342 y(can)i(tak)m(e)g(for)f Fx(n)p FA(\(\000\))h(the)g(pro)s(duct)g(of)f(the)h(largest)f(suc)m(h)i (dimension)d(with)h(the)h(order)g(of)f(\000.)754 b Fj(\003)1656 596 y Fz(References)-180 774 y FB([AD])42 b(N.)28 b(Andruskiewitsc)n(h) f(and)h(S.)g(D\025)-42 b(asc\025)g(alescu,)26 b Fp(On)j(quantum)g(gr)l (oups)h(at)g Ff(\000)p FB(1,)c(Algebr.)i(Represen)n(t.)f(Theory)-7 b(,)26 b(to)i(app)r(ear.)-180 893 y([AJS])42 b(H.)29 b(H.)h(Andersen,)g(J.)f(Jan)n(tzen)f(and)i(W.)f(So)r(ergel,)g Fp(R)l(epr)l(esentations)i(of)h(quantum)f(gr)l(oups)g(at)g(a)h Fe(p)p Fp(-th)f(r)l(o)l(ot)g(of)h(unity)f(and)h(of)-9 1013 y(semisimple)f(gr)l(oups)f(in)g(char)l(acteristic)h Fe(p)p Fp(:)39 b(Indep)l(endenc)l(e)30 b(of)h Fe(p)c FB(Ast)n(\023)-39 b(erisque)26 b Fd(220)p FB(,)h(1994.)-180 1133 y([A)n(G])42 b(N.)47 b(Andruskiewitsc)n(h)g(and)g(M.)g(Gra)r(~)-44 b(na,)51 b(Braided)c(Hopf)g(algebras)e(o)n(v)n(er)h(non-ab)r(elian)g (groups,)51 b Fc(Bol.)c(Acad.)g(Ciencias)-9 1252 y(\(C\023)-42 b(ordoba\))26 b Fd(63)h FB(\(1999\),)g(45-78,)e(a)n(v)-5 b(ailable)26 b(at)i Fo(www.mate.uncor.e)o(du)o(/an)o(dr)o(us/)o(ar)o (ti)o(cul)o(os)o(.ht)o(ml)o FB(.)-180 1372 y([AS1])42 b(N.)25 b(Andruskiewitsc)n(h)f(and)g(H.-J.)h(Sc)n(hneider,)f Fp(Lifting)k(of)g(Quantum)e(Line)l(ar)h(Sp)l(ac)l(es)g(and)h(Pointe)l (d)g(Hopf)g(A)n(lgebr)l(as)f(of)h(or)l(der)-9 1491 y Fe(p)33 1461 y Fb(3)70 1491 y FB(,)f(J.)h(Algebra)e Fd(209)h FB(\(1998\),)g(658{691.)-180 1611 y([AS2])42 b(N.)d(Andruskiewitsc)n(h) g(and)h(H.-J.)f(Sc)n(hneider,)j Fp(Finite)f(quantum)f(gr)l(oups)h(and)g (Cartan)g(matric)l(es)p FB(,)i(Adv.)d(in)g(Math.)g Fd(154)-9 1730 y FB(\(2000\),)26 b(1{45.)-180 1850 y([AS3])42 b(N.)32 b(Andruskiewitsc)n(h)f(and)g(H.-J.)g(Sc)n(hneider,)i Fp(Lifting)h(of)g(Nichols)h(algebr)l(as)g(of)f(typ)l(e)g Fe(A)2919 1862 y Fb(2)2990 1850 y Fp(and)g(Pointe)l(d)g(Hopf)g(A)n (lgebr)l(as)g(of)-9 1969 y(or)l(der)23 b Fe(p)244 1939 y Fb(4)281 1969 y FB(,)f(in)f("Hopf)f(algebras)e(and)i(quan)n(tum)h (groups",)f(Pro)r(ceedings)e(of)i(the)h(Collo)r(quium)f(in)h(Brussels)e (1998,)g(Marcel)h(Dekk)n(er,)-9 2089 y(ed.)27 b(S.)h(Caenep)r(eel,)g (1{18)d(\(2000\).)-180 2209 y([AS4])42 b(N.)22 b(Andruskiewitsc)n(h)g (and)h(H.-J.)f(Sc)n(hneider,)h Fp(Pointe)l(d)j(Hopf)g(algebr)l(as)p FB(,)f(to)d(app)r(ear)f(in)i("New)f(directions)g(in)h(Hopf)g (algebras",)-9 2328 y(MSRI)28 b(series)e(Cam)n(bridge)g(Univ.)i(Press.) -180 2448 y([BDG])42 b(M.)30 b(Beattie,)f(S.)h(D\025)-42 b(asc\025)g(alescu,)28 b(and)h(L.)h(Gr)r(\177)-44 b(unenfelder,)29 b(On)g(the)h(n)n(um)n(b)r(er)f(of)g(t)n(yp)r(es)g(of)h (\014nite-dimensional)e(Hopf)i(algebras,)-9 2567 y Fc(In)n(v)n(en)n (tiones)c(Math.)i Fd(136)f FB(\(1999\),)f(1-7.)-180 2687 y([BDR])42 b(M.)28 b(Beattie,)f(S.)h(D\025)-42 b(asc\025)g(alescu,)27 b(and)g(S.)h(Raian)n(u,)f Fp(Lifting)j(of)h(Nichols)g(algebr)l(as)g(of) g(typ)l(e)f Fe(B)2949 2699 y Fb(2)2986 2687 y FB(,)e(preprin)n(t)f (2001.)-180 2806 y([CD])42 b(S.)28 b(Caenep)r(eel)f(and)h(S.)g(D\025) -42 b(asc\025)g(alescu,)56 b Fp(Pointe)l(d)30 b(Hopf)h(algebr)l(as)g (of)g(dimension)g Fe(p)2613 2776 y Fb(3)2650 2806 y FB(,)c(J.)h (Algebra)e Fd(209)h FB(\(1998\),)g(622{634.)-180 2926 y([CDR])42 b(S.)23 b(Caenep)r(eel,)h(S.)f(D\025)-42 b(asc\025)g(alescu) 22 b(and)h(S.)g(Raian)n(u,)g Fp(Classifying)28 b(Pointe)l(d)e(Hopf)h (algebr)l(as)f(of)h(dimension)f FB(16,)d(Comm.)g(Algebra)-9 3045 y Fd(28)k FB(\(2000\),)f(pp.)i(541{568.)-180 3165 y([dCP])41 b(C.)25 b(de)h(Concini)e(and)h(C.)g(Pro)r(cesi)f Fp(Quantum)i(Gr)l(oups)p FB(,)g(in)f("D-mo)r(dules,)g(Represen)n (tation)f(theory)h(and)f(Quan)n(tum)h(Groups",)-9 3284 y(31{140,)f(Lecture)k(Notes)f(in)h(Maths.)f Fd(1565)g FB(\(1993\),)f(Springer-V)-7 b(erlag)25 b(.)-180 3404 y([Di])42 b(D.)28 b(Didt,)h Fp(Linkable)i(Dynkin)f(diagr)l(ams)p FB(,)f(preprin)n(t)e(\(2001\).)-180 3524 y([DT])42 b(Y.)26 b(Doi)f(and)h(M.)g(T)-7 b(ak)n(euc)n(hi,)24 b Fp(Multiplic)l(ation)30 b(alter)l(ation)f(by)f(two-c)l(o)l(cycles.)h(The)g(quantum)e(version)p FB(,)g(Comm)n(un.)e(Algebra)f Fd(22)p FB(,)-9 3643 y(No.14,)i (\(1994\),)h(5715-5732.)-180 3763 y([Dr])42 b(V.)28 b(G.)f(Drinfeld,)i Fp(Quasi-Hopf)h(algebr)l(as)p FB(,)f(Leningrad)e(Math.)h(J.)f Fd(1)g FB(\(1990\),)g(1419{1457.)-180 3882 y([G])60 b(S.)28 b(Gelaki,)f(On)g(p)r(oin)n(ted)h(Hopf)g(algebras)d(and)j(Kaplansky's)e (ten)n(th)i(conjecture,)f Fc(J.)g(Algebra)g Fd(209)g FB(\(1998\),)f(635-657.)-180 4002 y([G)r(~)-44 b(n1])41 b(M.)28 b(Gra)r(~)-44 b(na,)27 b Fp(Pointe)l(d)j(Hopf)h(algebr)l(as)g (of)g(dimension)g FB(32,)26 b(Comm.)i(Algebra)f Fd(28)g FB(\(2000\),)f(pp.)i(2935{2976.)-180 4121 y([G)r(~)-44 b(n2])41 b(M.)28 b(Gra)r(~)-44 b(na,)27 b Fp(On)i(Pointe)l(d)h(Hopf)h (algebr)l(as)g(of)g(dimension)g Fe(p)1965 4091 y Fb(5)2002 4121 y FB(,)c(Glasgo)n(w)f(Math.)i(J.)f Fd(42)h FB(\(2000\),)e (405{419.)-180 4241 y([G)r(~)-44 b(n3])41 b(M.)28 b(Gra)r(~)-44 b(na,)27 b Fp(On)i(Nichols)i(algebr)l(as)g(of)g(low)f(dimension)p FB(,)f(Con)n(temp.)f(Math.)f Fd(267)g FB(\(2000\),)g(pp.)h(111{134.) -180 4360 y([K])60 b(V.)28 b(Kac,)e Fp(In\014nite)j(dimensional)j(Lie)e (algebr)l(as)p FB(,)f(Cam)n(bridge)e(Univ.)h(Press,)e(Third)h(edition,) h(1995.)-180 4480 y([L1])41 b(G.)33 b(Lusztig,)g Fp(Finite)h (dimensional)i(Hopf)f(algebr)l(as)g(arising)h(fr)l(om)e(quantize)l(d)g (universal)h(enveloping)h(algebr)l(as)p FB(,)f(J.)d(of)g(Amer.)-9 4600 y(Math.)27 b(So)r(c.)h Fd(3)f FB(257{296.)-180 4719 y([L2])41 b(G.)28 b(Lusztig,)f Fp(Quantum)h(gr)l(oups)i(at)g(r)l(o)l (ots)g(of)g(1)p FB(,)e(Geom.)g(Dedicata)f Fd(35)g FB(\(1990\),)g (89{114.)-180 4839 y([L3])41 b(G.)28 b(Lusztig,)f Fp(Intr)l(o)l (duction)i(to)h(quantum)f(gr)l(oups)p FB(,)f(Birkh\177)-42 b(auser,)26 b(1993.)-180 4958 y([Mj])42 b(S.)28 b(Ma)5 b(jid,)27 b Fp(Cr)l(osse)l(d)k(pr)l(o)l(ducts)f(by)g(br)l(aide)l(d)h (gr)l(oups)f(and)h(b)l(osonization)p FB(,)e(J.)e(Algebra)g Fd(163)g FB(\(1994\),)f(165{190.)-180 5078 y([Ma])41 b(A.)28 b(Masuok)-5 b(a,)27 b Fp(Defending)j(the)g(ne)l(gate)l(d)g (Kaplansky)h(c)l(onje)l(ctur)l(e)p FB(,)d(Pro)r(c.)e(Amer.)i(Math.)g (So)r(c.)f Fd(129)g FB(\(2001\),)f(3185{3192.)-180 5197 y([Mo])41 b(S.)28 b(Mon)n(tgomery)-7 b(,)26 b Fp(Hopf)31 b(algebr)l(as)g(and)f(their)h(actions)f(on)g(rings)p FB(,)e(CBMS)f(Lecture)h(Notes)f(82,)g(Amer.)g(Math.)h(So)r(c.,)g(1993.) -180 5317 y([Mu])42 b(E.)27 b(M)r(\177)-44 b(uller,)28 b Fp(Some)h(topics)i(on)f(F)-6 b(r)l(ob)l(enius-Lusztig)29 b(kernels,)i(I)p FB(,)c(J.)h(Algebra)e Fd(206)h FB(\(1998\),)g (624{658.)p eop %%Page: 27 27 27 26 bop 446 0 a Fn(FINITE)33 b(QUANTUM)g(GR)n(OUPS)f(O)n(VER)g (ABELIAN)g(GR)n(OUPS)g(OF)i(PRIME)f(EXPONENT)549 b(27)-180 203 y FB([Ms])41 b(I.)28 b(Musson,)f Fp(Finite)k(Quantum)d(Gr)l(oups)i (and)g(Pointe)l(d)g(Hopf)h(A)n(lgebr)l(as)f FB(,)e(preprin)n(t)f (\(1999\).)-180 323 y([N])63 b(W.D.)28 b(Nic)n(hols,)f Fp(Bialgebr)l(as)32 b(of)f(typ)l(e)f(one)p FB(,)e(Comm)n(un.)f(Alg.)h Fd(6)f FB(\(1978\),)g(1521{1552.)-180 442 y([NZ])42 b(W.D.)28 b(Nic)n(hols)f(and)h(M.)f(B.)h(Zo)r(eller,)f Fp(A)i(Hopf)i(algebr)l(a)g (fr)l(e)l(eness)f(The)l(or)l(em)p FB(,)f(Amer.)e(J.)h(Math.)f Fd(111)g FB(\(1989\),)g(381{385.)-180 562 y([Ra])41 b(D.)28 b(Radford,)f Fp(Hopf)k(algebr)l(as)g(with)g(pr)l(oje)l(ction)p FB(,)e(J.)e(Algebra)g Fd(92)g FB(\(1985\),)f(322{347.)-180 681 y([Ri])42 b(C.)28 b(Ringel,)f Fp(PBW-b)l(ases)k(of)f(quantum)f(gr)l (oups)p FB(,)f(J.)f(reine)g(angew.)g(Math.)h Fd(470)f FB(\(1996\),)f(51{88.)-180 801 y([Ro1])41 b(M.)28 b(Rosso,)e Fp(Gr)l(oup)l(es)k(quantiques)f(et)h(algebr)l(es)h(de)f(b)l(attage)g (quantiques)p FB(,)e(C.R.A.S.)g(\(P)n(aris\))e Fd(320)h FB(\(1995\),)g(145{148)-180 920 y([Ro2])41 b(M.)28 b(Rosso,)e Fp(Quantum)i(gr)l(oups)i(and)g(quantum)f(shu\017es)p FB(,)f(In)n(v)n(en)n(tiones)e(Math.)i Fd(133)f FB(\(1998\),)f(399{416.) -180 1040 y([Sc)n(h])41 b(P)-7 b(.)35 b(Sc)n(hauen)n(burg,)h Fp(A)g(Char)l(acterization)j(of)e(the)g(Bor)l(el-like)i(sub)l(algebr)l (as)e(of)g(Quantum)e(Enveloping)k(algebr)l(as)p FB(,)f(Comm)n(un.)-9 1159 y(Alg.)27 b Fd(24)g FB(\(1996\),)g(2811{2823.)-180 1279 y([SvO])41 b(D.)35 b(Stefan)g(and)g(F.)g(v)-5 b(an)34 b(Oystaey)n(en,)h Fp(Ho)l(chschild)k(c)l(ohomolo)l(gy)f(and)f(c)l(or)l (adic)l(al)h(\014ltr)l(ation)e(of)h(p)l(ointe)l(d)g(Hopf)g(algebr)l(as) p FB(,)h(J.)-9 1399 y(Algebra)26 b Fd(210)h FB(\(1998\),)f(535{556.) -180 1518 y([W)-7 b(o])42 b(W)-7 b(orono)n(wicz,)34 b(S.)h(L.,)i Fp(Di\013er)l(ential)g(c)l(alculus)f(on)g(c)l(omp)l(act)h(matrix)f (pseudo)l(gr)l(oups)h(\(quantum)e(gr)l(oups\))p FB(,)i(Comm)n(un.)e (Math.)-9 1638 y(Ph)n(ys.)26 b Fd(122)h FB(\(1989\),)g(125{170.)-80 1862 y Fa(F)-10 b(a)n(cul)k(t)g(ad)34 b(de)g(Ma)-6 b(tem)754 1855 y(\023)751 1862 y(atica,)35 b(Astr)n(onom)1450 1855 y(\023)1460 1862 y(\020a)e(y)h(F)1703 1855 y(\023)1713 1862 y(\020sica,)g(Universid)n(ad)f(Na)n(cional)g(de)h(C)3124 1855 y(\023)3121 1862 y(ordoba,)f(\(5000\))h(Ciud)n(ad)-180 1981 y(Universit)-6 b(aria,)31 b(C)520 1974 y(\023)517 1981 y(ordoba,)g(Ar)n(gentina)-80 2101 y Fp(E-mail)f(addr)l(ess)7 b FB(:)38 b Fo(andrus@mate.uncor)o(.e)o(du)-80 2304 y Fa(Ma)-6 b(thema)g(tisches)26 b(Institut,)f(Universit)1426 2297 y(\177)1423 2304 y(at)48 b(M)1656 2297 y(\177)1653 2304 y(unchen,)26 b(Theresienstra\031e)i(39,)d(D-80333)f(M)3342 2297 y(\177)3339 2304 y(unchen,)h(Germany)-80 2424 y Fp(E-mail)30 b(addr)l(ess)7 b FB(:)38 b Fo(hanssch@rz.mathem)o(at)o (ik.)o(un)o(i-m)o(ue)o(nc)o(hen)o(.d)o(e)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF