%!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: Yetterfinfin.dvi %%Pages: 21 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips Yetterfinfin.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2001.01.25:1703 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] 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b(The)36 b(actions)456 1932 y(are)31 b(completely)f(determined)h(b)m(y)i(the)e (action)g(of)g(a)g(Borel)f(subalgebra,)i(that)f(is)456 2048 y(b)m(y)40 b(the)h(action)e(of)g Fq(E)46 b Fs(\(or)39 b Fq(F)14 b Fs(\))40 b(on)f Fq(u)h Fs(\(see)h(Theorem)f(3.1\).)65 b(As)40 b(noted)g(ab)s(o)m(v)m(e,)456 2164 y(this)46 b(result)h(can)g(b)s(e)g(view)m(ed)h(as)f(a)f(quan)m(tum)h(analog)f(of) g(classical)f(w)m(ork)j(of)456 2280 y(Jacobson)33 b([J)q(])g(on)g (restricted)h(Lie)e(algebras)h(\(see)h(Remark)f(3.3\);)g(it)f(is)h (also)f(the)456 2397 y(analog)i(at)i(a)f(ro)s(ot)g(of)h(1)g(of)f ([MSm],)i(whic)m(h)g(describ)s(ed)g(actions)e(of)h Fq(U)3077 2412 y Fo(q)3115 2397 y Fs(\()p Fq(sl)3228 2412 y Fn(2)3268 2397 y Fs(\))f(on)456 2513 y(the)j(p)s(olynomial)d(ring)i Fq(A)g Fs(=)g Fj(C)20 b Fs([)p Fq(X)8 b Fs(],)46 b(when)39 b Fq(q)j Fs(w)m(as)d(not)f(a)g(ro)s(ot)f(of)h(1)g(\(Remark)456 2629 y(3.3\).)555 2745 y(As)28 b(an)f(application)d(w)m(e)k(construct)g (smash-pro)s(ducts)g Fq(A)p Fs(#)p Fq(u)2786 2760 y Fo(q)2824 2745 y Fs(\()p Fq(sl)2937 2760 y Fn(2)2976 2745 y Fs(\))f(whic)m(h)g (are)456 2862 y(not)22 b(semisimple,)h(and)g(where)h Fq(A)e Fs(is)h(a)f(comm)m(utativ)m(e)g(\014eld)g(suc)m(h)j(that)d(the)h (action)456 2978 y(of)43 b Fq(u)634 2993 y Fo(q)672 2978 y Fs(\()p Fq(sl)785 2993 y Fn(2)824 2978 y Fs(\))h(is)f(outer)i(on)e Fq(A)i Fs(\(that)e(is,)k(the)d(action)f(of)h(an)m(y)g(non-trivial)d (Hopf)456 3094 y(subalgebra)28 b(of)h Fq(u)1104 3109 y Fo(q)1141 3094 y Fs(\()p Fq(sl)1254 3109 y Fn(2)1294 3094 y Fs(\))g(acts)g(non-trivially)d(on)i Fq(A)p Fs(,)i(hence)h(is)d (not)h(inner\).)42 b(Th)m(us)456 3210 y(the)h(old)g(result)g(of)g (Azuma)m(y)m(a)h(whic)m(h)g(sa)m(ys)h(that)e(if)g Fq(H)53 b Fs(=)46 b Fq(k)s(G)e Fs(is)f(the)h(group)456 3326 y(algebra)35 b(of)h(a)g(\014nite)g(group)g(acting)f(on)h(an)h(algebra)e Fq(A)h Fs(with)g(outer)h Fq(G)p Fs(-action,)456 3443 y(then)h Fq(A)p Fs(#)p Fq(H)45 b Fs(is)36 b(simple,)h(do)s(es)h(not)f (generalize)g(in)f(a)h(straigh)m(tforw)m(ard)g(w)m(a)m(y)i(to)456 3559 y(actions)30 b(of)g(p)s(oin)m(ted)g(Hopf)h(algebras.)42 b(Our)31 b(examples)f(seem)h(to)g(b)s(e)g(the)g(\014rst)g(of)456 3675 y(this)h(t)m(yp)s(e)h(where)h(the)f(smash)g(pro)s(duct)g(is)f(not) g(ev)m(en)i(semisimple.)555 3791 y(In)g(Section)f(4,)g(w)m(e)h(sho)m(w) g(that)f(an)m(y)g(action)g(of)f Fq(H)41 b Fs(on)33 b Fq(A)28 b Fs(=)h Fq(k)s Fs([)p Fq(u)f Fp(j)g Fq(u)3057 3755 y Fo(n)3132 3791 y Fs(=)g Fq(\014)6 b Fs(])33 b(as)456 3908 y(in)25 b(\(I\))h(extends)i(uniquely)e(to)g(an)g(action)f(of)h (the)g(Drinfel'd)f(double)h Fq(D)s Fs(\()p Fq(H)8 b Fs(\).)40 b(As)27 b(a)456 4024 y(consequence,)j(in)25 b(Section)h(5)f(w)m(e)i (see)g(that)f(there)g(is)g(a)f(unique)i(left)e Fq(H)8 b Fs(-como)s(dule)456 4140 y(algebra)33 b(structure)j(on)e Fq(A)p Fs(,)h(suc)m(h)h(that)e Fq(A)h Fs(is)f(a)g(Y)-8 b(etter-Drinfel'd)33 b(algebra)g(o)m(v)m(er)456 4256 y Fq(H)8 b Fs(.)43 b(In)32 b(particular,)f(if)g Fq(g)25 b Fp(\001)d Fq(u)27 b Fs(=)g Fq(!)t(u)32 b Fs(and)g Fq(x)22 b Fp(\001)g Fq(u)27 b Fs(=)g Fq(\015)5 b Fs(1)33 b(for)e(some)i Fq(\015)f Fp(2)c Fq(k)s Fs(,)33 b(then)g(the)456 4373 y(unique)d(left)g Fq(H)8 b Fs(-como)s(dule)28 b(algebra)h(structure)j (on)e Fq(A)g Fs(is)g(giv)m(en)h(b)m(y)g(the)g(explicit)456 4489 y(form)m(ula)870 4725 y Fq(\032)p Fs(\()p Fq(u)p Fs(\))d(=)1193 4600 y Fo(n)p Fk(\000)p Fn(1)1188 4630 y Fi(X)1183 4840 y Fo(m)p Fn(=0)1353 4725 y Fq(\015)1409 4684 y Fk(\000)p Fo(m)1530 4725 y Fs(\()p Fq(!)d Fp(\000)e Fs(1\))1841 4684 y Fo(m)1907 4725 y Fq(!)1982 4650 y Fh(m)p Fm(\()p Fh(m)p Fm(+1\))p 1982 4669 234 3 v 2083 4710 a(2)2230 4725 y Fq(x)2285 4684 y Fo(m)2352 4725 y Fq(g)2403 4684 y Fk(\000)p Fn(\()p Fo(m)p Fn(+1\))2691 4725 y Fp(\012)f Fq(u)2846 4684 y Fo(m)p Fn(+1)3003 4725 y Fq(;)456 4983 y Fs(answ)m(ering)33 b(a)f(question)h(in)f([CFM,)h (Remark)f(2.9].)555 5099 y(F)-8 b(or)31 b(general)f(references)j(on)e (Hopf)g(algebras,)g(see)i([M])e(or)g([K].)43 b(Throughout)456 5216 y Fq(H)52 b Fs(denotes)45 b(a)f(Hopf)h(algebra)e(with)h(com)m (ultiplication)c(\001)48 b(:)g Fq(H)38 b Fp(\012)31 b Fq(H)55 b Fp(!)48 b Fq(H)8 b Fs(,)p eop %%Page: 3 3 3 2 bop 3406 233 a Fl(3)456 425 y Fs(counit)45 b Fq(")51 b Fs(:)f Fq(H)58 b Fp(!)51 b Fq(k)s Fs(,)e(and)e(an)m(tip)s(o)s(de)e Fq(S)6 b Fs(.)84 b(W)-8 b(e)46 b(use)h(the)g(usual)e(summation)456 541 y(notation)31 b(for)h(\001)h(and)f(denote)i(the)f(in)m(v)m(erse)h (of)e Fq(S)38 b Fs(\(under)33 b(comp)s(ostion\))e(b)m(y)p 3319 461 66 4 v 34 w Fq(S)6 b Fs(.)581 790 y(2.)55 b Fr(Skew)38 b(deriv)-9 b(a)i(tions)38 b(and)g(a)n(ctions)f(of)i(the)e(T) -9 b(aft)38 b(algebra)555 965 y Fs(In)c(this)g(section)g(w)m(e)h (determine)f(the)g(sk)m(ew)i(deriv)-5 b(ations)33 b(asso)s(ciated)h(to) f(cer-)456 1081 y(tain)26 b(automorphisms)g(of)h(algebras)g Fq(A)h Fs(=)g Fq(k)s Fs(\()p Fq(u)p Fs(\),)g(and)f(then)i(apply)e (these)i(results)456 1197 y(to)38 b(the)g(actions)g(of)g(the)h(T)-8 b(aft)38 b(algebras.)61 b(Recall)36 b(that)j(for)f(a)g(giv)m(en)g (automor-)456 1313 y(phism)30 b Fq(\033)35 b Fs(of)c(an)g(algebra)f Fq(A)p Fs(,)i Fq(D)h Fs(is)e(called)f(a)h Fq(\033)t Fs(-)p Fg(skew)h(derivation)f Fs(of)f Fq(A)i Fs(if)e(for)g(all)456 1430 y Fq(a;)17 b(b)28 b Fp(2)g Fq(A)p Fs(,)1365 1564 y Fq(D)s Fs(\()p Fq(ab)p Fs(\))g(=)f Fq(D)s Fs(\()p Fq(a)p Fs(\))p Fq(b)c Fs(+)f Fq(\033)t Fs(\()p Fq(a)p Fs(\))p Fq(D)s Fs(\()p Fq(b)p Fs(\))p Fq(:)456 1718 y Fs(W)-8 b(e)33 b(usually)e(write)i Fq(\033)t Fs(\()p Fq(a)p Fs(\))27 b(=)h Fq(\033)e Fp(\001)c Fq(a)33 b Fs(and)f Fq(D)s Fs(\()p Fq(a)p Fs(\))c(=)f Fq(D)e Fp(\001)d Fq(a)p Fs(.)555 1834 y(W)-8 b(e)33 b(\014rst)g(need)h(a)e(tec)m(hnical)g(lemma.)456 2019 y Fu(Lemma)37 b(2.1.)49 b Fg(L)-5 b(et)42 b Fq(A)g Fs(=)f Fq(k)s Fs([)p Fq(u)p Fp(j)p Fq(u)1704 1983 y Fo(n)1791 2019 y Fs(=)g Fq(\014)6 b Fs(])p Fg(,)43 b(for)f(some)g Fq(\014)k Fp(2)c Fq(k)s Fg(,)i(and)e(let)g Fq(\026)g Fg(b)-5 b(e)42 b(a)456 2135 y(primitive)d Fq(n)936 2099 y Fo(th)1046 2135 y Fg(r)-5 b(o)g(ot)39 b(of)h(1)f(in)g Fq(k)s Fg(.)59 b(Cho)-5 b(ose)39 b Fs(0)d Fp(6)p Fs(=)g Fq(\016)41 b Fp(2)c Fq(k)42 b Fg(and)d Fs(0)d Fp(\024)h Fq(s;)17 b(t)37 b Fp(\024)g Fq(n)25 b Fp(\000)i Fs(1)456 2251 y Fg(such)j(that)h Fq(t)p Fs(\(1)12 b Fp(\000)g Fq(s)p Fs(\))1204 2224 y Fp(\030)1205 2256 y Fs(=)1309 2251 y(1\(mo)s(d)26 b Fq(n)p Fs(\))p Fg(.)44 b(L)-5 b(et)31 b Fq(\033)g Fp(2)d Fq(Aut)2288 2266 y Fo(k)2331 2251 y Fs(\()p Fq(A)p Fs(\))i Fg(b)-5 b(e)31 b(given)e(by)i Fq(\033)16 b Fp(\001)c Fq(u)28 b Fs(=)f Fq(\013)q(u)p Fg(,)456 2368 y(wher)-5 b(e)39 b Fq(\013)f Fs(=)g Fq(\026)1009 2332 y Fo(t)1038 2368 y Fg(.)60 b(De\014ne)39 b Fq(D)k Fg(on)d Fq(u)g Fg(by)g Fq(D)29 b Fp(\001)c Fq(u)37 b Fs(=)h Fq(\016)t(u)2411 2332 y Fo(s)2447 2368 y Fg(.)61 b(Then)39 b Fq(D)k Fg(extends)c(to)i(a)456 2484 y Fq(\033)t Fg(-skew)d(derivation)g(of)g Fq(A)h Fg(satisfying)f Fq(D)s(\033)h Fs(=)c Fq(\026\033)t(D)41 b Fg(and)e Fq(D)2723 2448 y Fo(n)2804 2484 y Fs(=)c(0)p Fg(.)57 b(Mor)-5 b(e)g(over)456 2600 y Fq(D)540 2564 y Fo(n)p Fk(\000)p Fn(1)704 2600 y Fp(6)p Fs(=)28 b(0)62 b Fp(\()-17 b(\))62 b Fq(\014)33 b Fp(6)p Fs(=)28 b(0)p Fg(.)555 2716 y(In)k(addition)f(the)h(fol)5 b(lowing)31 b(identities)g(hold,)h(for)g(al)5 b(l)32 b Fq(p)g Fg(with)g Fs(0)27 b Fp(\024)h Fq(p)g Fp(\024)g Fq(n)16 b Fp(\000)g Fs(1)456 2833 y Fg(:)456 2978 y(\(a\))34 b Fq(D)25 b Fp(\001)d Fq(u)832 2942 y Fo(p)899 2978 y Fs(=)27 b Fq(\016)t Fs(\()1087 2903 y Fi(P)1192 2930 y Fo(p)p Fk(\000)p Fn(1)1192 3007 y Fo(i)p Fn(=0)1339 2978 y Fq(\013)1402 2942 y Fo(i)1429 2978 y Fs(\))p Fq(u)1523 2942 y Fo(p)p Fn(+)p Fo(s)p Fk(\000)p Fn(1)1740 2978 y Fg(;)456 3124 y(\(b\))34 b Fq(D)699 3088 y Fo(q)759 3124 y Fp(\001)22 b Fq(u)865 3088 y Fo(p)932 3124 y Fs(=)555 3255 y Fq(\016)602 3219 y Fo(q)640 3255 y Fs(\()678 3180 y Fi(P)783 3207 y Fo(p)p Fk(\000)p Fn(1)783 3284 y Fo(i)807 3293 y Fm(1)842 3284 y Fn(=0)953 3255 y Fq(\013)1016 3219 y Fo(i)1040 3228 y Fm(1)1078 3255 y Fs(\)\()1154 3180 y Fi(P)1259 3207 y Fn(\()p Fo(p)p Fk(\000)p Fn(1\)+\()p Fo(s)p Fk(\000)p Fn(1\))1259 3284 y Fo(i)1283 3293 y Fm(2)1318 3284 y Fn(=0)1693 3255 y Fq(\013)1756 3219 y Fo(i)1780 3228 y Fm(2)1819 3255 y Fs(\))17 b Fp(\001)g(\001)g(\001)d Fs(\()2044 3180 y Fi(P)2149 3207 y Fn(\()p Fo(p)p Fk(\000)p Fn(1\)+\()p Fo(q)r Fk(\000)p Fn(1\)\()p Fo(s)p Fk(\000)p Fn(1\))2149 3284 y Fo(i)2173 3292 y Fh(q)2208 3284 y Fn(=0)2762 3255 y Fq(\013)2825 3219 y Fo(i)2849 3227 y Fh(q)2887 3255 y Fs(\))p Fq(u)2981 3219 y Fo(p)p Fn(+)p Fo(q)r Fn(\()p Fo(s)p Fk(\000)p Fn(1\))456 3385 y Fg(\(c\))40 b(L)-5 b(et)41 b Fq(q)838 3400 y Fn(0)919 3385 y Fg(b)-5 b(e)40 b(the)h(\(unique\))g Fq(q)k Fg(such)40 b(that)i Fs(1)c Fp(\024)i Fq(q)2380 3400 y Fn(0)2458 3385 y Fq(<)f(n)i Fg(and)f Fq(q)2910 3400 y Fn(0)2989 3358 y Fp(\030)2990 3389 y Fs(=)3105 3385 y Fq(pt)h Fg(\(mo)-5 b(d)456 3501 y Fq(n)p Fg(\).)55 b(Then)38 b Fq(D)981 3465 y Fo(q)1044 3501 y Fp(\001)24 b Fq(u)1152 3465 y Fo(p)1226 3501 y Fs(=)34 b(0)k Fg(for)h Fq(q)1626 3516 y Fn(0)1700 3501 y Fq(<)34 b(q)39 b Fp(\024)34 b Fq(n)p Fg(.)56 b(If)38 b Fq(\014)i Fp(6)p Fs(=)35 b(0)p Fg(,)k(then)f Fq(D)2881 3465 y Fo(q)2944 3501 y Fp(\001)24 b Fq(u)3052 3465 y Fo(p)3126 3501 y Fp(6)p Fs(=)34 b(0)39 b Fg(for)456 3618 y Fs(1)27 b Fp(\024)h Fq(q)k Fp(\024)c Fq(q)860 3633 y Fn(0)927 3618 y Fq(<)g(n)p Fg(;)35 b(if)f Fq(\014)g Fs(=)27 b(0)p Fg(,)35 b(then)f Fq(D)1854 3582 y Fo(q)1886 3591 y Fm(0)1947 3618 y Fp(\001)21 b Fq(u)2052 3582 y Fo(p)2119 3618 y Fs(=)28 b(0)p Fg(.)456 3802 y(Pr)-5 b(o)g(of.)41 b Fs(Clearly)e Fq(\033)44 b Fs(acts)d(as)f(an)g (automorphism)e(of)i Fq(A)p Fs(.)67 b(T)-8 b(o)40 b(see)h(that)f Fq(D)j Fs(acts)456 3919 y(as)37 b(a)g Fq(\033)t Fs(-sk)m(ew)i(deriv)-5 b(ation,)37 b(where)h(in)f(fact)g Fq(D)28 b Fp(\001)d Fq(u)2307 3882 y Fo(p)2383 3919 y Fs(is)37 b(as)g(in)g(\(a\),)h(it)e (su\016ces)k(to)456 4096 y(sho)m(w)34 b(that)g(the)g(map)f(\010)c(:)h Fq(A)g Fp(!)f Fq(M)1780 4111 y Fn(2)1819 4096 y Fs(\()p Fq(A)p Fs(\))34 b(giv)m(en)g(b)m(y)g(\010\()p Fq(a)p Fs(\))c(=)2727 3956 y Fi(\022)2800 4036 y Fq(\033)c Fp(\001)c Fq(a)83 b(D)25 b Fp(\001)d Fq(a)2867 4153 y Fs(0)227 b Fq(a)3272 3956 y Fi(\023)3379 4096 y Fs(is)456 4268 y(an)36 b(algebra)g(map.)54 b(Since)37 b(the)g(only)f(relation)f(in)h Fq(A)h Fs(is)f Fq(u)2604 4232 y Fo(n)2685 4268 y Fs(=)e Fq(\014)6 b Fs(,)37 b(it)f(su\016ces)j(to)456 4440 y(c)m(hec)m(k)g (that)d(\010\()p Fq(u)p Fs(\))1140 4404 y Fo(n)1222 4440 y Fs(=)e Fq(\014)6 b(I)i Fs(.)56 b(W)-8 b(e)37 b(set)g Fq(M)46 b Fs(:=)35 b(\010\()p Fq(u)p Fs(\))f(=)2480 4300 y Fi(\022)2553 4381 y Fq(\013)q(u)83 b(\016)t(u)2858 4345 y Fo(s)2588 4497 y Fs(0)160 b Fq(u)2894 4300 y Fi(\023)2967 4440 y Fs(.)56 b(It)37 b(is)f(easy)456 4613 y(to)c(c)m(hec)m(k)j(b)m(y) e(induction)f(that)1043 4845 y(\010\()p Fq(u)p Fs(\))1245 4804 y Fo(p)1312 4845 y Fs(=)c Fq(M)1520 4804 y Fo(p)1588 4845 y Fs(=)1691 4704 y Fi(\022)1765 4790 y Fq(\013)1828 4754 y Fo(p)1867 4790 y Fq(u)1923 4754 y Fo(p)2045 4790 y Fq(\016)t Fs(\()2130 4715 y Fi(P)2235 4742 y Fo(p)p Fk(\000)p Fn(1)2235 4819 y Fo(i)p Fn(=0)2382 4790 y Fq(\013)2445 4754 y Fo(i)2473 4790 y Fs(\))p Fq(u)2567 4754 y Fo(p)p Fn(+)p Fo(s)p Fk(\000)p Fn(1)1839 4906 y Fs(0)479 b Fq(u)2423 4870 y Fo(p)2784 4704 y Fi(\023)456 5099 y Fs(Th)m(us)29 b(\010\()p Fq(u)p Fs(\))900 5063 y Fo(n)975 5099 y Fs(=)e Fq(\014)6 b(I)36 b Fs(since)28 b Fq(\013)1515 5063 y Fo(n)1590 5099 y Fs(=)f(1)h(implies)2097 5025 y Fi(P)2202 5051 y Fo(n)p Fk(\000)p Fn(1)2202 5128 y Fo(i)p Fn(=0)2356 5099 y Fq(\013)2419 5063 y Fo(i)2474 5099 y Fs(=)g(0.)42 b(Moreo)m(v)m(er)29 b(form)m(ula)456 5216 y(\(a\))j(is)g(also)g(clear)g (from)f(this)h(computation.)p eop %%Page: 4 4 4 3 bop 456 236 a Fl(4)661 b(S.)33 b(MONTGOMER)-6 b(Y)33 b(AND)f(H.-J.)i(SCHNEIDER)555 425 y Fs(Since)e Fq(D)i Fs(acts)e(as)g(a)f Fq(\033)t Fs(-sk)m(ew)i(deriv)-5 b(ation,)30 b(also)h Fq(\033)2385 389 y Fk(\000)p Fn(1)2479 425 y Fq(D)s(\033)k Fs(is)c(a)g Fq(\033)t Fs(-sk)m(ew)i(deriv)-5 b(a-)456 541 y(tion.)42 b(No)m(w)33 b(b)m(y)h(\(a\),)e(for)g(all)e Fq(p)p Fs(,)1072 815 y(\()p Fq(\033)1169 774 y Fk(\000)p Fn(1)1264 815 y Fq(D)s(\033)t Fs(\))22 b Fp(\001)f Fq(u)1572 774 y Fo(p)1639 815 y Fs(=)28 b Fq(\013)1806 774 y Fo(p)1845 815 y Fq(\033)1904 774 y Fk(\000)p Fn(1)1998 815 y Fs(\()p Fq(D)d Fp(\001)d Fq(u)2248 774 y Fo(p)2287 815 y Fs(\))28 b(=)f Fq(\013)2519 774 y Fo(p)2559 815 y Fq(\016)t Fs(\()2653 686 y Fo(p)p Fk(\000)p Fn(1)2644 720 y Fi(X)2659 930 y Fo(i)p Fn(=0)2804 815 y Fq(\013)2867 774 y Fo(i)2895 815 y Fs(\))p Fq(\033)2992 774 y Fk(\000)p Fn(1)3108 815 y Fp(\001)22 b Fq(u)3214 774 y Fo(p)p Fn(+)p Fo(s)p Fk(\000)p Fn(1)1155 1082 y Fs(=)83 b Fq(\013)1377 1041 y Fo(p)p Fk(\000)p Fn(\()p Fo(p)p Fn(+)p Fo(s)p Fk(\000)p Fn(1\))1740 1082 y Fq(\016)t Fs(\()1825 988 y Fi(X)1985 1082 y Fq(\013)2048 1041 y Fo(i)2076 1082 y Fs(\))p Fq(u)2170 1041 y Fo(p)p Fn(+)p Fo(s)p Fk(\000)p Fn(1)2414 1082 y Fs(=)28 b Fq(\026D)d Fp(\001)c Fq(u)2788 1041 y Fo(p)456 1283 y Fs(since)31 b Fq(\013)756 1247 y Fn(1)p Fk(\000)p Fo(s)910 1283 y Fs(=)d Fq(\026)p Fs(,)j(using)g(\(a\).)42 b(Th)m(us)33 b Fq(\033)1884 1247 y Fk(\000)p Fn(1)1978 1283 y Fq(D)s(\033)f Fs(=)27 b Fq(\026D)34 b Fs(on)d(all)e(of)h Fq(A)p Fs(,)i(whic)m(h)f(is)g(the)456 1399 y(desired)i(relation)d(b)s (et)m(w)m(een)35 b Fq(\033)i Fs(and)32 b Fq(D)s Fs(.)555 1516 y(T)-8 b(o)28 b(see)i(\(b\))e(and)g(\(c\),)i(w)m(e)f(\014x)g Fq(p)f Fs(and)g(pro)s(ceed)h(b)m(y)g(induction)f(on)g Fq(q)t Fs(;)h(note)g(that)456 1632 y(\(a\))j(is)g(the)h(case)g Fq(q)f Fs(=)27 b(1.)44 b(Assume)33 b(the)g(result)g(is)f(true)h(for)f Fq(q)26 b Fp(\000)c Fs(1.)43 b(Then)456 1905 y Fq(D)540 1864 y Fo(q)599 1905 y Fp(\001)22 b Fq(u)705 1864 y Fo(p)772 1905 y Fs(=)28 b Fq(D)960 1864 y Fo(q)r Fk(\000)p Fn(1)1110 1905 y Fp(\001)22 b Fs(\()p Fq(D)i Fp(\001)e Fq(u)1409 1864 y Fo(p)1448 1905 y Fs(\))28 b(=)g Fq(\016)t Fs(\()1712 1777 y Fo(p)p Fk(\000)p Fn(1)1703 1811 y Fi(X)1718 2021 y Fo(i)p Fn(=0)1863 1905 y Fq(\013)1926 1864 y Fo(i)1954 1905 y Fs(\))p Fq(D)2076 1864 y Fo(q)r Fk(\000)p Fn(1)2226 1905 y Fp(\001)22 b Fq(u)2332 1864 y Fo(p)p Fn(+)p Fo(s)p Fk(\000)p Fn(1)539 2260 y Fs(=)83 b Fq(\016)t Fs(\()792 2131 y Fo(p)p Fk(\000)p Fn(1)783 2165 y Fi(X)798 2375 y Fo(i)p Fn(=0)943 2260 y Fq(\013)1006 2219 y Fo(i)1034 2260 y Fs(\))p Fq(\016)1119 2219 y Fo(q)r Fk(\000)p Fn(1)1247 2260 y Fs(\()1285 2128 y Fn(\()p Fo(p)p Fn(+)p Fo(s)p Fk(\000)p Fn(2\))1347 2165 y Fi(X)1342 2375 y Fo(j)1371 2384 y Fm(1)1405 2375 y Fn(=0)1569 2260 y Fq(\013)1632 2219 y Fo(j)1661 2228 y Fm(1)1699 2260 y Fs(\))17 b Fp(\001)g(\001)g (\001)e Fs(\()1925 2128 y Fn(\()p Fo(p)p Fn(+)p Fo(s)p Fk(\000)p Fn(2\)+\()p Fo(q)r Fk(\000)p Fn(2\)\()p Fo(s)p Fk(\000)p Fn(1\))2193 2165 y Fi(X)2149 2375 y Fo(j)2178 2384 y Fh(q)r Ff(\000)p Fm(1)2290 2375 y Fn(=0)2621 2260 y Fq(\013)2684 2219 y Fo(j)2713 2228 y Fh(q)r Ff(\000)p Fm(1)2829 2260 y Fs(\))p Fq(u)2923 2219 y Fn(\()p Fo(p)p Fn(+)p Fo(s)p Fk(\000)p Fn(1\)+\()p Fo(q)r Fk(\000)p Fn(1\)\()p Fo(s)p Fk(\000)p Fn(1\))539 2634 y Fs(=)83 b Fq(\016)745 2593 y Fo(q)783 2634 y Fs(\()833 2505 y Fo(p)p Fk(\000)p Fn(1)823 2539 y Fi(X)821 2749 y Fo(i)845 2758 y Fm(1)879 2749 y Fn(=0)986 2634 y Fq(\013)1049 2593 y Fo(i)1073 2602 y Fm(1)1111 2634 y Fs(\)\()1187 2501 y Fn(\()p Fo(p)p Fk(\000)p Fn(1\)+\()p Fo(s)p Fk(\000)p Fn(1\))1322 2539 y Fi(X)1319 2749 y Fo(i)1343 2758 y Fm(2)1378 2749 y Fn(=0)1617 2634 y Fq(\013)1680 2593 y Fo(i)1704 2602 y Fm(2)1742 2634 y Fs(\))17 b Fp(\001)g(\001)g(\001)e Fs(\()1968 2501 y Fo(p)p Fk(\000)p Fn(1+\()p Fo(q)r Fk(\000)p Fn(1\)\()p Fo(s)p Fk(\000)p Fn(1\))2164 2539 y Fi(X)2162 2749 y Fo(i)2186 2757 y Fh(q)2220 2749 y Fn(=0)2522 2634 y Fq(\013)2585 2593 y Fo(i)2609 2601 y Fh(q)2647 2634 y Fs(\))p Fq(u)2741 2593 y Fo(p)p Fn(+)p Fo(q)r Fn(\()p Fo(s)p Fk(\000)p Fn(1\))456 2913 y Fs(whic)m(h)46 b(is)g(the)g(desired) g(form.)83 b(Ho)m(w)m(ev)m(er,)51 b(the)c(di\016cult)m(y)f(with)f(this) h(formal)456 3030 y(computation)34 b(is)h(that)h(it)e(ma)m(y)i(equal)g (zero)g(for)f(some)h Fq(q)t Fs('s.)53 b(T)-8 b(o)36 b(see)h(when)g (this)456 3146 y(happ)s(ens,)j(w)m(e)g(\014rst)e(consider)h(the)g(set)g (of)e(p)s(ossible)h(exp)s(onen)m(ts)i(of)e Fq(u)p Fs(,)h(namely)456 3262 y Fp(f)p Fq(p)22 b Fs(+)g Fq(q)t Fs(\()p Fq(s)f Fp(\000)i Fs(1\))k Fp(j)h Fs(0)f Fp(\024)h Fq(q)k Fp(\024)c Fq(n)22 b Fp(\000)h Fs(1)p Fp(g)p Fs(.)555 3378 y(Since)42 b Fq(s)28 b Fp(\000)h Fs(1)41 b(is)g(in)m(v)m(ertible)g(mo)s(d)f Fq(n)i Fs(and)g Fq(p)f Fs(is)g(\014xed,)k(this)c(set)i(con)m(tains)e Fq(n)456 3495 y Fs(distinct)h(elemen)m(ts)h(mo)s(d)f Fq(n)p Fs(.)75 b(Th)m(us)45 b(for)d(exactly)i(one)f Fq(q)t Fs(,)i Fq(p)30 b Fs(+)f Fq(q)t Fs(\()p Fq(s)g Fp(\000)h Fs(1\))3273 3467 y Fp(\030)3273 3499 y Fs(=)3396 3495 y(0)456 3614 y(\(mo)s(d)i Fq(n)p Fs(\),)i(namely)f(for)g Fq(q)h Fs(=)29 b Fq(q)1586 3629 y Fn(0)1656 3586 y Fp(\030)1657 3618 y Fs(=)1763 3614 y Fq(pt)34 b Fs(\(mo)s(d)e Fq(n)p Fs(\).)47 b(F)-8 b(or)33 b(this)h Fq(q)2720 3629 y Fn(0)2759 3614 y Fs(,)g Fq(u)2876 3578 y Fo(p)p Fn(+)p Fo(q)2999 3587 y Fm(0)3032 3578 y Fn(\()p Fo(s)p Fk(\000)p Fn(1\))3244 3614 y Fs(=)29 b Fq(\014)3410 3578 y Fo(b)456 3730 y Fs(for)k(some)g(b;)i(th)m(us)f Fq(D)1266 3694 y Fo(q)1298 3703 y Fm(0)1359 3730 y Fp(\001)23 b Fq(u)1466 3694 y Fo(p)1534 3730 y Fs(=)30 b Fq(\016)1687 3694 y Fo(q)1719 3703 y Fm(0)1757 3730 y Fq(\025)1814 3745 y Fo(q)1846 3754 y Fm(0)1880 3745 y Fo(;p)1939 3730 y Fq(\014)2000 3694 y Fo(b)2034 3730 y Fs(,)k(where)h Fq(\025)2435 3745 y Fo(q)r(;p)2562 3730 y Fs(is)e(the)h(pro)s(duct)g(of)f(the)456 3846 y(summation)d(terms.)555 3962 y(W)-8 b(e)29 b(m)m(ust)f(also)e (consider)j(the)f(co)s(e\016cien)m(ts)h Fq(\025)2231 3977 y Fo(q)r(;p)2324 3962 y Fs(.)42 b(A)28 b(giv)m(en)g(summation)e (term)456 4022 y Fi(P)561 4048 y Fn(\()p Fo(p)p Fk(\000)p Fn(1\)+\()p Fo(r)r Fk(\000)p Fn(1\)\()p Fo(s)p Fk(\000)p Fn(1\))561 4126 y Fo(i)585 4134 y Fh(r)619 4126 y Fn(=0)1174 4097 y Fq(\013)1237 4061 y Fo(i)1261 4069 y Fh(r)1330 4097 y Fs(equals)32 b(0)e(only)h(if)f(the)i(n)m(um)m(b)s(er)f(of)g (summands)g(is)g(divis-)456 4217 y(ible)d(b)m(y)i Fq(n)p Fs(;)h(equiv)-5 b(alen)m(tly)29 b Fq(p)16 b Fs(+)g(\()p Fq(r)j Fp(\000)d Fs(1\)\()p Fq(s)g Fp(\000)g Fs(1\))2170 4190 y Fp(\030)2171 4221 y Fs(=)2275 4217 y(0)30 b(\(mo)s(d)e Fq(n)p Fs(\).)43 b(By)30 b(the)g(previous)456 4333 y(computation,)41 b(this)f(means)g(that)g Fq(r)30 b Fp(\000)e Fs(1)41 b(=)g Fq(q)2214 4348 y Fn(0)2254 4333 y Fs(,)h(or)e(that)g Fq(r)k Fs(=)d Fq(q)2917 4348 y Fn(0)2984 4333 y Fs(+)27 b(1.)67 b(Th)m(us)456 4450 y Fq(\025)513 4465 y Fo(q)r(;p)649 4450 y Fp(6)p Fs(=)43 b(0)e(for)g Fq(q)47 b Fp(\024)d Fq(q)1270 4465 y Fn(0)1310 4450 y Fs(,)f(and)f Fq(\025)1636 4465 y Fo(q)1668 4474 y Fm(0)1702 4465 y Fn(+1)p Fo(;p)1895 4450 y Fs(=)h(0.)70 b(Consequen)m(tly)44 b Fq(D)2858 4413 y Fo(q)2924 4450 y Fp(\001)28 b Fq(u)3036 4413 y Fo(p)3119 4450 y Fs(=)43 b(0)e(for)456 4566 y Fq(q)499 4581 y Fn(0)566 4566 y Fq(<)28 b(q)j Fp(\024)d Fq(n)p Fs(,)33 b(and)g(so)f Fq(D)1360 4530 y Fo(n)1429 4566 y Fp(\001)22 b Fq(A)28 b Fs(=)f(0.)555 4682 y(No)m(w)32 b(if)d Fq(\014)34 b Fs(=)27 b(0,)k(then)h Fq(D)1467 4646 y Fo(q)1499 4655 y Fm(0)1555 4682 y Fp(\001)18 b Fq(u)1657 4646 y Fo(p)1724 4682 y Fs(=)28 b(0)i(;)i(in)e(particular)f Fq(D)2610 4646 y Fo(n)p Fk(\000)p Fn(1)2774 4682 y Fs(=)f(0.)42 b(Ho)m(w)m(ev)m(er)33 b(if)456 4798 y Fq(\014)g Fp(6)p Fs(=)27 b(0,)33 b(then)g Fq(D)1062 4762 y Fo(q)1122 4798 y Fp(\001)22 b Fq(u)1228 4762 y Fo(p)1295 4798 y Fp(6)p Fs(=)27 b(0)32 b(for)g Fq(q)g Fp(\024)c Fq(q)1851 4813 y Fn(0)1891 4798 y Fs(.)43 b(This)33 b(\014nishes)g(the)g(pro)s(of.)p 3374 4915 4 66 v 3378 4852 59 4 v 3378 4915 V 3436 4915 4 66 v 456 5099 a Fu(Theorem)k(2.2.)49 b Fg(Assume)36 b(that)g Fq(A)29 b Fs(=)g Fq(k)s Fs(\()p Fq(u)p Fs(\))36 b Fg(is)f(an)g Fq(n)p Fg(-dimensional)f Fq(k)s Fg(-algebr)-5 b(a.)456 5216 y(L)g(et)36 b Fq(\033)j Fg(b)-5 b(e)36 b(an)f(automorphism)g(of)g Fq(A)h Fg(such)g(that)g Fq(\033)2326 5179 y Fo(n)2402 5216 y Fs(=)29 b Fq(id)36 b Fg(and)f Fq(\033)27 b Fp(\001)c Fq(u)28 b Fs(=)i Fq(\013)q(u)p Fg(,)35 b(for)p eop %%Page: 5 5 5 4 bop 3406 233 a Fl(5)456 425 y Fg(some)41 b Fq(\013)g Fp(2)g Fq(k)s Fg(,)i(and)f(let)g Fq(D)i Fg(b)-5 b(e)42 b(a)f(non-trivial)g Fq(\033)t Fg(-skew)g(derivation)g(of)h Fq(A)g Fg(such)456 541 y(that)i Fq(D)s(\033)49 b Fs(=)c Fq(\026\033)t(D)s Fg(,)h(for)e Fq(\026)g Fg(a)g(primitive)g Fq(n)2098 505 y Fo(th)2213 541 y Fg(r)-5 b(o)g(ot)44 b(of)g(1.)73 b(Then)43 b(ther)-5 b(e)45 b(exists)456 658 y Fq(\014)6 b(;)17 b(\016)31 b Fp(2)d Fq(k)36 b Fg(with)c Fq(\016)g Fp(6)p Fs(=)c(0)k Fg(and)h Fs(0)27 b Fp(\024)h Fq(s;)17 b(t)28 b Fp(\024)g Fq(n)18 b Fp(\000)h Fs(1)32 b Fg(with)h Fq(t)p Fs(\(1)18 b Fp(\000)g Fq(s)p Fs(\))2722 630 y Fp(\030)2723 662 y Fs(=)2827 658 y(1\(mo)s(d)26 b Fq(n)p Fs(\))34 b Fg(such)456 774 y(that)555 890 y(\(a\))h Fq(u)776 854 y Fo(n)850 890 y Fs(=)27 b Fq(\014)555 1006 y Fg(\(b\))35 b Fq(\013)28 b Fs(=)g Fq(\026)968 970 y Fo(t)1032 1006 y Fg(and)34 b Fq(\033)39 b Fg(has)34 b(or)-5 b(der)35 b Fq(n)g Fg(in)f Fs(Aut)2123 1021 y Fo(k)2166 1006 y Fs(\()p Fq(A)p Fs(\))p Fg(;)555 1123 y(\(c\))h Fq(D)24 b Fp(\001)e Fq(u)27 b Fs(=)h Fq(\016)t(u)1160 1086 y Fo(s)1196 1123 y Fg(.)555 1239 y(\(d\))35 b Fq(D)804 1203 y Fo(n)878 1239 y Fs(=)28 b(0)p Fg(,)34 b(and)g Fq(D)1368 1203 y Fo(n)p Fk(\000)p Fn(1)1533 1239 y Fp(6)p Fs(=)27 b(0)63 b Fp(\()-17 b(\))62 b Fq(\014)33 b Fp(6)p Fs(=)28 b(0)p Fg(.)456 1355 y(Conversely,)46 b(if)e Fq(u)1155 1319 y Fo(n)1248 1355 y Fs(=)i Fq(\014)51 b Fp(2)c Fq(k)s Fg(,)g(then)d(for)h(any)g(such)f(choic)-5 b(e)44 b(of)g Fs(0)i Fp(6)p Fs(=)g Fq(\016)k Fp(2)c Fq(k)456 1471 y Fg(and)38 b Fq(s;)17 b(t)38 b Fg(with)g Fq(t)p Fs(\(1)25 b Fp(\000)h Fq(s)p Fs(\))1395 1444 y Fp(\030)1396 1475 y Fs(=)1507 1471 y(1)39 b Fg(\(mo)-5 b(d)37 b Fq(n)p Fg(\),)j(ther)-5 b(e)38 b(is)h(an)f(automorphism)f Fq(\033)43 b Fg(of)38 b Fq(A)456 1588 y Fg(given)32 b(by)h Fq(\033)23 b Fp(\001)18 b Fq(u)27 b Fs(=)h Fq(\026)1203 1551 y Fo(t)1232 1588 y Fq(u)33 b Fg(and)f(a)h Fq(\033)t Fg(-skew)g(derivation)f(of)h Fq(A)g Fg(given)g(by)g Fq(D)21 b Fp(\001)d Fq(u)28 b Fs(=)f Fq(\016)t(u)3408 1551 y Fo(s)456 1704 y Fg(satisfying)34 b Fq(D)s(\033)d Fs(=)d Fq(\026\033)t(D)37 b Fg(and)e Fq(D)1673 1668 y Fo(n)1747 1704 y Fs(=)27 b(0)p Fq(:)456 1860 y Fg(Pr)-5 b(o)g(of.)41 b Fs(First)29 b(note)h(that)g(since)g (dim\()p Fq(A)p Fs(\))d(=)g Fq(n)p Fs(,)k(the)g(elemen)m(ts)f Fp(f)p Fs(1)p Fq(;)17 b(u;)g(:)g(:)g(:)31 b(;)17 b(u)3258 1824 y Fo(n)p Fk(\000)p Fn(1)3395 1860 y Fp(g)456 1985 y Fs(are)32 b(a)g(basis)h(for)f Fq(A)p Fs(.)44 b(Th)m(us)34 b(for)e(some)g Fq(\016)1915 2000 y Fo(i)1971 1985 y Fp(2)c Fq(k)s Fs(,)33 b Fq(D)25 b Fp(\001)d Fq(u)27 b Fs(=)2521 1910 y Fi(P)2627 1936 y Fo(n)p Fk(\000)p Fn(1)2627 2014 y Fo(i)p Fn(=0)2780 1985 y Fq(\016)2823 2000 y Fo(i)2852 1985 y Fq(u)2908 1949 y Fo(i)2935 1985 y Fs(.)555 2102 y(No)m(w)47 b Fq(D)s(\033)54 b Fs(=)c Fq(\026\033)t(D)f Fs(implies)43 b(that)j Fq(\013)q Fs(\()2028 2027 y Fi(P)2133 2131 y Fo(i)2177 2102 y Fq(\016)2220 2117 y Fo(i)2249 2102 y Fq(u)2305 2066 y Fo(i)2333 2102 y Fs(\))k(=)g Fq(\026)p Fs(\()2644 2027 y Fi(P)2749 2131 y Fo(i)2794 2102 y Fq(\016)2837 2117 y Fo(i)2865 2102 y Fq(\013)2928 2066 y Fo(i)2956 2102 y Fq(u)3012 2066 y Fo(i)3040 2102 y Fs(\),)f(and)d(so)456 2220 y Fq(\016)499 2235 y Fo(i)527 2220 y Fs(\()p Fq(\013)628 2183 y Fn(1)p Fk(\000)p Fo(i)758 2220 y Fp(\000)12 b Fq(\026)p Fs(\))28 b(=)g(0)f(for)g(all)f Fq(i)p Fs(.)42 b(Since)27 b Fq(D)15 b Fp(\001)d Fq(u)27 b Fp(6)p Fs(=)h(0,)g(w)m(e)h(m)m(ust)f(ha)m(v)m(e)h Fq(\016)2844 2235 y Fo(s)2908 2220 y Fp(6)p Fs(=)f(0)f(for)g(some)456 2336 y Fq(s)p Fs(,)38 b(and)f(so)h Fq(\026)d Fs(=)g Fq(\013)1154 2300 y Fn(1)p Fk(\000)p Fo(s)1281 2336 y Fs(.)58 b(Since)37 b Fq(\033)1684 2300 y Fo(n)1766 2336 y Fs(=)f Fq(id)p Fs(,)i Fq(\013)2090 2300 y Fo(n)2173 2336 y Fs(=)d(1)i(and)g(so)h(1)25 b Fp(\000)h Fq(s)37 b Fs(is)f(relativ)m(ely)456 2452 y(prime)30 b(to)g Fq(n)p Fs(.)44 b(Th)m(us)32 b Fq(o)p Fs(\()p Fq(\013)q Fs(\))27 b(=)h Fq(n)j Fs(and)g Fq(\033)k Fs(has)d(order)f Fq(n)h Fs(in)e(Aut)2699 2467 y Fo(k)2742 2452 y Fs(\()p Fq(A)p Fs(\).)43 b(Moreo)m(v)m(er)32 b(if)456 2568 y Fq(t)g Fs(is)g(the)h(in)m(v)m(erse)h(of)f(1)21 b Fp(\000)i Fq(s)33 b Fs(mo)s(d)e Fq(n)p Fs(,)i(then)g Fq(\013)28 b Fs(=)g Fq(\026)2285 2532 y Fo(t)2314 2568 y Fs(.)555 2685 y(Also)36 b Fq(\026)d Fs(=)g Fq(\013)1043 2648 y Fn(1)p Fk(\000)p Fo(s)1203 2685 y Fs(=)g Fq(\013)1375 2648 y Fn(1)p Fk(\000)p Fo(i)1529 2685 y Fs(is)j(imp)s(ossible)d(if)i Fq(i)f Fp(6)p Fs(=)f Fq(s)p Fs(,)k(and)f(so)g Fq(\016)2847 2700 y Fo(i)2909 2685 y Fs(=)d(0)j(if)e Fq(i)g Fp(6)p Fs(=)f Fq(s)p Fs(.)456 2801 y(Th)m(us)h Fq(D)25 b Fp(\001)c Fq(u)28 b Fs(=)f Fq(\016)t(u)1148 2765 y Fo(s)1184 2801 y Fs(,)33 b(where)h Fq(\016)d Fs(=)d Fq(\016)1747 2816 y Fo(s)1784 2801 y Fs(.)555 2917 y(W)-8 b(e)46 b(claim)c(that)j Fq(u)1288 2881 y Fo(n)1383 2917 y Fs(=)k Fq(\014)h Fs(for)45 b(some)f Fq(\014)55 b Fp(2)49 b Fq(k)s Fs(.)81 b(T)-8 b(o)45 b(see)h(this,)h(let)e Fq(f)11 b Fs(\()p Fq(x)p Fs(\))49 b(=)456 3033 y Fq(x)511 2997 y Fo(n)570 3033 y Fs(+)12 b Fq(\013)720 3048 y Fo(n)p Fk(\000)p Fn(1)857 3033 y Fq(x)912 2997 y Fo(n)p Fk(\000)p Fn(1)1061 3033 y Fs(+)g Fp(\001)17 b(\001)g(\001)10 b Fs(+)i Fq(\013)1427 3048 y Fn(0)1466 3033 y Fs(,)29 b(where)g(all)c Fq(\013)1991 3048 y Fo(i)2047 3033 y Fp(2)j Fq(k)s Fs(,)h(b)s(e)f(the)g(minim)m(um)c (p)s(olynomial)456 3149 y(of)32 b Fq(u)p Fs(.)43 b(Then)33 b Fq(f)11 b Fs(\()p Fq(\033)t Fs(\()p Fq(u)p Fs(\)\))27 b(=)g Fq(f)11 b Fs(\()p Fq(u)p Fs(\))27 b(=)h(0.)43 b(Since)33 b Fq(\013)g Fs(is)f(a)g(primitiv)m(e)e Fq(n)2848 3113 y Fo(th)2951 3149 y Fs(ro)s(ot)i(of)g(1,)g(it)456 3266 y(follo)m(ws)f(that)h Fq(\013)1049 3281 y Fn(1)1089 3266 y Fq(;)17 b(:)g(:)g(:)32 b(;)17 b(\013)1386 3281 y Fo(n)p Fk(\000)p Fn(1)1551 3266 y Fs(=)27 b(0.)43 b(Th)m(us)35 b Fq(f)11 b Fs(\()p Fq(x)p Fs(\))27 b(=)h Fq(x)22 b Fs(+)g Fq(\013)2579 3281 y Fn(0)2619 3266 y Fs(;)33 b(no)m(w)g(set)g Fq(\014)g Fs(=)28 b Fp(\000)p Fq(\013)3365 3281 y Fn(0)3405 3266 y Fs(.)555 3382 y(P)m(art)43 b(\(d\))g(follo)m(ws)f(from)f(Lemma)h (2.1)g(\(c\),)j(and)e(the)h(con)m(v)m(erse)h(statemen)m(t)456 3498 y(also)31 b(follo)m(ws)h(from)f(Lemma)g(2.1.)p 3374 3614 4 66 v 3378 3552 59 4 v 3378 3614 V 3436 3614 4 66 v 456 3780 a(W)-8 b(e)22 b(see)h(next)f(that)g(when)h Fq(A)e Fs(has)i(no)e(nilp)s(oten)m(t)g(elemen)m(ts)h(and)g(is)f Fq(n)p Fs(-dimensional,)456 3897 y(w)m(e)34 b(ma)m(y)f(assume)h(that)f Fq(A)g Fs(is)g(of)g(the)h(form)e Fq(A)d Fs(=)g Fq(k)s Fs(\()p Fq(u)p Fs(\))k(where)h Fq(\033)27 b Fp(\001)22 b Fq(u)29 b Fs(=)f Fq(\013)q(u)p Fs(,)33 b(and)456 4013 y(so)f(Theorem)h(2.2)f(applies.)456 4169 y Fu(Theorem)37 b(2.3.)49 b Fg(L)-5 b(et)28 b Fq(A)g Fg(b)-5 b(e)28 b(an)g(arbitr)-5 b(ary)28 b Fq(k)s Fg(-algebr)-5 b(a)28 b(with)g(no)f(non-zer)-5 b(o)27 b(nilp)-5 b(o-)456 4286 y(tent)36 b(elements)f(and)h(assume)f (that)h Fq(\033)e Fp(2)i Fg(A)n(ut)2136 4301 y Fo(k)2180 4286 y Fs(\()p Fq(A)p Fs(\))g Fg(such)g(that)g Fq(\033)2847 4250 y Fo(n)2924 4286 y Fs(=)29 b Fq(id)p Fg(.)48 b(L)-5 b(et)37 b Fq(D)456 4402 y Fg(b)-5 b(e)41 b(a)g(non-trivial)g Fq(\033)t Fg(-skew)g(derivation)f(of)i Fq(A)f Fg(such)h(that)g Fq(D)s(\033)i Fs(=)39 b Fq(\026\033)t(D)s Fg(,)k(wher)-5 b(e)456 4518 y Fq(\026)44 b Fg(is)h(a)g(primitive)f Fq(n)1254 4482 y Fo(th)1370 4518 y Fg(r)-5 b(o)g(ot)45 b(of)g(1.)75 b(Then)44 b Fq(D)2206 4482 y Fo(n)p Fk(\000)p Fn(1)2389 4518 y Fp(6)p Fs(=)j(0)p Fg(,)g(and)e(al)5 b(l)44 b(the)h(p)-5 b(owers)456 4634 y Fq(\026)515 4598 y Fo(i)542 4634 y Fq(;)17 b Fs(0)28 b Fp(\024)g Fq(i)g Fp(\024)g Fq(n)22 b Fp(\000)h Fs(1)p Fg(,)34 b(ar)-5 b(e)35 b(eigenvalues)f(of)g Fq(\033)t Fg(.)555 4751 y(If)46 b(mor)-5 b(e)g(over)45 b Fq(A)h Fg(is)f Fq(n)p Fg(-dimensional,)i(then)f(ther)-5 b(e)46 b(exist)f Fq(u)j Fp(2)g Fq(A)e Fg(and)g Fs(0)i Fp(6)p Fs(=)456 4867 y Fq(\014)6 b(;)17 b(\015)32 b Fp(2)c Fq(k)38 b Fg(such)c(that)456 4983 y(\(a\))g Fq(A)28 b Fs(=)f Fq(k)s Fs(\()p Fq(u)p Fs(\))35 b Fg(and)f Fq(u)1290 4947 y Fo(n)1364 4983 y Fs(=)28 b Fq(\014)33 b Fp(2)28 b Fq(k)s Fg(;)456 5099 y(\(b\))34 b Fq(\033)26 b Fp(\001)c Fq(u)27 b Fs(=)h Fq(\026u)p Fg(,)34 b(and)g(so)h Fq(\033)j Fg(has)d(or)-5 b(der)34 b Fq(n)h Fg(in)g(A)n(ut)2317 5114 y Fo(k)2360 5099 y Fs(\()p Fq(A)p Fs(\))p Fg(;)456 5216 y(\(c\))f Fq(D)25 b Fp(\001)d Fq(u)27 b Fs(=)g Fq(\015)5 b Fs(1)p Fg(,)35 b Fq(D)1211 5179 y Fo(n)p Fk(\000)p Fn(1)1375 5216 y Fp(6)p Fs(=)28 b(0)p Fg(,)35 b(and)f Fq(D)1866 5179 y Fo(n)1940 5216 y Fs(=)28 b(0)p Fg(.)p eop %%Page: 6 6 6 5 bop 456 236 a Fl(6)661 b(S.)33 b(MONTGOMER)-6 b(Y)33 b(AND)f(H.-J.)i(SCHNEIDER)456 425 y Fg(In)g(p)-5 b(articular)35 b Fq(A)g Fg(is)f(c)-5 b(ommutative.)555 541 y(Mor)g(e)g(over)35 b(if)g Fq(u)1135 505 y Fk(0)1186 541 y Fp(2)28 b Fq(A)35 b Fg(and)g Fq(\014)1639 505 y Fk(0)1662 541 y Fq(;)17 b(\015)1762 505 y Fk(0)1813 541 y Fp(2)28 b Fq(k)38 b Fg(also)d(satisfy)g(\(a\),)f(\(b\),)h(and)f(\(c\),)h(then)456 658 y(for)f(some)g Fs(0)28 b Fp(6)p Fs(=)f Fq(\027)34 b Fp(2)29 b Fq(k)s Fg(,)34 b(we)h(have)f Fq(u)1760 622 y Fk(0)1811 658 y Fs(=)27 b Fq(\027)6 b(u;)17 b(\014)2129 622 y Fk(0)2179 658 y Fs(=)28 b Fq(\027)2337 622 y Fo(n)2384 658 y Fq(\014)6 b Fg(,)35 b(and)f Fq(\015)2755 622 y Fk(0)2806 658 y Fs(=)28 b Fq(\027)6 b(\015)f Fg(.)555 907 y Fs(Before)37 b(pro)m(ving)f(the)h(theorem,)g(w)m(e)h(\014rst)e (mak)m(e)h(some)f(preliminary)e(obser-)456 1023 y(v)-5 b(ations.)46 b(Since)34 b Fq(\033)1146 987 y Fo(n)1223 1023 y Fs(=)c Fq(id)k Fs(and)g Fq(k)i Fs(con)m(tains)e(a)g(primitiv)m (e)e Fq(n)2669 987 y Fo(th)2773 1023 y Fs(ro)s(ot)h(of)g(1,)h Fq(\033)k Fs(can)456 1140 y(b)s(e)32 b(diagonalized)f(in)g(End)1434 1155 y Fo(k)1478 1140 y Fs(\()p Fq(A)p Fs(\).)43 b(Th)m(us)1676 1429 y Fq(A)28 b Fs(=)g Fp(\010)1958 1388 y Fo(n)p Fk(\000)p Fn(1)1958 1455 y Fo(i)p Fn(=0)2095 1429 y Fq(A)2168 1444 y Fo(i)2197 1429 y Fq(;)456 1718 y Fs(where)38 b Fq(A)815 1733 y Fo(i)879 1718 y Fs(:=)e Fp(f)p Fq(a)f Fp(2)h Fq(A)p Fp(j)p Fq(\033)29 b Fp(\001)c Fq(a)36 b Fs(=)g Fq(\026)1752 1682 y Fo(i)1779 1718 y Fq(a)p Fp(g)p Fs(.)58 b(Note)37 b(that)g Fq(D)28 b Fp(\001)d Fq(A)2656 1733 y Fo(i)2720 1718 y Fp(\022)36 b Fq(A)2906 1733 y Fo(i)p Fk(\000)p Fn(1)3025 1718 y Fs(,)i(for)f(all)e Fq(i)p Fs(.)456 1834 y(F)-8 b(or,)42 b(if)e Fq(a)j Fp(2)f Fq(A)1041 1849 y Fo(i)1070 1834 y Fs(,)h(then)f Fq(\033)32 b Fp(\001)27 b Fs(\()p Fq(D)k Fp(\001)d Fq(a)p Fs(\))42 b(=)g Fq(\026)2027 1798 y Fk(\000)p Fn(1)2121 1834 y Fq(D)31 b Fp(\001)c Fs(\()p Fq(\033)32 b Fp(\001)c Fq(a)p Fs(\))42 b(=)g Fq(\026)2777 1798 y Fo(i)p Fk(\000)p Fn(1)2895 1834 y Fs(\()p Fq(D)31 b Fp(\001)d Fq(a)p Fs(\);)45 b(th)m(us)456 1950 y Fq(D)24 b Fp(\001)e Fq(a)28 b Fp(2)g Fq(A)857 1965 y Fo(i)p Fk(\000)p Fn(1)976 1950 y Fs(.)456 2200 y Fu(Lemma)37 b(2.4.)49 b Fg(L)-5 b(et)35 b Fq(a)29 b Fp(2)g Fq(A)1492 2215 y Fo(i)1555 2200 y Fg(such)35 b(that)h Fq(D)25 b Fp(\001)d Fq(a)28 b Fp(6)p Fs(=)h(0)35 b Fg(but)g Fq(D)2649 2164 y Fn(2)2711 2200 y Fp(\001)22 b Fq(a)28 b Fs(=)h(0)p Fg(.)45 b(Then)35 b(for)456 2316 y(al)5 b(l)34 b Fq(m)h Fg(with)g Fs(1)27 b Fp(\024)i Fq(m)f(<)f(n)p Fg(,)555 2432 y(\(i\))35 b Fq(D)784 2396 y Fo(m)872 2432 y Fp(\001)22 b Fq(a)973 2396 y Fo(m)1067 2432 y Fs(=)28 b Fq(\015)1222 2447 y Fo(m)1288 2432 y Fs(\()p Fq(D)d Fp(\001)d Fq(a)p Fs(\))1571 2396 y Fo(m)1637 2432 y Fg(,)35 b(for)g(some)f Fs(0)27 b Fp(6)p Fs(=)h Fq(\015)2338 2447 y Fo(m)2432 2432 y Fp(2)g Fq(k)s Fg(;)555 2548 y(\(ii\))35 b Fq(D)814 2512 y Fo(m)902 2548 y Fp(\001)22 b Fq(a)1003 2512 y Fo(m)p Fk(\000)p Fn(1)1187 2548 y Fs(=)28 b(0)p Fq(:)456 2665 y Fg(Conse)-5 b(quently)34 b Fq(D)1130 2629 y Fo(n)p Fk(\000)p Fn(1)1289 2665 y Fp(\001)22 b Fq(A)27 b Fp(6)p Fs(=)h(0)35 b Fg(and)f Fq(A)1889 2680 y Fo(j)1953 2665 y Fp(6)p Fs(=)28 b(0)34 b Fg(for)h(al)5 b(l)35 b(j.)456 2914 y(Pr)-5 b(o)g(of.)41 b Fs(By)h(induction)f(on)h Fq(m)p Fs(.)73 b(The)43 b(case)g Fq(m)h Fs(=)g(1)e(is)f(giv)m(en,)k(as) d(is)g(the)g(case)456 3030 y Fq(m)35 b Fs(=)g(2)i(for)g(\(ii\).)54 b(W)-8 b(e)38 b(\014rst)f(sho)m(w)i(that)d(if)g(\(i\)is)g(true)h(for)g Fq(m)25 b Fp(\000)h Fs(1)37 b(and)g Fq(m)p Fs(,)h(and)456 3147 y(\(ii\))30 b(is)i(true)h(for)f Fq(m)p Fs(,)h(then)g(\(ii\))e(is)h (true)h(for)f Fq(m)23 b Fs(+)f(1.)43 b(No)m(w)521 3436 y Fq(D)605 3395 y Fo(m)p Fn(+1)784 3436 y Fp(\001)22 b Fq(a)885 3395 y Fo(m)979 3436 y Fs(=)28 b Fq(D)c Fp(\001)e Fs(\()p Fq(D)1360 3395 y Fo(m)1449 3436 y Fp(\001)f Fq(a)1549 3395 y Fo(m)1616 3436 y Fs(\))28 b(=)f Fq(\015)1836 3451 y Fo(m)1902 3436 y Fq(D)e Fp(\001)d Fs(\()p Fq(D)j Fp(\001)d Fq(a)p Fs(\))2341 3395 y Fo(m)604 3582 y Fs(=)83 b Fq(\015)814 3597 y Fo(m)880 3582 y Fs([\()p Fq(D)25 b Fp(\001)d Fs(\()p Fq(D)j Fp(\001)d Fq(a)p Fs(\)\)\()p Fq(D)i Fp(\001)e Fq(a)p Fs(\))1704 3541 y Fo(m)p Fk(\000)p Fn(1)1883 3582 y Fs(+)g(\()p Fq(\033)k Fp(\001)c Fs(\()p Fq(D)j Fp(\001)d Fq(a)p Fs(\)\)\()p Fq(D)j Fp(\001)c Fs(\()p Fq(D)k Fp(\001)d Fq(a)p Fs(\))2947 3541 y Fo(m)p Fk(\000)p Fn(1)3104 3582 y Fs(\)])604 3729 y(=)83 b Fq(\015)814 3744 y Fo(m)880 3729 y Fs([\()p Fq(D)1029 3687 y Fn(2)1091 3729 y Fp(\001)21 b Fq(a)p Fs(\)\()p Fq(D)k Fp(\001)d Fq(a)p Fs(\))1512 3687 y Fo(m)p Fk(\000)p Fn(1)1669 3729 y Fs(\))g(+)g(\()p Fq(\033)k Fp(\001)c Fs(\()p Fq(D)j Fp(\001)d Fq(a)p Fs(\)\))p Fq(\015)2373 3687 y Fk(\000)p Fn(1)2368 3753 y Fo(m)p Fk(\000)p Fn(1)2524 3729 y Fs(\()p Fq(D)j Fp(\001)d Fs(\()p Fq(D)2840 3687 y Fo(m)p Fk(\000)p Fn(1)3018 3729 y Fp(\001)g Fq(a)3119 3687 y Fo(m)p Fk(\000)p Fn(1)3276 3729 y Fs(\)\)])604 3874 y(=)83 b(0)p Fq(:)456 4162 y Fs(Here)35 b(w)m(e)g(ha)m(v)m(e)g (used)h(\(i\))d(for)h Fq(m)g Fs(in)f(the)i(\014rst)g(line,)e(\(i\))g (for)h Fq(m)24 b Fp(\000)f Fs(1)34 b(in)g(the)g(next)456 4279 y(to)f(last)g(line,)g Fq(D)1059 4242 y Fn(2)1121 4279 y Fp(\001)23 b Fq(a)29 b Fs(=)h(0,)k(and)g(\(ii\))e(for)h Fq(m)h Fs(to)f(see)i(that)f Fq(D)25 b Fp(\001)e Fs(\()p Fq(D)2863 4242 y Fo(m)p Fk(\000)p Fn(1)3042 4279 y Fp(\001)g Fq(a)3144 4242 y Fo(m)p Fk(\000)p Fn(1)3301 4279 y Fs(\))30 b(=)456 4395 y Fq(D)540 4359 y Fo(m)628 4395 y Fp(\001)22 b Fq(a)729 4359 y Fo(m)p Fk(\000)p Fn(1)913 4395 y Fs(=)28 b(0.)555 4511 y(W)-8 b(e)31 b(no)m(w)g(sho)m(w)h(that)e(if)g(\(i\))f (is)h(true)h(for)f Fq(m)h Fs(and)g(\(ii\))e(is)h(true)h(for)f Fq(m)18 b Fs(+)g(1,)31 b(then)456 4627 y(\(i\))36 b(is)i(true)g(for)g Fq(m)26 b Fs(+)f(1.)60 b(Since)38 b Fq(D)i Fs(is)e(a)g Fq(\033)t Fs(-deriv)-5 b(ation)35 b(with)j Fq(D)s(\033)i Fs(=)d Fq(\026\033)t(D)s Fs(,)i(for)456 4743 y Fq(b;)17 b(c)27 b Fp(2)i Fq(A)j Fs(w)m(e)i(ha)m(v)m(e)1245 5111 y Fq(D)1329 5069 y Fo(p)1369 5111 y Fs(\()p Fq(bc)p Fs(\))28 b(=)1713 4982 y Fo(p)1659 5016 y Fi(X)1670 5226 y Fo(j)t Fn(=0)1820 5030 y Fi(\000)1865 5066 y Fo(p)1867 5145 y(j)1901 5030 y Fi(\001)1947 5162 y Fo(\026)1993 5111 y Fq(\033)2052 5074 y Fo(j)2089 5111 y Fq(D)2173 5074 y Fo(p)p Fk(\000)p Fo(j)2299 5111 y Fs(\()p Fq(b)p Fs(\))p Fq(D)2500 5074 y Fo(j)2537 5111 y Fs(\()p Fq(c)p Fs(\))p eop %%Page: 7 7 7 6 bop 3406 233 a Fl(7)456 427 y Fs(where)737 346 y Fi(\000)783 383 y Fo(p)785 461 y(j)819 346 y Fi(\001)864 478 y Fo(\026)943 427 y Fs(is)32 b(the)h Fq(q)t 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1724 y Fs(where)33 b Fq(\015)788 1739 y Fo(m)p Fn(+1)972 1724 y Fs(=)1076 1643 y Fi(\000)1122 1680 y Fo(m)p Fn(+1)1180 1758 y(1)1274 1643 y Fi(\001)1320 1763 y Fo(\026)1366 1724 y Fq(\015)1417 1739 y Fo(m)1484 1724 y Fq(\026)1543 1688 y Fo(m)p Fn(\()p Fo(i)p Fk(\000)p Fn(1\))1806 1724 y Fp(6)p Fs(=)27 b(0)p Fq(;)g Fs(since)33 b Fq(D)25 b Fp(\001)d Fq(a)28 b Fp(2)g Fq(A)2653 1739 y Fo(i)p Fk(\000)p Fn(1)2771 1724 y Fs(.)555 1859 y(This)33 b(pro)m(v)m(es)h(parts)f (\(i\))f(and)g(\(ii\).)555 1975 y(No)m(w)45 b(if)e Fq(D)974 1939 y Fo(n)p Fk(\000)p Fn(1)1141 1975 y Fp(\001)29 b Fq(A)47 b Fs(=)h(0,)e(w)m(e)f(ma)m(y)f(c)m(ho)s(ose)h Fq(a)f Fs(suc)m(h)i(that)e Fq(D)32 b Fp(\001)e Fq(a)47 b Fp(6)p Fs(=)g(0)d(but)456 2091 y Fq(D)540 2055 y Fn(2)587 2091 y Fp(\001)8 b Fq(a)27 b Fs(=)h(0,)f(where)g Fq(a)g Fp(2)i Fq(A)1429 2106 y Fo(i)1482 2091 y Fs(for)d(some)f Fq(i)p Fs(.)41 b(By)27 b(\(i\))d(ab)s(o)m(v)m(e,)k Fq(D)2620 2055 y Fo(m)2694 2091 y Fp(\001)8 b Fq(a)2781 2055 y Fo(m)2876 2091 y 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b(e)30 b(ma)m(y)e(th)m(us)i(c)m(ho)s(ose)g Fq(a)e Fp(2)g Fq(A)1690 2571 y Fo(i)1719 2556 y Fs(,)h(for)g(some)g Fq(i)p Fs(,)h(suc)m(h)g(that)f Fq(D)2760 2520 y Fo(n)p Fk(\000)p Fn(1)2912 2556 y Fp(\001)15 b Fq(a)27 b Fp(6)p Fs(=)h(0.)42 b(No)m(w)456 2673 y(0)27 b Fp(6)p Fs(=)h Fq(D)720 2636 y Fo(j)771 2673 y Fp(\001)15 b Fq(a)27 b Fp(2)h Fq(A)1059 2688 y Fo(i)p Fk(\000)p Fo(j)1175 2673 y Fs(,)h(for)g(all)e Fq(j)33 b Fs(=)28 b(0)p Fq(;)17 b(:)g(:)g(:)32 b(;)17 b(n)e Fp(\000)g Fs(1,)30 b(and)f(so)g Fq(A)2616 2688 y Fo(j)2680 2673 y Fp(6)p Fs(=)f(0)h(for)f(all)f Fq(j)6 b Fs(.)42 b(This)456 2789 y(pro)m(v)m(es)34 b(the)f(lemma.)p 3374 2789 4 66 v 3378 2726 59 4 v 3378 2789 V 3436 2789 4 66 v 555 2990 a(T)-8 b(o)43 b(\014nish)h(the)f(pro)s(of)f(of)h (Theorem)g(2.3)g(w)m(e)h(can)f(no)m(w)h(assume)g(that)f Fq(A)g Fs(is)456 3106 y Fq(n)p Fs(-dimensional.)456 3307 y Fg(Pr)-5 b(o)g(of.)41 b Fs(By)c(Lemma)f(2.4,)i Fq(A)1527 3322 y Fo(j)1601 3307 y Fs(is)f(1-dimensional)d(for)j(all)e Fq(j)6 b 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b(Mor)-5 b(e)g(over)36 b Fq(A)h Fg(is)g(also)f(an)456 5216 y Fq(H)8 b Fg(-mo)-5 b(dule)33 b(algebr)-5 b(a.)p eop %%Page: 8 8 8 7 bop 456 236 a Fl(8)661 b(S.)33 b(MONTGOMER)-6 b(Y)33 b(AND)f(H.-J.)i(SCHNEIDER)555 425 y Fg(If)40 b(for)g(some)f Fq(v)i Fp(2)d Fq(A)p Fg(,)j(also)e Fq(A)f Fs(=)f Fq(k)s Fs(\()p Fq(v)t Fs(\))j Fg(and)f Fq(g)29 b Fp(\001)d Fq(v)41 b Fs(=)c Fq(\013)q(v)44 b Fg(for)c(some)f Fq(\013)f Fp(2)f Fq(k)s Fg(,)456 541 y(then)k Fq(\013)f Fs(=)g Fq(!)962 505 y Fo(t)1032 541 y Fg(for)h(some)g Fq(t)h Fg(r)-5 b(elatively)41 b(prime)g(to)g Fq(n)h Fg(and)47 b Fs(~)-55 b Fq(x)27 b Fp(\001)g Fq(v)43 b Fs(=)d Fq(\016)t(v)3095 505 y Fo(s)3131 541 y Fg(,)j(wher)-5 b(e)456 658 y Fq(t)p Fs(\(1)22 b Fp(\000)g Fq(s)p Fs(\))811 630 y Fp(\030)812 662 y Fs(=)916 658 y(1)35 b Fg(\()p Fs(mo)s(d)e Fq(n)p Fg(\),)i(for)g(some)f Fs(0)27 b Fp(6)p Fs(=)h Fq(\016)k Fp(2)c Fq(k)s Fg(.)555 834 y Fs(The)g(theorem)e(follo)m(ws)f(directly)h (from)f(Theorems)i(2.2)f(and)h(2.3,)g(noting)f(that)462 950 y(~)-55 b Fq(x)511 914 y Fo(n)580 950 y Fp(\001)22 b Fq(u)27 b Fs(=)h(0)k(giv)m(es)h(an)f(induced)i(action)d(of)h Fq(H)40 b Fs(on)33 b Fq(A)p Fs(.)555 1066 y(W)-8 b(e)34 b(close)f(this)g(section)h(with)f(an)g(example)g(to)g(sho)m(w)h(that)f (the)h(assumption)456 1182 y(of)i(no)i(non-zero)f(nilp)s(oten)m(t)f (elemen)m(ts)i(is)e(essen)m(tial)i(in)e(Theorem)i(2.5)f(\(and)g(in)456 1299 y(Theorem)32 b(2.3\))h(in)e(order)i(to)f(sho)m(w)i Fq(A)28 b Fs(=)f Fq(k)s Fs(\()p Fq(u)p Fs(\).)456 1475 y Fu(Example)36 b(2.6.)49 b Fs(\(1\))40 b(Inner)i(actions)e(of)g(the)h (T)-8 b(aft)41 b(Hopf)f(algebra)g Fq(T)3079 1492 y Fo(n)3122 1473 y Fm(2)3161 1475 y Fs(\()p Fq(!)t Fs(\))f(on)456 1591 y(an)30 b(arbitrary)g(algebra)f Fq(A)i Fs(are)f(giv)m(en)h(as)g (follo)m(ws.)41 b(Let)31 b Fq(g)g Fp(2)d Fq(A)j Fs(b)s(e)g(an)f(in)m(v) m(ertible)456 1707 y(elemen)m(t)f(and)g Fq(\033)j Fs(:)c Fq(A)f Fp(!)h Fq(A)p Fs(,)i Fq(\033)t Fs(\()p Fq(b)p Fs(\))e(=)f Fq(g)t(bg)1949 1671 y Fk(\000)p Fn(1)2072 1707 y Fs(for)i(all)e Fq(b)h Fp(2)g Fq(A)p Fs(,)i(the)g(inner)f (automor-)456 1824 y(phism)i(de\014ned)j(b)m(y)f Fq(g)t Fs(.)43 b(Then)34 b(the)f(order)f(of)g Fq(\033)k Fs(is)c(the)h (smallest)e(in)m(teger)h Fq(m)c Fp(\025)h Fs(1)456 1940 y(suc)m(h)34 b(that)f Fq(g)939 1904 y Fo(m)1038 1940 y Fs(is)f(cen)m(tral)h(in)g Fq(A)p Fs(.)45 b(F)-8 b(or)32 b(an)h(elemen)m(t)g Fq(a)c Fp(2)g Fq(A)p Fs(,)k(let)f Fq(D)g Fs(:)c Fq(A)h Fp(!)f Fq(A)33 b Fs(b)s(e)456 2056 y(the)28 b(inner)f(sk)m(ew-deriv)-5 b(ation)28 b(de\014ned)h(b)m(y)g Fq(a)f Fs(and)f Fq(\033)t Fs(,)i(that)f(is)f Fq(D)s Fs(\()p Fq(b)p Fs(\))h(=)f Fq(ab)12 b Fp(\000)g Fq(\033)t Fs(\()p Fq(b)p Fs(\))p Fq(a)456 2172 y Fs(for)32 b(all)e Fq(b)e Fp(2)g Fq(A)p Fs(.)44 b Fq(D)35 b Fs(is)d(non-zero)h(if)e(and)i(only)f (if)f Fq(g)2290 2136 y Fk(\000)p Fn(1)2384 2172 y Fq(a)i Fs(is)f(not)g(cen)m(tral)h(in)f Fq(A)p Fs(.)555 2288 y(Assume)i(that)e Fq(\033)t Fs(\()p Fq(a)p Fs(\))c(=)g Fq(!)1511 2252 y Fk(\000)p Fn(1)1605 2288 y Fq(a)p Fs(.)44 b(Then)34 b Fq(D)s(\033)d Fs(=)d Fq(!)t(\033)t(D)s Fs(,)k(and)h(if)e Fq(n)i Fs(is)g(o)s(dd,)f Fq(\033)37 b Fs(has)456 2405 y(order)32 b Fq(n)h Fs(and)g Fq(a)1042 2369 y Fo(n)1121 2405 y Fs(is)g(cen)m(tral,)f(then)h Fq(D)1875 2369 y Fo(n)1950 2405 y Fs(=)27 b(0.)456 2575 y Fg(Pr)-5 b(o)g(of.)41 b Fs(The)33 b(equalit)m(y)g Fq(D)s(\033)e Fs(=)d Fq(!)t(\033)t(D)35 b Fs(follo)m(ws)c(easily)-8 b(.)43 b(W)-8 b(rite)32 b Fq(D)f Fs(=)c Fq(L)3011 2590 y Fo(a)3076 2575 y Fp(\000)22 b Fq(R)3249 2590 y Fo(a)3291 2575 y Fq(\033)32 b Fp(2)456 2691 y Fs(End)p Fq(A)p Fs(,)42 b(where)e Fq(L)1126 2706 y Fo(a)1168 2691 y Fq(;)17 b(R)1286 2706 y Fo(a)1367 2691 y Fs(denote)40 b(left)f(and)g(righ)m(t)g(m)m(ultiplication)c(b)m (y)40 b Fq(a)p Fs(.)64 b(Then)456 2807 y Fq(!)t Fs(\()p Fq(R)633 2822 y Fo(a)674 2807 y Fq(\033)t Fs(\))p Fq(L)837 2822 y Fo(a)918 2807 y Fs(=)40 b Fq(L)1100 2822 y Fo(a)1142 2807 y Fs(\()p Fq(R)1254 2822 y Fo(a)1296 2807 y Fq(\033)t Fs(\),)h(and)f(w)m(e)h(see)g(from)d(the)i Fq(q)t Fs(-bimomial)35 b(form)m(ula)j(that)456 2923 y(for)32 b(all)e Fq(b)e Fp(2)g Fq(A)p Fs(,)581 3107 y Fq(D)665 3066 y Fo(n)712 3107 y Fs(\()p Fq(b)p Fs(\))g(=)f Fq(L)1026 3066 y Fo(n)1026 3131 y(a)1074 3107 y Fs(\()p Fq(b)p Fs(\))22 b(+)g(\()p Fp(\000)p Fq(R)1500 3122 y Fo(a)1543 3107 y Fq(\033)t Fs(\))1640 3066 y Fo(n)1686 3107 y Fs(\()p Fq(b)p Fs(\))28 b(=)g Fq(a)1986 3066 y Fo(n)2033 3107 y Fq(b)23 b Fs(+)f(\()p Fp(\000)p Fs(1\))2397 3066 y Fo(n)2444 3107 y Fq(!)2509 3066 y Fk(\000)2574 3032 y Fh(n)p Fm(\()p Fh(n)p Ff(\000)p Fm(1\))p 2573 3051 204 3 v 2659 3092 a(2)2790 3107 y Fq(\033)2849 3066 y Fo(n)2896 3107 y Fs(\()p Fq(b)p Fs(\))p Fq(a)3064 3066 y Fo(n)3139 3107 y Fs(=)28 b(0)p Fq(:)p 3374 3267 4 66 v 3378 3204 59 4 v 3378 3267 V 3436 3267 4 66 v 555 3437 a Fs(\(2\))34 b(All)f(the)i(assumptions)g(in)f(\(1\))g (can)h(b)s(e)f(realized)g(for)g(matrix)f(rings)h Fq(A)e Fs(=)456 3553 y Fq(M)550 3568 y Fo(r)588 3553 y Fs(\()p Fq(k)s Fs(\).)71 b(Assume)42 b Fq(n)i(>)f Fs(2)e(is)h(o)s(dd.)71 b(Let)41 b Fq(e)2091 3568 y Fo(ij)2152 3553 y Fs(,)j(1)f Fp(\024)h Fq(i;)17 b(j)49 b Fp(\024)44 b Fq(r)s Fs(,)g(b)s(e)e(the)g (matrix)456 3669 y(units)e(of)g Fq(A)p Fs(.)67 b(T)-8 b(ak)m(e)42 b Fq(g)j Fs(=)c Fq(!)t(e)1557 3684 y Fn(11)1659 3669 y Fs(+)1762 3594 y Fi(P)1867 3621 y Fo(r)1867 3698 y(i)p Fn(=2)2002 3669 y Fq(e)2047 3684 y Fo(ii)2100 3669 y Fs(,)h(and)f Fq(a)g Fs(=)g Fq(e)2621 3684 y Fn(1)p Fo(r)2695 3669 y Fs(.)67 b(Then)42 b(the)f(inner)456 3785 y(automorphism)24 b Fq(\033)30 b Fs(de\014ned)e(b)m(y)f Fq(g)i Fs(has)e(order)f Fq(n)p Fs(,)i Fq(\033)t Fs(\()p Fq(a)p Fs(\))g(=)f Fq(!)t(a)p Fs(,)g Fq(a)2782 3749 y Fo(n)2857 3785 y Fs(=)h(0)e(is)f(cen)m(tral,)456 3902 y(and)36 b Fq(g)700 3866 y Fk(\000)p Fn(1)794 3902 y Fq(a)f Fs(=)f Fq(!)1055 3866 y Fk(\000)p Fn(1)1149 3902 y Fq(a)j Fs(is)f(not)h(cen)m(tral.Th)m(us)h Fq(T)2142 3918 y Fo(n)2185 3899 y Fm(2)2224 3902 y Fs(\()p Fq(!)t Fs(\))d(acts)j(on)e Fq(A)h Fs(with)f(non-zero)456 4018 y(sk)m(ew-deriv)-5 b(ation)34 b Fq(x)p Fp(\001)p Fs(.)51 b(Ho)m(w)m(ev)m(er)37 b Fq(A)32 b Fp(6)p Fs(=)g Fq(k)s Fs(\()p Fq(u)p Fs(\))i(whenev)m(er)k Fq(r)c(>)e Fs(1.)50 b(Hence)36 b(w)m(e)g(see)456 4134 y(\(with)30 b Fq(n)e Fs(=)f Fq(r)950 4098 y Fn(2)989 4134 y Fs(\))k(that)f(Theorem)h(2.5)f (is)g(not)g(true)h(if)f Fq(A)g Fs(has)h(non-zero)g(nilp)s(oten)m(t)456 4250 y(elemen)m(ts.)731 4457 y(3.)55 b Fr(An)38 b(applica)-7 b(tion)38 b(to)g(Fr)n(obenius-Lusztig)e(kernels)555 4631 y Fs(W)-8 b(e)42 b(consider)h(the)f(sp)s(ecial)f(case)i(of)e(the)h(F)-8 b(rob)s(enius-Lusztig)41 b(k)m(ernel)h(asso-)456 4748 y(ciated)e(to)g Fq(U)944 4763 y Fo(q)982 4748 y Fs(\()p Fq(sl)1095 4763 y Fn(2)1135 4748 y Fs(\),)i(when)g Fq(q)i Fs(is)d(a)f(primitiv)m(e)e(2)p Fq(n)2320 4711 y Fo(th)2431 4748 y Fs(ro)s(ot)i(of)g(1,)i(for)e Fq(n)i(>)f Fs(2;)j(it)456 4864 y(is)34 b(a)i(\014nite-dimensional)c(quotien)m(t)j(of)g Fq(U)1997 4879 y Fo(q)2036 4864 y Fs(\()p Fq(sl)2149 4879 y Fn(2)2188 4864 y Fs(\).)52 b(See)36 b([K,)g(IV.5.6].)52 b(This)35 b(Hopf)456 4980 y(algebra)c(is)h(also)g(called)f(the)i Fg(r)-5 b(estricte)g(d)33 b Fs(quan)m(tum)g(en)m(v)m(eloping)f(algebra) g(of)g Fq(sl)3378 4995 y Fn(2)3417 4980 y Fs(.)456 5096 y(Sp)s(eci\014cally)-8 b(,)1390 5216 y Fq(u)1446 5231 y Fo(q)1511 5216 y Fs(=)28 b Fq(u)1671 5231 y Fo(q)1708 5216 y Fs(\()p Fq(sl)1821 5231 y Fn(2)1861 5216 y Fs(\))f(:=)h Fq(k)s Fp(h)p Fq(E)6 b(;)17 b(F)s(;)g(K)7 b Fp(i)p eop %%Page: 9 9 9 8 bop 3406 233 a Fl(9)456 425 y Fs(where)33 b Fq(E)6 b(;)17 b(F)s(;)g(K)39 b Fs(satisfy)33 b(the)g(relations)e Fq(E)2038 389 y Fo(n)2113 425 y Fs(=)c(0)p Fq(;)45 b(F)2414 389 y Fo(n)2488 425 y Fs(=)27 b(0)p Fq(;)45 b(K)2802 389 y Fo(n)2876 425 y Fs(=)28 b(1,)k(and)815 644 y Fq(K)7 b(E)34 b Fs(=)27 b Fq(q)1161 603 y Fn(2)1200 644 y Fq(E)6 b(K)r(;)45 b(K)7 b(F)41 b Fs(=)28 b Fq(q)1780 603 y Fk(\000)p Fn(2)1874 644 y Fq(F)14 b(K)r(;)44 b(E)6 b(F)35 b Fp(\000)23 b Fq(F)14 b(E)33 b Fs(=)2679 577 y Fq(K)c Fp(\000)23 b Fq(K)2981 541 y Fk(\000)p Fn(1)p 2679 622 397 4 v 2722 713 a Fq(q)j Fp(\000)d Fq(q)2938 684 y Fk(\000)p Fn(1)456 865 y Fs(with)38 b(coaction)g(\001)p Fq(E)44 b Fs(=)38 b Fq(E)32 b Fp(\012)26 b Fq(K)34 b Fs(+)26 b(1)g Fp(\012)h Fq(E)6 b Fs(,)40 b(\001)p Fq(F)52 b Fs(=)37 b Fq(F)j Fp(\012)27 b Fs(1)f(+)g Fq(K)2919 828 y Fk(\000)p Fn(1)3040 865 y Fp(\012)g Fq(F)14 b Fs(,)40 b(and)456 981 y(\001)p Fq(K)35 b Fs(=)27 b Fq(K)j Fp(\012)22 b Fq(K)7 b Fs(.)555 1097 y(Note)31 b(that)g Fq(u)1055 1112 y Fo(q)1124 1097 y Fs(is)f(generated)i(b)m(y)g(t)m(w)m(o)g(di\013eren)m(t)f(copies)h(of) e(the)i(T)-8 b(aft)30 b(algebra)456 1213 y(as)36 b(in)g(\(1.1\),)h (although)e(with)h(t)m(w)m(o)h(di\013eren)m(t)g(c)m(hoices)g(for)f Fq(!)t Fs(.)54 b(Namely)-8 b(,)36 b Fq(H)3268 1228 y Fn(1)3341 1213 y Fs(:=)456 1329 y Fq(k)s Fp(h)p Fq(K)639 1293 y Fk(\000)p Fn(1)733 1329 y Fq(;)17 b(F)d Fp(i)929 1302 y(\030)929 1334 y Fs(=)1043 1329 y Fq(T)1100 1346 y Fo(n)1143 1327 y Fm(2)1181 1329 y Fs(\()p Fq(q)1266 1293 y Fk(\000)p Fn(2)1360 1329 y Fs(\))38 b(and)g Fq(H)1712 1344 y Fn(2)1788 1329 y Fs(:=)e Fq(k)s Fp(h)p Fq(K)2110 1293 y Fk(\000)p Fn(1)2205 1329 y Fq(;)17 b(E)6 b(K)2417 1293 y Fk(\000)p Fn(1)2511 1329 y Fp(i)2586 1302 y(\030)2587 1334 y Fs(=)2700 1329 y Fq(T)2757 1346 y Fo(n)2800 1327 y Fm(2)2839 1329 y Fs(\()p Fq(q)2924 1293 y Fn(2)2963 1329 y Fs(\).)59 b(One)38 b(can)456 1446 y(think)30 b(of)g Fq(H)901 1461 y Fn(1)971 1446 y Fs(as)g Fq(b)1129 1410 y Fk(\000)1189 1446 y Fs(,)h(the)f(\\Borel")g(subalgebra)g(of)g Fq(u)2418 1461 y Fo(q)2455 1446 y Fs(,)h(and)g(similarly)c(of)i Fq(H)3287 1461 y Fn(2)3357 1446 y Fs(as)456 1562 y Fq(b)497 1526 y Fn(+)556 1562 y Fs(.)44 b(W)-8 b(e)33 b(can)f(no)m(w)i(apply)e (our)g(results)h(from)f(Section)g(2)g(to)h Fq(u)2757 1577 y Fo(q)2794 1562 y Fs(.)555 1678 y(In)f(the)f(spirit)f(of)h (Theorem)g(2.2,)h(w)m(e)g(giv)m(e)f(the)h(form)d(of)i(all)e(p)s (ossible)i(actions)456 1794 y(of)g Fq(u)622 1809 y Fo(q)660 1794 y Fs(\()p Fq(sl)773 1809 y Fn(2)812 1794 y Fs(\))h(on)g Fq(A)c Fs(=)g Fq(k)s Fs(\()p Fq(u)p Fs(\))j(so)h(that)g(the)h(action)e (of)h Fq(K)39 b Fs(stabilizes)31 b(the)h(space)i Fq(k)s(u)p Fs(.)456 1971 y Fu(Theorem)j(3.1.)49 b Fg(L)-5 b(et)38 b Fq(A)33 b Fs(=)g Fq(k)s Fs(\()p Fq(u)p Fs(\))k Fg(b)-5 b(e)37 b(an)h Fq(n)p Fg(-dimensional)d Fq(k)s Fg(-algebr)-5 b(a,)38 b(for)f Fq(n)d(>)456 2088 y Fs(2)p Fg(.)46 b(Assume)36 b(that)g Fq(A)g Fg(is)f(a)g Fq(u)1501 2103 y Fo(q)1539 2088 y Fs(\()p Fq(sl)1652 2103 y Fn(2)1691 2088 y Fs(\))p Fg(-mo)-5 b(dule)35 b(algebr)-5 b(a)35 b(such)g(that)h Fq(K)30 b Fp(\001)22 b Fq(u)29 b Fs(=)g Fq(\013)3265 2051 y Fk(\000)p Fn(1)3359 2088 y Fq(u)p Fg(,)456 2204 y(for)42 b(some)f Fq(\013)i Fp(2)f Fq(k)s Fg(,)i(and)e(that)h Fq(F)e Fp(\001)28 b Fq(u)41 b Fp(6)p Fs(=)g(0)h Fg(or)h Fq(E)33 b Fp(\001)28 b Fq(u)41 b Fp(6)p Fs(=)h(0)p Fg(.)67 b(Then)41 b(ther)-5 b(e)43 b(exist)456 2320 y Fq(\014)6 b(;)17 b(\015)5 b(;)17 b(\016)31 b Fp(2)e Fq(k)39 b Fg(with)c Fq(\015)5 b(;)17 b(\016)33 b Fp(6)p Fs(=)28 b(0)36 b Fg(and)e Fs(0)29 b Fp(\024)g Fq(s;)17 b(t;)g(l)31 b Fp(\024)e Fq(n)23 b Fp(\000)g Fs(1)35 b Fg(with)h Fq(t)p Fs(\(1)22 b Fp(\000)h Fq(s)p Fs(\))3040 2292 y Fp(\030)3041 2324 y Fs(=)3146 2320 y(1)35 b Fg(\(mo)-5 b(d)456 2436 y Fq(n)p Fg(\))35 b(and)f Fq(s)22 b Fs(+)g Fq(l)1003 2409 y Fp(\030)1004 2440 y Fs(=)1108 2436 y(2)35 b Fg(\(mo)-5 b(d)34 b Fq(n)p Fg(\))h(such)f(that)555 2552 y(\(a\))h Fq(u)776 2516 y Fo(n)850 2552 y Fs(=)27 b Fq(\014)34 b Fp(2)28 b Fq(k)s Fg(,)35 b Fq(\013)28 b Fs(=)f Fq(q)1495 2516 y Fk(\000)p Fn(2)p Fo(t)1650 2552 y Fg(and)34 b Fq(K)42 b Fg(has)35 b(or)-5 b(der)34 b Fq(n)h Fg(in)g(A)n(ut)p Fs(\()p Fq(A)p Fs(\))p Fg(;)555 2669 y(\(b\))g Fq(F)g Fp(\001)22 b Fq(u)27 b Fs(=)h Fq(\015)5 b(u)1162 2633 y Fo(s)1233 2669 y Fg(and)35 b Fq(E)28 b Fp(\001)22 b Fq(u)27 b Fs(=)g Fq(\016)t(u)1862 2633 y Fo(l)1888 2669 y Fg(;)555 2785 y(\(c\))38 b(If)h Fq(s)34 b Fs(=)h(0)j Fg(and)g Fq(l)f Fs(=)e(2)p Fg(,)k(or)g Fq(s)34 b Fs(=)h(2)j Fg(and)g Fq(l)f Fs(=)e(0)p Fg(,)k(then)g Fq(\016)t(\015)g Fs(=)c Fp(\000)p Fq(q)t Fg(.)56 b(If)39 b(b)-5 b(oth)456 2901 y Fq(s;)17 b(l)29 b Fp(\025)f Fs(2)p Fg(,)35 b(then)g Fq(s)22 b Fs(+)g Fq(l)30 b Fs(=)d Fq(n)22 b Fs(+)h(2)p Fg(,)34 b Fq(\014)f Fp(6)p Fs(=)28 b(0)p Fg(,)34 b(and)872 3120 y Fq(\016)t(\015)f Fs(=)1302 3053 y Fq(\013)1365 3017 y Fo(s)p Fk(\000)p Fn(2)1492 3053 y Fs(\(1)22 b Fp(\000)g Fq(\013)q Fs(\))p 1116 3097 871 4 v 1116 3201 a Fq(\014)6 b Fs(\()p Fq(q)26 b Fp(\000)d Fq(q)1431 3172 y Fk(\000)p Fn(1)1525 3201 y Fs(\)\()1601 3126 y Fi(P)1706 3152 y Fo(s)p Fk(\000)p Fn(2)1706 3230 y Fo(j)t Fn(=0)1849 3201 y Fq(\013)1912 3172 y Fo(j)1949 3201 y Fs(\))2024 3120 y(=)2400 3053 y(1)f Fp(\000)h Fq(\013)2634 3017 y Fk(\000)p Fn(1)p 2138 3097 854 4 v 2138 3204 a Fq(\014)6 b Fs(\()p Fq(q)25 b Fp(\000)e Fq(q)2452 3175 y Fk(\000)p Fn(1)2546 3204 y Fs(\)\()2622 3129 y Fi(P)2727 3156 y Fo(l)q Fk(\000)p Fn(2)2727 3233 y Fo(i)p Fn(=0)2862 3204 y Fq(\013)2925 3175 y Fo(i)2953 3204 y Fs(\))3001 3120 y Fq(:)456 3365 y Fg(Conversely,)28 b(any)g(choic)-5 b(e)28 b(of)g Fq(\014)6 b(;)17 b(\015)5 b(;)17 b(\016)31 b Fg(and)c Fq(s;)17 b(t;)g(l)31 b Fg(satisfying)c(the) i(ab)-5 b(ove)27 b(r)-5 b(elations)456 3482 y(de\014nes)33 b(an)i Fq(u)975 3497 y Fo(q)1012 3482 y Fs(\()p Fq(sl)1125 3497 y Fn(2)1165 3482 y Fs(\))p Fg(-mo)-5 b(dule)34 b(algebr)-5 b(a)34 b(structur)-5 b(e)36 b(on)e Fq(A)p Fg(.)456 3658 y(Pr)-5 b(o)g(of.)41 b Fs(W)-8 b(e)32 b(assume)g(that)g Fq(F)i Fp(\001)20 b Fq(A)28 b Fp(6)p Fs(=)f(0;)32 b(the)g(case)h Fq(E)26 b Fp(\001)20 b Fq(A)28 b Fp(6)p Fs(=)g(0)j(is)g(similar.)40 b(Apply)456 3775 y(Theorem)47 b(2.2)g(to)g Fq(H)1269 3790 y Fn(1)1356 3775 y Fs(with)f Fq(\033)57 b Fs(=)c Fq(K)1923 3739 y Fk(\000)p Fn(1)2017 3775 y Fs(,)e Fq(D)56 b Fs(=)c Fq(F)14 b Fs(,)51 b(and)c Fq(\026)53 b Fs(=)f Fq(q)3006 3739 y Fk(\000)p Fn(2)3148 3775 y Fs(to)47 b(\014nd)456 3891 y Fq(\014)6 b(;)17 b(\015)45 b Fp(2)c Fq(k)j Fs(with)c Fq(\015)46 b Fp(6)p Fs(=)41 b(0)f(and)g Fq(t;)17 b(s)40 b Fs(with)g Fq(t)p Fs(\(1)28 b Fp(\000)g Fq(s)p Fs(\))2364 3863 y Fp(\030)2364 3895 y Fs(=)2482 3891 y(1)40 b(\(mo)s(d)f Fq(n)p Fs(\))i(to)f(see)i(that)456 4007 y Fq(u)512 3971 y Fo(n)586 4007 y Fs(=)27 b Fq(\014)6 b Fs(,)33 b Fq(\013)28 b Fs(=)f Fq(q)1050 3971 y Fk(\000)p Fn(2)p Fo(t)1203 4007 y Fs(and)32 b Fq(F)k Fp(\001)22 b Fq(u)27 b Fs(=)h Fq(\015)5 b(u)1840 3971 y Fo(s)1876 4007 y Fs(.)43 b(This)33 b(pro)m(v)m(es)h(\(a\).)555 4135 y(No)m(w)g(since)g Fq(E)6 b(F)36 b Fp(\000)23 b Fq(F)14 b(E)35 b Fs(=)1593 4096 y Fo(K)5 b Fk(\000)p Fo(K)1778 4072 y Ff(\000)p Fm(1)p 1593 4112 266 4 v 1624 4170 a Fo(q)r Fk(\000)p Fo(q)1747 4151 y Ff(\000)p Fm(1)1902 4135 y Fs(and)34 b Fq(\013)g Fs(is)e(a)h(primitiv)m(e)e Fq(n)2845 4099 y Fo(th)2949 4135 y Fs(ro)s(ot)h(of)h(1,)g(it)456 4265 y(follo)m(ws)e(that)h(\()p Fq(E)6 b(F)36 b Fp(\000)23 b Fq(F)14 b(E)6 b Fs(\))22 b Fp(\001)f Fq(u)28 b Fp(6)p Fs(=)f(0,)32 b(and)h(so)g(also)f Fq(E)c Fp(\001)22 b Fq(A)27 b Fp(6)p Fs(=)h(0.)555 4382 y(Next,)46 b(w)m(e)d(apply)f (Theorem)g(2.2)g(to)g Fq(H)2062 4397 y Fn(2)2101 4382 y Fs(,)i(with)e Fq(\033)48 b Fs(=)c Fq(K)2717 4346 y Fk(\000)p Fn(1)2854 4382 y Fs(but)e(no)m(w)h(with)456 4498 y Fq(D)540 4462 y Fk(0)594 4498 y Fs(=)31 b Fq(E)6 b(K)869 4462 y Fk(\000)p Fn(1)999 4498 y Fs(and)35 b Fq(\026)c Fs(=)g Fq(q)1435 4462 y Fn(2)1474 4498 y Fs(;)36 b(this)f(giv)m(es)g(us)h(0)31 b Fp(6)p Fs(=)g Fq(\016)2333 4462 y Fk(0)2388 4498 y Fp(2)h Fq(k)38 b Fs(and)d(0)c Fp(\024)h Fq(r)m(;)17 b(l)34 b Fp(\024)e Fq(n)23 b Fp(\000)i Fs(1)456 4614 y(with)47 b Fq(\013)53 b Fs(=)g Fq(q)984 4578 y Fn(2)p Fo(r)1105 4614 y Fs(and)47 b Fq(D)1393 4578 y Fk(0)1449 4614 y Fp(\001)32 b Fq(u)52 b Fs(=)h Fq(\016)1793 4578 y Fk(0)1816 4614 y Fq(u)1872 4578 y Fo(l)1898 4614 y Fs(.)88 b(Since)48 b Fq(E)59 b Fs(=)53 b Fq(D)2627 4578 y Fk(0)2650 4614 y Fq(K)7 b Fs(,)51 b(it)46 b(follo)m(ws)h(that)456 4730 y Fq(E)33 b Fp(\001)27 b Fq(u)40 b Fs(=)h Fq(\016)t(u)932 4694 y Fo(l)957 4730 y Fs(,)h(where)g Fq(\016)i Fs(:=)d Fq(\013)1610 4694 y Fk(\000)p Fn(1)1704 4730 y Fq(\016)1751 4694 y Fk(0)1774 4730 y Fs(.)67 b(Moreo)m(v)m(er)41 b Fq(q)2350 4694 y Fk(\000)p Fn(2)p Fo(t)2511 4730 y Fs(=)f Fq(q)2674 4694 y Fn(2)p Fo(r)2787 4730 y Fs(implies)e Fq(r)3214 4703 y Fp(\030)3214 4735 y Fs(=)3332 4730 y Fp(\000)p Fq(t)456 4847 y Fs(\(mo)s(d)p Fq(n)p Fs(\))d(and)i(th)m(us)h(that)f Fq(s)25 b Fp(\000)g Fs(1)1699 4819 y Fp(\030)1700 4851 y Fs(=)1811 4847 y(1)g Fp(\000)g Fq(l)39 b Fs(\(mo)s(d)d Fq(n)p Fs(\);)j(equiv)-5 b(alen)m(tly)d(,)37 b Fq(s)25 b Fs(+)g Fq(l)3283 4819 y Fp(\030)3284 4851 y Fs(=)3396 4847 y(2)456 4963 y(\(mo)s(d)31 b Fq(n)p Fs(\).)44 b(Th)m(us)34 b(\(b\))e(is)g(pro)m(v)m(ed.)555 5079 y(It)27 b(remains)f(only)g(to)h (c)m(hec)m(k)i(the)e(relation)e(b)s(et)m(w)m(een)k Fq(\015)j Fs(and)27 b Fq(\016)t Fs(.)41 b(T)-8 b(o)27 b(see)h(this)e(w)m(e)456 5202 y(use)k(the)h(fact)e(that)h(the)g(actions)f(of)g Fq(E)6 b(F)31 b Fp(\000)16 b Fq(F)e(E)36 b Fs(and)2433 5162 y Fo(K)5 b Fk(\000)p Fo(K)2618 5139 y Ff(\000)p Fm(1)p 2433 5179 V 2463 5236 a Fo(q)r Fk(\000)p Fo(q)2586 5217 y Ff(\000)p Fm(1)2738 5202 y Fs(m)m(ust)30 b(agree)g(on)g Fq(u)p Fs(.)p eop %%Page: 10 10 10 9 bop 456 236 a Fl(10)623 b(S.)33 b(MONTGOMER)-6 b(Y)33 b(AND)f(H.-J.)i(SCHNEIDER)456 437 y Fs(T)-8 b(rivially)860 398 y Fo(K)5 b Fk(\000)p Fo(K)1045 374 y Ff(\000)p Fm(1)p 860 414 266 4 v 890 471 a Fo(q)r Fk(\000)p Fo(q)1013 452 y Ff(\000)p Fm(1)1158 437 y Fp(\001)23 b Fq(u)29 b Fs(=)1409 398 y Fo(\013)1454 374 y Ff(\000)p Fm(1)1537 398 y Fk(\000)p Fo(\013)p 1409 414 229 4 v 1420 471 a(q)r Fk(\000)p Fo(q)1543 452 y Ff(\000)p Fm(1)1647 437 y Fq(u)p Fs(,)k(so)h(w)m(e)h(consider)f(the)g(action)e(of)i Fq(E)6 b(F)36 b Fp(\000)24 b Fq(F)14 b(E)6 b Fs(.)456 571 y(Applying)31 b(Lemma)g(2.1)i(with)f Fq(\033)g Fs(=)27 b Fq(K)1886 535 y Fk(\000)p Fn(1)2013 571 y Fs(and)33 b Fq(D)d Fs(=)e Fq(F)14 b Fs(,)32 b(w)m(e)i(see)f(that)1435 836 y Fq(F)i Fp(\001)22 b Fq(u)1639 795 y Fo(l)1692 836 y Fs(=)28 b Fq(\015)5 b Fs(\()1906 712 y Fo(l)q Fk(\000)p Fn(1)1890 742 y Fi(X)1905 951 y Fo(i)p Fn(=0)2050 836 y Fq(\013)2113 795 y Fo(i)2141 836 y Fs(\))p Fq(u)2235 795 y Fo(l)q Fn(+)p Fo(s)p Fk(\000)p Fn(1)2438 836 y Fq(:)456 1105 y Fs(Similarly)-8 b(,)28 b(use)34 b(Lemma)d(2.1)h(with)g Fq(\033)g Fs(=)27 b Fq(K)2065 1069 y Fk(\000)p Fn(1)2192 1105 y Fs(and)33 b Fq(D)2466 1069 y Fk(0)2517 1105 y Fs(=)27 b Fq(E)6 b(K)2788 1069 y Fk(\000)p Fn(1)2915 1105 y Fs(to)33 b(see)g(that)1334 1367 y Fq(E)28 b Fp(\001)22 b Fq(u)1540 1325 y Fo(s)1604 1367 y Fs(=)28 b Fq(\013)1771 1325 y Fn(1)p Fk(\000)p Fo(s)1897 1367 y Fq(\016)t Fs(\()1993 1242 y Fo(s)p Fk(\000)p Fn(1)1982 1272 y Fi(X)1993 1482 y Fo(j)t Fn(=0)2143 1367 y Fq(\013)2206 1325 y Fo(j)2242 1367 y Fs(\))p Fq(u)2336 1325 y Fo(s)p Fn(+)p Fo(l)q Fk(\000)p Fn(1)2539 1367 y Fq(:)456 1639 y Fs(Com)m(bining)i(these,)k (w)m(e)g(ha)m(v)m(e)1106 1903 y(\()p Fq(E)6 b(F)36 b Fp(\000)23 b Fq(F)14 b(E)6 b Fs(\))22 b Fp(\001)g Fq(y)30 b Fs(=)e Fq(\015)5 b(\016)t Fs([)p Fq(\013)2061 1862 y Fn(1)p Fk(\000)p Fo(s)2215 1779 y(s)p Fk(\000)p Fn(1)2204 1809 y Fi(X)2215 2019 y Fo(j)t Fn(=0)2365 1903 y Fq(\013)2428 1862 y Fo(j)2486 1903 y Fp(\000)2602 1779 y Fo(l)q Fk(\000)p Fn(1)2586 1809 y Fi(X)2601 2019 y Fo(i)p Fn(=0)2746 1903 y Fq(\013)2809 1862 y Fo(i)2837 1903 y Fs(])p Fq(u)2920 1862 y Fo(s)p Fn(+)p Fo(l)q Fk(\000)p Fn(1)1190 2260 y Fs(=)82 b Fq(\015)5 b(\016)t(\013)1514 2218 y Fn(1)p Fk(\000)p Fo(s)1641 2260 y Fs([)1679 2135 y Fo(s)p Fk(\000)p Fn(1)1668 2165 y Fi(X)1679 2375 y Fo(j)t Fn(=0)1829 2260 y Fq(\013)1892 2218 y Fo(j)1950 2260 y Fp(\000)23 b Fq(\013)2113 2218 y Fo(s)p Fk(\000)p Fn(1)2272 2135 y Fo(l)q Fk(\000)p Fn(1)2256 2165 y Fi(X)2271 2375 y Fo(i)p Fn(=0)2416 2260 y Fq(\013)2479 2218 y Fo(i)2507 2260 y Fs(])p Fq(u)2590 2218 y Fo(s)p Fn(+)p Fo(l)q Fk(\000)p Fn(1)1190 2612 y Fs(=)82 b Fq(\015)5 b(\016)t(\013)1514 2571 y Fn(1)p Fk(\000)p Fo(s)1641 2612 y Fs([)1679 2488 y Fo(s)p Fk(\000)p Fn(2)1668 2518 y Fi(X)1679 2728 y Fo(j)t Fn(=0)1829 2612 y Fq(\013)1892 2571 y Fo(j)1950 2612 y Fp(\000)23 b Fq(\013)q Fs(\()2185 2488 y Fo(n)p Fk(\000)p Fn(1)2179 2518 y Fi(X)2151 2728 y Fo(i)p Fn(=)p Fo(s)p Fk(\000)p Fn(1)2369 2612 y Fq(\013)2432 2571 y Fo(i)2459 2612 y Fs(\)])p Fq(u)2580 2571 y Fo(s)p Fn(+)p Fo(l)q Fk(\000)p Fn(1)1190 2965 y Fs(=)82 b Fq(\015)5 b(\016)t(\013)1514 2924 y Fn(1)p Fk(\000)p Fo(s)1641 2965 y Fs(\(1)22 b(+)g Fq(\013)q Fs(\)\()1998 2841 y Fo(s)p Fk(\000)p Fn(2)1987 2871 y Fi(X)1998 3081 y Fo(j)t Fn(=0)2147 2965 y Fq(\013)2210 2924 y Fo(j)2246 2965 y Fs(\))p Fq(u)2340 2924 y Fo(s)p Fn(+)p Fo(l)q Fk(\000)p Fn(1)2543 2965 y Fq(;)456 3237 y Fs(assuming)39 b(for)g(the)h(momen)m(t)f(that)h Fq(s;)17 b(l)41 b Fp(\025)g Fs(2,)g(hence)h Fq(s)27 b Fs(+)g Fq(l)42 b Fs(=)e Fq(n)g Fs(=)g(2)f(and)h(so)456 3354 y Fq(u)512 3317 y Fo(s)p Fn(+)p Fo(l)q Fk(\000)p Fn(1)742 3354 y Fs(=)28 b Fq(\014)6 b(u)p Fs(.)43 b(Th)m(us)34 b(w)m(e)f(ha)m(v)m(e) 1202 3548 y Fq(\013)1265 3511 y Fk(\000)p Fn(1)1381 3548 y Fp(\000)23 b Fq(\013)p 1202 3592 342 4 v 1218 3683 a(q)j Fp(\000)c Fq(q)1433 3655 y Fk(\000)p Fn(1)1581 3615 y Fs(=)28 b Fq(\015)5 b(\016)t(\014)h(\013)1912 3574 y Fn(1)p Fk(\000)p Fo(s)2038 3615 y Fs(\(1)22 b(+)g Fq(\013)q Fs(\)\()2395 3490 y Fo(s)p Fk(\000)p Fn(2)2384 3520 y Fi(X)2395 3730 y Fo(j)t Fn(=0)2544 3615 y Fq(\013)2607 3574 y Fo(j)2643 3615 y Fs(\))p Fq(:)456 3897 y Fs(Since)38 b Fq(\013)779 3861 y Fk(\000)p Fn(1)899 3897 y Fp(\000)27 b Fq(\013)38 b Fs(=)f Fq(\013)1279 3861 y Fk(\000)p Fn(1)1373 3897 y Fs(\(1)26 b Fp(\000)g Fq(\013)1652 3861 y Fn(2)1691 3897 y Fs(\),)40 b(the)f(\014rst)g(form)e(of)g Fq(\015)5 b(\016)43 b Fs(in)37 b(\(c\))i(no)m(w)g(follo)m(ws)456 4013 y(b)m(y)44 b(cancelling)d(1)29 b(+)g Fq(\013)43 b Fs(from)f(b)s(oth)h(sides)h(\(note)f(that)f Fq(\013)2626 3977 y Fn(2)2711 4013 y Fp(6)p Fs(=)j(1)e(implies)e(that)456 4141 y Fq(\014)e Fp(6)p Fs(=)34 b(0\).)55 b(The)37 b(second)h(form)d (then)i(follo)m(ws)e(b)m(y)i(noting)e(that)h Fq(\013)2860 4105 y Fo(s)p Fk(\000)p Fn(1)3003 4066 y Fi(P)3109 4092 y Fo(l)q Fk(\000)p Fn(2)3109 4170 y Fo(i)p Fn(=0)3244 4141 y Fq(\013)3307 4105 y Fo(i)3369 4141 y Fs(=)456 4269 y Fp(\000)550 4195 y Fi(P)655 4221 y Fo(s)p Fk(\000)p Fn(2)655 4298 y Fo(j)t Fn(=0)798 4269 y Fq(\013)861 4233 y Fo(j)898 4269 y Fs(.)555 4402 y(No)m(w)f(if)d Fq(s)e Fs(=)f(0,)34 b(and)g(so)g Fq(t)c Fs(=)g(1)j(and)h Fq(l)e Fs(=)e(2,)j Fq(u)2248 4366 y Fo(s)p Fn(+)p Fo(l)q Fk(\000)p Fn(1)2481 4402 y Fs(=)d Fq(u)j Fs(and)h(so)g Fq(\014)39 b Fs(do)s(es)34 b(not)456 4518 y(app)s(ear.)68 b(In)41 b(this)f(case)i(the)f(second)i(form)c(of)h(the)i(form)m(ula)d(for)h Fq(\015)5 b(\016)t Fs(,)43 b(without)456 4634 y(the)31 b Fq(\014)6 b Fs(,)30 b(mak)m(es)i(sense)g(and)f(so,)g(using)f(that)h (no)m(w)g Fq(\013)d Fs(=)g Fq(q)2518 4598 y Fk(\000)p Fn(2)p Fo(t)2665 4634 y Fs(=)g Fq(q)2816 4598 y Fk(\000)p Fn(2)2910 4634 y Fs(,)j(w)m(e)h(see)g(that)456 4751 y Fq(\015)5 b(\016)43 b Fs(=)c Fp(\000)p Fq(q)t Fs(.)64 b(The)40 b(case)h Fq(s)e Fs(=)g(2)p Fq(;)17 b(l)41 b Fs(=)e(0)g(is)g(similar,)f(using)h Fq(\013)h Fs(=)f Fq(q)2902 4714 y Fn(2)2980 4751 y Fs(in)g(the)h(\014rst)456 4867 y(form)31 b(of)h(the)h(form)m(ula)e(for)h Fq(\015)5 b(\016)t Fs(.)555 4983 y(Con)m(v)m(ersely)-8 b(,)48 b(supp)s(ose)d Fq(\014)6 b(;)17 b(\015)5 b(;)17 b(\016)47 b Fs(and)c Fq(s;)17 b(t;)g(l)45 b Fs(are)f(giv)m(en)f(as)h(in)f(the)g(theorem.)456 5099 y(They)38 b(de\014ne)g(actions)f(of)f Fq(H)1530 5114 y Fn(1)1606 5099 y Fs(and)i Fq(H)1882 5114 y Fn(2)1958 5099 y Fs(on)f Fq(A)e Fs(=)g Fq(k)s Fs([)p Fq(u)g Fp(j)g Fq(u)2608 5063 y Fo(n)2689 5099 y Fs(=)h Fq(\014)6 b Fs(])36 b(b)m(y)i(Theorem)456 5216 y(2.2.)71 b(It)42 b(remains)f(to)g(sho)m(w)i(that)f Fq(A)g Fs(is)f(in)g(fact)h(an)g Fq(u)2515 5231 y Fo(q)2552 5216 y Fs(\()p Fq(sl)2665 5231 y Fn(2)2705 5216 y Fs(\)-mo)s(dule,)g(that)g(is)p eop %%Page: 11 11 11 10 bop 3368 233 a Fl(11)456 437 y Fs(\()p Fq(E)6 b(F)37 b Fp(\000)24 b Fq(F)14 b(E)6 b Fs(\))24 b Fp(\001)f Fq(a)32 b Fs(=)g(\()1280 398 y Fo(K)5 b Fk(\000)p Fo(K)1465 374 y Ff(\000)p Fm(1)p 1279 414 266 4 v 1310 471 a Fo(q)r Fk(\000)p Fo(q)1433 452 y Ff(\000)p Fm(1)1555 437 y Fs(\))24 b Fp(\001)f Fq(a)35 b Fs(for)g(all)e Fq(a)f Fp(2)g Fq(A)p Fs(.)50 b(This)35 b(is)g(true)g(for)g Fq(a)c Fs(=)h Fq(u)i Fs(b)m(y)456 562 y(the)f(previous)g(argumen)m(t.)45 b(Hence)34 b(it)e(is)g(true)i(for)e(all)f Fq(a)d Fp(2)h Fq(A)p Fs(,)k(since)h Fq(E)6 b(F)36 b Fp(\000)23 b Fq(F)14 b(E)456 684 y Fs(and)655 645 y Fo(K)5 b Fk(\000)p Fo(K)840 621 y Ff(\000)p Fm(1)p 655 661 V 686 718 a Fo(q)r Fk(\000)p Fo(q)809 700 y Ff(\000)p Fm(1)964 684 y Fs(b)s(oth)32 b(act)g(as)h(\()p Fq(\033)n(;)17 b(\033)1670 648 y Fk(\000)p Fn(1)1765 684 y Fs(\)-sk)m(ew)34 b(deriv)-5 b(ations.)p 3374 684 4 66 v 3378 622 59 4 v 3378 684 V 3436 684 4 66 v 456 887 a Fu(Corollary)37 b(3.2.)49 b Fg(L)-5 b(et)30 b Fq(A)g Fg(b)-5 b(e)30 b(an)f Fq(n)p Fg(-dimensional)f Fq(k)s Fg(-algebr)-5 b(a)29 b(with)h(no)g(non-zer)-5 b(o)456 1003 y(nilp)g(otent)30 b(elements,)h(and)f(assume)g(that)h Fq(A)g Fg(is)g(a)g Fq(u)2342 1018 y Fo(q)2379 1003 y Fs(\()p Fq(sl)2492 1018 y Fn(2)2532 1003 y Fs(\))p Fg(-mo)-5 b(dule)30 b(algebr)-5 b(a)30 b(such)456 1119 y(that)46 b Fq(F)d Fp(\001)30 b Fq(A)47 b Fp(6)p Fs(=)h(0)d Fg(\(or)g(that)h Fq(E)36 b Fp(\001)30 b Fq(A)48 b Fp(6)p Fs(=)f(0)p Fg(\).)76 b(Then)45 b(ther)-5 b(e)45 b(exists)h Fq(u)g Fp(2)i Fq(A)e Fg(and)456 1235 y Fs(0)27 b Fp(6)p Fs(=)h Fq(\014)6 b(;)17 b(\015)5 b(;)17 b(\016)30 b Fp(2)e Fq(k)38 b Fg(such)d(that)555 1352 y(\(a\))g Fq(A)27 b Fs(=)h Fq(k)s Fs(\()p Fq(u)p Fs(\))p Fg(,)34 b Fq(u)1230 1316 y Fo(n)1304 1352 y Fs(=)28 b Fq(\014)6 b Fg(,)34 b(and)h Fq(K)29 b Fp(\001)22 b Fq(u)27 b Fs(=)g Fq(q)2118 1316 y Fn(2)2158 1352 y Fq(u)p Fg(;)555 1468 y(\(b\))35 b Fq(F)g Fp(\001)22 b Fq(u)27 b Fs(=)h Fq(\015)5 b Fs(1)35 b Fg(and)f Fq(E)28 b Fp(\001)22 b Fq(u)27 b Fs(=)h Fq(\016)t(u)1819 1432 y Fn(2)1858 1468 y Fg(;)555 1584 y(\(c\))35 b Fq(\015)5 b(\016)31 b Fs(=)d Fp(\000)p Fq(q)t Fg(.)555 1700 y(Mor)-5 b(e)g(over)35 b Fq(u)f Fg(is)h(unique)g(up)g(to)g(a)f(sc)-5 b(alar)35 b(multiple.)456 1894 y(Pr)-5 b(o)g(of.)41 b Fs(As)26 b(b)s(efore)f(w)m(e)i(ma)m(y)e(assume)h(that)g Fq(F)21 b Fp(\001)8 b Fq(A)27 b Fp(6)p Fs(=)h(0.)41 b(W)-8 b(e)26 b(apply)f(Theorem)h(2.5)456 2011 y(to)35 b Fq(H)659 2026 y Fn(1)733 2011 y Fs(with)h Fq(!)g Fs(=)c Fq(q)1211 1975 y Fk(\000)p Fn(2)1305 2011 y Fq(;)17 b(g)36 b Fs(=)c Fq(K)1630 1975 y Fk(\000)p Fn(1)1725 2011 y Fs(,)k(and)g Fq(x)d Fs(=)f Fq(F)50 b Fs(to)35 b(\014nd)h Fq(u)c Fp(2)h Fq(A)j Fs(and)f Fq(\014)6 b(;)17 b(\015)37 b Fp(2)c Fq(k)456 2127 y Fs(suc)m(h)f(that)e Fq(A)e Fs(=)f Fq(k)s Fs(\()p Fq(u)p Fs(\),)k Fq(K)1421 2091 y Fk(\000)p Fn(1)1533 2127 y Fp(\001)17 b Fq(u)28 b Fs(=)f Fq(q)1812 2091 y Fk(\000)p Fn(2)1906 2127 y Fq(u)p Fs(,)k(and)f Fq(F)i Fp(\001)18 b Fq(u)27 b Fs(=)g Fq(\015)5 b Fs(1.)43 b(Theorem)31 b(3.1)f(no)m(w)456 2243 y(applies,)h(as)i(w)m(e)h(are)e(in)g(the)h (case)h Fq(s)27 b Fs(=)h(0)p Fq(;)17 b(l)29 b Fs(=)f(2.)p 3374 2359 V 3378 2297 59 4 v 3378 2359 V 3436 2359 4 66 v 456 2553 a Fu(Remark)37 b(3.3.)49 b Fs(Theorem)37 b(3.1)g(giv)m(es)h(a)f(quan)m(tum)h(analog)e(of)h(classical)e(w)m(ork) 456 2670 y(of)29 b(Jacobson)h([J])f(on)h(the)g(Witt)e(algebra.)42 b(See)30 b(also)f([Z)o(].)43 b(That)30 b(is,)g(assume)g(that)456 2786 y Fq(k)39 b Fs(has)e(c)m(haracteristic)g Fq(p)e Fp(6)p Fs(=)f(0)p Fq(;)17 b Fs(2,)37 b(and)g(let)f Fq(A)f Fs(=)g Fq(k)s Fs([)p Fq(v)t Fp(j)p Fq(v)2497 2750 y Fo(p)2571 2786 y Fs(=)f(0].)56 b(W)-8 b(e)37 b(ma)m(y)g(write)456 2902 y Fq(A)29 b Fs(=)g Fq(k)s Fs([)p Fq(u)p Fp(j)p Fq(u)884 2866 y Fo(p)952 2902 y Fs(=)g(1])34 b(b)m(y)g(setting)f Fq(u)c Fs(=)g Fq(v)e Fp(\000)c Fs(1.)46 b(Then)35 b(Der)2522 2917 y Fo(k)2564 2902 y Fs(\()p Fq(a)p Fs(\))f(is)f(spanned)i(b)m(y)f (all)456 3018 y Fq(e)501 3033 y Fo(i)529 3018 y Fs(,)29 b Fq(i)f Fs(=)g(0)p Fq(;)17 b(:)g(:)g(:)32 b(;)17 b(p)c Fp(\000)g Fs(1,)28 b(where)i Fq(e)1613 3033 y Fo(i)1669 3018 y Fs(is)e(determined)g(b)m(y)h(setting)e Fq(e)2759 3033 y Fo(i)2801 3018 y Fp(\001)13 b Fq(u)27 b Fs(=)g Fq(u)3084 2982 y Fo(i)p Fn(+1)3202 3018 y Fs(.)42 b(One)456 3135 y(ma)m(y)36 b(c)m(hec)m(k)j(that)d([)p Fq(e)1225 3150 y Fo(i)1253 3135 y Fq(;)17 b(e)1342 3150 y Fo(j)1379 3135 y Fs(])34 b(=)g(\()p Fq(i)25 b Fp(\000)h Fq(j)6 b Fs(\))p Fq(e)1878 3150 y Fo(i)p Fn(+)p Fo(j)1993 3135 y Fs(.)55 b(Th)m(us,)39 b(setting)d Fq(h)f Fs(:=)f Fq(e)2951 3150 y Fn(0)2991 3135 y Fs(,)j(it)f(follo)m(ws)456 3251 y(that)26 b([)p Fq(e)733 3266 y Fo(i)762 3251 y Fq(;)17 b(h)p Fs(])27 b(=)h Fq(ie)1098 3266 y Fo(i)1154 3251 y Fs(and)f(that)f([)p Fq(e)1615 3266 y Fo(i)1644 3251 y Fq(;)17 b(e)1733 3266 y Fk(\000)p Fo(i)1816 3251 y Fs(])28 b(=)f(2)p Fq(ih)p Fs(.)42 b(Th)m(us)28 b(Der)2578 3266 y Fo(k)2621 3251 y Fs(\()p Fq(A)p Fs(\))f(con)m(tains)g(a)f(cop)m (y)456 3367 y(of)32 b Fq(sl)642 3382 y Fn(2)714 3367 y Fs(for)g(eac)m(h)h Fq(i)28 b Fp(6)p Fs(=)g(0)f(mo)s(d)g Fq(p)p Fs(.)456 3561 y Fu(Remark)37 b(3.4.)49 b Fs(Corollary)e(3.2)i (also)f(giv)m(es)i(an)f(analog)f(of)g([MSm,)54 b(3.8],)f(in)456 3677 y(whic)m(h)42 b(it)f(w)m(as)h(sho)m(wn)h(that)f(when)h Fq(q)i Fs(is)d(not)f(a)h(ro)s(ot)f(of)g(1,)j(there)e(are)g(essen-)456 3794 y(tially)h(t)m(w)m(o)k(mo)s(dule-algebra)c(actions)j(of)f Fq(U)2153 3809 y Fo(q)2192 3794 y Fs(\()p Fq(sl)2305 3809 y Fn(2)2344 3794 y Fs(\))h(on)g(the)h(p)s(olynomial)42 b(ring)456 3910 y Fj(C)20 b Fs([)p Fq(X)8 b Fs(].)49 b(T)-8 b(o)31 b(state)g(this)g(result)g(precisely)-8 b(,)32 b(w)m(e)g(recall)e(the)i(Drinfel'd-)d(Jim)m(b)s(o)g(for-)456 4034 y(m)m(ulation)g(of)991 4009 y(~)974 4034 y Fq(U)39 b Fs(=)27 b Fq(U)1248 4049 y Fo(q)1287 4034 y Fs(\()p Fq(sl)1400 4049 y Fn(2)1439 4034 y Fs(\))32 b([Ji)o(],)g([D].)43 b(Assume)33 b Fq(k)e Fs(=)c Fj(C)58 b Fs(and)32 b(that)f(0)d Fp(6)p Fs(=)35 b(~)-57 b Fq(q)36 b Fs(is)31 b(not)456 4151 y(a)h(ro)s(ot)g(of)g(1.)43 b(Then)1494 4276 y(~)1478 4301 y Fq(U)38 b Fs(:=)28 b Fj(C)19 b Fp(h)1846 4276 y Fs(~)1817 4301 y Fq(E)12 b(;)1967 4276 y Fs(~)1945 4301 y Fq(F)i(;)2091 4276 y Fs(~)2066 4301 y Fq(K)7 b(;)2225 4276 y Fs(~)2200 4301 y Fq(K)2290 4260 y Fk(\000)p Fn(1)2384 4301 y Fp(i)456 4481 y Fs(where)760 4456 y(~)737 4481 y Fq(E)f(;)881 4456 y Fs(~)859 4481 y Fq(F)14 b(;)1006 4456 y Fs(~)980 4481 y Fq(K)39 b Fs(satisfy)33 b(the)g(relations)816 4717 y(~)790 4742 y Fq(K)902 4717 y Fs(~)880 4742 y Fq(E)g Fs(=)i(~)-56 b Fq(q)1136 4701 y Fn(2)1198 4717 y Fs(~)1175 4742 y Fq(E)1279 4717 y Fs(~)1253 4742 y Fq(K)7 b(;)1441 4717 y Fs(~)1415 4742 y Fq(K)1527 4717 y Fs(~)1505 4742 y Fq(F)41 b Fs(=)35 b(~)-56 b Fq(q)1760 4701 y Fk(\000)p Fn(2)1876 4717 y Fs(~)1854 4742 y Fq(F)1957 4717 y Fs(~)1931 4742 y Fq(K)7 b(;)2115 4717 y Fs(~)2092 4742 y Fq(E)2192 4717 y Fs(~)2170 4742 y Fq(F)36 b Fp(\000)2391 4717 y Fs(~)2369 4742 y Fq(F)2468 4717 y Fs(~)2445 4742 y Fq(E)e Fs(=)2691 4650 y(~)2665 4675 y Fq(K)2755 4639 y Fn(2)2816 4675 y Fp(\000)2942 4650 y Fs(~)2916 4675 y Fq(K)3006 4639 y Fk(\000)p Fn(2)p 2665 4719 436 4 v 2715 4811 a Fs(~)-56 b Fq(q)2755 4782 y Fn(2)2816 4811 y Fp(\000)30 b Fs(~)-56 b Fq(q)2963 4782 y Fk(\000)p Fn(2)456 5005 y Fs(with)31 b(coaction)g(\001\()1203 4980 y(~)1180 5005 y Fq(E)6 b Fs(\))28 b(=)1450 4980 y(~)1427 5005 y Fq(E)f Fp(\012)1650 4980 y Fs(~)1624 5005 y Fq(K)1714 4969 y Fk(\000)p Fn(1)1829 5005 y Fs(+)1951 4980 y(~)1925 5005 y Fq(K)h Fp(\012)2156 4980 y Fs(~)2134 5005 y Fq(E)6 b Fs(,)32 b(\001\()2412 4980 y(~)2390 5005 y Fq(F)14 b Fs(\))27 b(=)2658 4980 y(~)2636 5005 y Fq(F)34 b Fp(\012)2857 4980 y Fs(~)2831 5005 y Fq(K)2921 4969 y Fk(\000)p Fn(1)3036 5005 y Fs(+)3158 4980 y(~)3132 5005 y Fq(K)28 b Fp(\012)3363 4980 y Fs(~)3341 5005 y Fq(F)13 b Fs(,)456 5129 y(and)32 b(\001)753 5104 y(~)726 5129 y Fq(K)j Fs(=)974 5104 y(~)948 5129 y Fq(K)29 b Fp(\012)1186 5104 y Fs(~)1160 5129 y Fq(K)7 b Fs(.)p eop %%Page: 12 12 12 11 bop 456 236 a Fl(12)623 b(S.)33 b(MONTGOMER)-6 b(Y)33 b(AND)f(H.-J.)i(SCHNEIDER)456 425 y Fu(Theorem)j(3.5.)49 b Fs([MSm,)29 b(3.8])i Fg(Assume)g(that)h(the)f(p)-5 b(olynomial)30 b(ring)h Fj(C)20 b Fs([)p Fq(X)8 b Fs(])37 b Fg(is)31 b(a)472 525 y Fs(~)456 550 y Fq(U)10 b Fg(-mo)-5 b(dule)29 b(algebr)-5 b(a.)42 b(Then)29 b(ther)-5 b(e)30 b(exists)f Fq(Y)49 b Fp(2)28 b Fj(C)20 b Fs([)p Fq(X)8 b Fs(])36 b Fg(such)29 b(that)h Fj(C)20 b Fs([)p Fq(X)8 b Fs(])34 b(=)27 b Fj(C)20 b Fs([)p Fq(Y)27 b Fs(])p Fg(,)456 666 y(and)34 b(one)g(of)h(the)g(fol)5 b(lowing)33 b(two)i(p)-5 b(ossibilities)34 b(holds:)555 785 y(\(a\))746 760 y Fs(~)720 785 y Fq(K)29 b Fp(\001)22 b Fq(Y)49 b Fs(=)34 b(~)-56 b Fq(q)1138 749 y Fk(\000)p Fn(2)1232 785 y Fq(Y)22 b Fg(,)1398 760 y Fs(~)1375 785 y Fq(E)29 b Fp(\001)21 b Fq(Y)49 b Fs(=)35 b(~)-56 b Fq(q)1782 749 y Fn(2)1821 785 y Fs(1)p Fg(,)35 b(and)2146 760 y Fs(~)2124 785 y Fq(F)h Fp(\001)22 b Fq(Y)49 b Fs(=)27 b Fp(\000)7 b Fs(~)-56 b Fq(q)2606 749 y Fk(\000)p Fn(2)2701 785 y Fq(Y)2779 749 y Fn(2)2818 785 y Fg(;)555 909 y(\(b\))741 884 y Fs(~)715 909 y Fq(K)29 b Fp(\001)22 b Fq(Y)49 b Fs(=)34 b(~)-56 b Fq(q)1133 873 y Fn(2)1172 909 y Fq(Y)22 b Fg(,)1338 884 y Fs(~)1316 909 y Fq(E)28 b Fp(\001)22 b Fq(Y)48 b Fs(=)28 b Fp(\000)7 b Fs(~)-56 b Fq(q)1799 873 y Fn(2)1839 909 y Fq(Y)1917 873 y Fn(2)1956 909 y Fg(,)35 b(and)2233 884 y Fs(~)2211 909 y Fq(F)g Fp(\001)22 b Fq(Y)49 b Fs(=)35 b(~)-56 b Fq(q)2616 873 y Fk(\000)p Fn(2)2710 909 y Fs(1)p Fg(.)456 1034 y(Her)-5 b(e)35 b(b)-5 b(oth)918 1009 y Fs(~)895 1034 y Fq(E)41 b Fg(and)1220 1009 y Fs(~)1198 1034 y Fq(F)48 b Fg(act)35 b(as)g Fs(\()1660 1009 y(~)1634 1034 y Fq(K)1724 998 y Fk(\000)p Fn(1)1818 1034 y Fq(;)1888 1009 y Fs(~)1862 1034 y Fq(K)7 b Fs(\))p Fg(-derivations)34 b(of)g Fj(C)20 b Fs([)p Fq(Y)27 b Fs(])p Fg(.)555 1222 y Fs(W)-8 b(e)37 b(can)f(translate)1336 1197 y(~)1319 1222 y Fq(U)47 b Fs(in)m(to)35 b(the)h(curren)m(tly)h (more)e(standard)h(de\014nition)f(\(as)456 1346 y(in)i([K]\))i(of)f Fq(U)965 1361 y Fo(q)1004 1346 y Fs(\()p Fq(sl)1117 1361 y Fn(2)1156 1346 y Fs(\))g(as)h(follo)m(ws:)55 b(set)39 b Fq(F)52 b Fs(:=)2165 1321 y(~)2143 1346 y Fq(F)2246 1321 y Fs(~)2219 1346 y Fq(K)8 b Fs(,)40 b Fq(E)k Fs(:=)2660 1321 y(~)2634 1346 y Fq(K)2724 1310 y Fk(\000)p Fn(1)2841 1321 y Fs(~)2818 1346 y Fq(E)6 b Fs(,)41 b Fq(K)k Fs(:=)3259 1321 y(~)3233 1346 y Fq(K)3323 1310 y Fk(\000)p Fn(2)3417 1346 y Fs(,)456 1463 y(and)32 b Fq(q)g Fs(:=)j(~)-57 b Fq(q)897 1426 y Fk(\000)p Fn(2)992 1463 y Fs(.)43 b(It)33 b(is)f(then)h(straigh)m(tforw)m(ard)f(to)g(c)m(hec)m(k)j(that)815 1687 y Fq(K)7 b(E)34 b Fs(=)27 b Fq(q)1161 1645 y Fn(2)1200 1687 y Fq(E)6 b(K)r(;)45 b(K)7 b(F)41 b Fs(=)28 b Fq(q)1780 1645 y Fk(\000)p Fn(2)1874 1687 y Fq(F)14 b(K)r(;)44 b(E)6 b(F)35 b Fp(\000)23 b Fq(F)14 b(E)33 b Fs(=)2679 1619 y Fq(K)c Fp(\000)23 b Fq(K)2981 1583 y Fk(\000)p Fn(1)p 2679 1664 397 4 v 2722 1755 a Fq(q)j Fp(\000)d Fq(q)2938 1726 y Fk(\000)p Fn(1)456 1911 y Fs(and)43 b(that)f(\001\()p Fq(E)6 b Fs(\))46 b(=)f Fq(E)36 b Fp(\012)29 b Fq(K)37 b Fs(+)29 b(1)g Fp(\012)h Fq(E)49 b Fs(and)43 b(\001\()p Fq(F)14 b Fs(\))45 b(=)g Fq(F)e Fp(\012)30 b Fs(1)f(+)g Fq(K)3110 1875 y Fk(\000)p Fn(1)3234 1911 y Fp(\012)h Fq(F)14 b Fs(,)456 2028 y(as)39 b(in)g Fq(u)759 2043 y Fo(q)797 2028 y Fs(.)64 b(After)39 b(doing)g(this,)i(w)m(e)f (see)h(that)e(part)h(\(b\))f(of)g(Theorem)h(3.5)f(sa)m(ys)456 2144 y(that)c Fq(K)d Fp(\001)24 b Fq(Y)54 b Fs(=)33 b Fq(q)1104 2108 y Fn(2)1144 2144 y Fq(Y)21 b Fs(,)37 b Fq(E)30 b Fp(\001)24 b Fq(Y)54 b Fs(=)33 b Fp(\000)p Fq(q)t(Y)1863 2108 y Fn(2)1936 2144 y Fs(=)g Fq(\016)t(Y)2170 2108 y Fn(2)2209 2144 y Fs(,)k(and)f Fq(F)i Fp(\001)24 b Fq(Y)55 b Fs(=)33 b(1)g(=)g Fq(\015)5 b Fs(1.)53 b(As)36 b(in)456 2260 y(Corollary)30 b(3.2,)j Fq(\016)t(\015)f Fs(=)c Fp(\000)p Fq(q)t Fs(.)555 2376 y(It)41 b(is)e(in)m(teresting)h (to)g(note)g(that)g(in)g(the)h(generic)f(case,)j(there)e(are)f(t)m(w)m (o)h(es-)456 2493 y(sen)m(tially)g(distinct)h(actions,)j(whereas)g(in)d (the)h(ro)s(ot)f(of)h(unit)m(y)g(case,)j(there)e(is)456 2609 y(only)39 b(one.)66 b(The)40 b(case)h(corresp)s(onding)f(to)f (part)h(\(a\))f(of)h(Theorem)g(3.5)f(in)g(our)456 2725 y(Corollary)33 b(3.2)h(is)g(obtained)g(from)g(\(b\))g(b)m(y)i(setting)e Fq(u)2464 2689 y Fk(0)2518 2725 y Fs(:=)e Fq(u)2709 2689 y Fo(n)p Fk(\000)p Fn(1)2845 2725 y Fs(.)50 b(Of)34 b(course)i(in)456 2841 y(the)d(generic)f(case,)i(one)f(cannot)f(replace)h Fq(Y)54 b Fs(b)m(y)33 b Fq(Y)2341 2805 y Fo(n)p Fk(\000)p Fn(1)2479 2841 y Fs(.)555 3021 y(W)-8 b(e)25 b(close)g(this)f(section)h (with)f(an)h(application)d(to)i(ideals)g(in)g(smash)h(pro)s(ducts.)456 3137 y(When)42 b Fq(H)49 b Fs(=)42 b Fq(T)1054 3154 y Fo(n)1097 3135 y Fm(2)1136 3137 y Fs(\()p Fq(!)t Fs(\))e(acts)i(on)f Fq(A)g Fs(as)g(in)g(Theorem)g(2.5)g(and)g Fq(A)g Fs(is)f(a)h(\014eld,)i (it)456 3253 y(w)m(as)32 b(sho)m(wn)i(in)d([CFM])h(that)g Fq(A)p Fs(#)p Fq(H)40 b Fs(is)31 b(semisimple)e(\(in)i(fact,)h(the)h (pro)s(of)e(there)456 3370 y(assumed)f(that)g Fq(\014)j Fs(=)28 b Fq(!)33 b Fs(and)d(that)g Fq(\015)j Fs(=)27 b(1,)j(but)h(the)f(argumen)m(ts)g(w)m(ork)h(similarly)456 3486 y(for)k(the)g(more)g(general)g(constan)m(ts\).)54 b(Here)36 b(the)g(action)e(is)h(outer)h(in)f(the)h(sense)456 3602 y(that)e(the)i(action)e(of)g(no)h(non-trivial)d(Hopf)j(subalgebra) g(is)f(inner.)50 b(Hence)37 b(this)456 3718 y(result)22 b(ma)m(y)g(b)s(e)g(view)m(ed)i(as)e(a)g(w)m(eak)i(form)d(of)h (classical)e(results)j(on)f(outer)h(actions,)456 3835 y(as)33 b(for)f(example)h(Azuma)m(y)m(a's)h(theorem,)f(whic)m(h)h(sa)m (ys)g(that)f Fq(A)p Fs(#)p Fq(H)41 b Fs(is)33 b(simple)e(if)456 3951 y(a)h(group)h(algebra)f Fq(H)41 b Fs(acts)33 b(on)g(a)g(simple)f (algebra)g Fq(A)h Fs(and)g(the)h(group)f(action)f(is)456 4067 y(outer.)555 4183 y(One)j(migh)m(t)d(hop)s(e)i(that)g(the)g(same)g (is)f(true)i(for)e(the)i(action)e(of)g Fq(u)2997 4198 y Fo(q)3035 4183 y Fs(\()p Fq(sl)3148 4198 y Fn(2)3187 4183 y Fs(\))h(on)g(a)456 4300 y(\014eld)39 b Fq(A)g Fs(as)g(in)f(Corollary)g(3.2.)62 b(The)41 b(action)d(is)g(again)g (outer)h(but)h(the)f(smash)456 4416 y(pro)s(duct)32 b Fq(A)p Fs(#)p Fq(u)1031 4431 y Fo(q)1069 4416 y Fs(\()p Fq(sl)1182 4431 y Fn(2)1222 4416 y Fs(\))g(is)g(not)h(ev)m(en)h (semisimple,)c(as)j(w)m(e)h(will)c(see.)555 4543 y(First,)23 b Fq(u)867 4558 y Fo(q)927 4543 y Fs(is)e(unimo)s(dular,)g(with)h (non-zero)f(in)m(tegral)f(\003)28 b(=)f Fq(E)2772 4507 y Fo(n)p Fk(\000)p Fn(1)2910 4543 y Fq(F)2987 4507 y Fo(n)p Fk(\000)p Fn(1)3123 4543 y Fs(\()3161 4468 y Fi(P)3267 4495 y Fo(n)p Fk(\000)p Fn(1)3267 4572 y Fo(i)p Fn(=0)3420 4543 y Fq(K)3510 4507 y Fo(i)3539 4543 y Fs(\).)456 4659 y(This)37 b(can)h(b)s(e)g(v)m(eri\014ed)g(directly;)i(alternativ)m(ely) -8 b(,)37 b(it)f(can)i(b)s(e)g(seen)h(as)f(a)f(conse-)456 4776 y(quence)42 b(of)d(the)i(fact)f(that)f Fq(u)1554 4791 y Fo(q)1632 4776 y Fs(is)h(a)f(Hopf)h(image)f(of)g(the)i(Drinfeld) d(double)i(of)456 4892 y Fq(T)513 4909 y Fo(n)556 4890 y Fm(2)594 4892 y Fs(,)33 b(as)g(in)f(4.7.)456 5071 y Fu(Corollary)37 b(3.6.)49 b Fg(L)-5 b(et)32 b Fq(A)f Fg(b)-5 b(e)32 b(a)f Fq(u)1701 5086 y Fo(q)1738 5071 y Fs(\()p Fq(sl)1851 5086 y Fn(2)1891 5071 y Fs(\))p Fg(-mo)-5 b(dule)30 b(algebr)-5 b(a)31 b(as)g(in)g(Cor)-5 b(ol)5 b(lary)31 b(3.2.)456 5188 y(Then)j Fq(A)p Fs(#)p Fq(u)920 5203 y Fo(q)958 5188 y Fs(\()p Fq(sl)1071 5203 y Fn(2)1110 5188 y Fs(\))h Fg(has)f(a)h(non-zer)-5 b(o)34 b(nilp)-5 b(otent)34 b(ide)-5 b(al)34 b Fq(I)i Fs(:=)27 b Fq(A)p Fs(\003)p Fq(A)p Fg(.)p eop %%Page: 13 13 13 12 bop 3368 233 a Fl(13)456 425 y Fg(Pr)-5 b(o)g(of.)41 b Fs(First,)28 b(for)h(an)m(y)g Fq(a)f Fp(2)g Fq(A;)17 b(h)28 b Fp(2)g Fq(H)8 b Fs(,)29 b(it)f(is)g(easy)i(to)f(see)h(that)e Fq(ha)p Fs(\003)g(=)g(\()p Fq(h)15 b Fp(\001)g Fq(a)p Fs(\)\003)456 546 y(and)29 b(\003)p Fq(ah)e Fs(=)h(\003\()p 1054 466 66 4 v Fq(S)5 b Fs(\()p Fq(h)p Fs(\))15 b Fp(\001)g Fq(a)p Fs(\).)42 b(It)28 b(follo)m(ws)g(that)h Fq(I)36 b Fs(is)29 b(an)g(ideal)e(of)h Fq(A)p Fs(#)p Fq(u)2945 561 y Fo(q)2983 546 y Fs(.)43 b(W)-8 b(e)29 b(claim)456 663 y(that)j(\003)22 b Fp(\001)g Fq(A)27 b Fs(=)h(0.)555 779 y(T)-8 b(o)29 b(see)i(this,)e(note)g(that)g(Lemma)f(2.1\(b\))h (implies)e(that)h Fq(F)2726 743 y Fo(n)p Fk(\000)p Fn(1)2878 779 y Fp(\001)15 b Fq(A)28 b Fp(\022)g Fq(k)s Fs(,)i(since)456 895 y Fq(s)39 b Fs(=)g(0.)63 b(But)39 b(then)h Fq(E)6 b(F)1379 859 y Fo(n)p Fk(\000)p Fn(1)1543 895 y Fp(\001)26 b Fq(A)40 b Fs(=)f(0)g(and)g(so)g(certainly)g(\003)26 b Fp(\001)h Fq(A)39 b Fs(=)g(0.)63 b(It)40 b(no)m(w)456 1011 y(follo)m(ws)31 b(that)h Fq(I)1038 975 y Fn(2)1105 1011 y Fs(=)c Fq(A)p Fs(\003)p Fq(A)p Fs(\003)p Fq(A)f Fs(=)h(0.)p 3374 1011 4 66 v 3378 949 59 4 v 3378 1011 V 3436 1011 4 66 v 906 1249 a(4.)55 b Fr(A)m(ctions)38 b(of)g(the)f(Drinfel)-7 b('d)38 b(double)g Fq(D)s Fs(\()p Fq(H)8 b Fs(\))555 1423 y(F)-8 b(or)33 b(an)m(y)h(\014nite-dimensional) 29 b(Hopf)34 b(algebra)e Fq(H)8 b Fs(,)33 b(w)m(e)h(recall)e(that)h (the)g(Drin-)456 1539 y(fel'd)i(double)h Fq(D)s Fs(\()p Fq(H)8 b Fs(\))32 b(=)h(\()p Fq(H)1508 1503 y Fk(\003)1547 1539 y Fs(\))1585 1503 y Fo(cop)1723 1539 y Fq(.)-17 b(/)33 b(H)43 b Fs(is)36 b(giv)m(en)g(as)g(follo)m(ws:)49 b(as)36 b(a)f(coalgebra,)456 1656 y(it)27 b(is)h(simply)f Fq(H)1040 1620 y Fk(\003)p Fo(cop)1192 1656 y Fp(\012)14 b Fq(H)8 b Fs(.)42 b(The)29 b(algebra)e(structure)j(is)e(more)g (complicated,)f(and)456 1772 y(for)h(our)h(purp)s(oses)h(w)m(e)g(use)g (a)f(form)m(ula)e(from)h([R];)i(see)g(also)e([M)q(,)i(10.3.11].)41 b(That)456 1888 y(is,)32 b(for)g Fq(f)5 b(;)17 b(f)886 1852 y Fk(0)937 1888 y Fp(2)28 b Fq(H)1120 1852 y Fk(\003)1191 1888 y Fs(and)33 b Fq(h;)17 b(h)1537 1852 y Fk(0)1588 1888 y Fp(2)28 b Fq(H)8 b Fs(,)895 2075 y(\()p Fq(f)38 b(.)-17 b(/)28 b(h)p Fs(\)\()p Fq(f)1319 2034 y Fk(0)1370 2075 y Fq(.)-17 b(/)27 b(h)1534 2034 y Fk(0)1558 2075 y Fs(\))g(=)1727 1980 y Fi(X)1887 2075 y Fq(f)11 b Fs(\()p Fq(h)2040 2090 y Fn(1)2107 2075 y Fq(*)27 b(f)2291 2034 y Fk(0)2342 2075 y Fq(\()p 2467 1995 66 4 v 27 w(S)6 b(h)2589 2090 y Fn(3)2629 2075 y Fs(\))27 b Fq(.)-17 b(/)28 b(h)2859 2090 y Fn(2)2899 2075 y Fq(h)2955 2034 y Fk(0)2978 2075 y Fq(:)456 2262 y Fs(In)33 b(particular)d(w)m(e)k(ha)m (v)m(e)985 2449 y(\()p Fq(")27 b(.)-17 b(/)28 b(h)p Fs(\)\()p Fq(f)38 b(.)-17 b(/)28 b Fs(1\))f(=)1750 2354 y Fi(X)1894 2449 y Fp(h)p Fq(f)1981 2464 y Fn(3)2020 2449 y Fq(;)17 b(h)2120 2464 y Fn(1)2159 2449 y Fp(ih)p Fq(f)2285 2464 y Fn(1)2324 2449 y Fq(;)p 2368 2369 V 17 w(S)6 b(h)2490 2464 y Fn(3)2530 2449 y Fp(i)p Fq(f)2617 2464 y Fn(2)2684 2449 y Fq(.)-17 b(/)27 b(h)2848 2464 y Fn(2)2888 2449 y Fq(:)-2459 b Fs(\(4.1\))555 2642 y(F)-8 b(or)32 b(simplicit)m(y)e(w)m (e)k(write)e Fq(h)c Fs(=)f(\()p Fq(")g(.)-17 b(/)28 b(h)p Fs(\))33 b(and)f Fq(f)39 b Fs(=)27 b(\()p Fq(f)39 b(.)-17 b(/)27 b Fs(1\).)555 2758 y(W)-8 b(e)29 b(no)m(w)g(sp)s(ecialize)e(to)h (the)g Fq(n)1681 2722 y Fn(2)1721 2758 y Fs(-dimensional)d(T)-8 b(aft)28 b(algebra)f Fq(H)36 b Fs(=)27 b Fq(T)3107 2774 y Fo(n)3150 2756 y Fm(2)3189 2758 y Fs(\()p Fq(!)t Fs(\))g(as)456 2874 y(in)k(1.1.)44 b(In)32 b(this)h(case)g(it)f(is)g(kno)m(wn)i(that)e Fq(H)2088 2838 y Fk(\003)2155 2846 y Fp(\030)2155 2878 y Fs(=)2260 2874 y Fq(H)8 b Fs(;)32 b(th)m(us)i(w)m(e)f(ma)m(y)g(write) 1007 3045 y Fq(H)1096 3004 y Fk(\003)1162 3045 y Fs(=)28 b Fq(k)s Fp(h)p Fq(G;)17 b(X)8 b Fp(j)27 b Fq(G)1701 3004 y Fo(n)1775 3045 y Fs(=)h Fq(";)17 b(X)2058 3004 y Fo(n)2131 3045 y Fs(=)28 b(0)p Fq(;)17 b(X)8 b(G)27 b Fs(=)g Fq(!)t(GX)8 b Fp(i)-2438 b Fs(\(4.2\))456 3222 y(where)46 b(\001\()p Fq(G)p Fs(\))k(=)f Fq(G)31 b Fp(\012)g Fq(G)p Fs(,)49 b(\001\()p Fq(X)8 b Fs(\))50 b(=)f Fq(X)39 b Fp(\012)31 b Fq(")g Fs(+)f Fq(G)h Fp(\012)h Fq(X)8 b Fs(,)48 b Fp(h)p Fq(G;)17 b Fs(1)p Fp(i)48 b Fs(=)i(1,)e(and)456 3338 y Fp(h)p Fq(X)r(;)17 b Fs(1)p Fp(i)27 b Fs(=)g Fq(")886 3353 y Fo(H)949 3334 y Ff(\003)989 3338 y Fs(\()p Fq(X)8 b Fs(\))27 b(=)h(0.)555 3454 y(The)34 b(dual)e(pairing)e(b)s(et)m(w)m (een)35 b Fq(H)40 b Fs(and)32 b Fq(H)2079 3418 y Fk(\003)2151 3454 y Fs(is)g(determined)g(b)m(y)914 3625 y Fp(h)p Fq(G;)17 b(g)t Fp(i)26 b Fs(=)i Fq(!)1359 3584 y Fk(\000)p Fn(1)1452 3625 y Fq(;)45 b Fp(h)p Fq(G;)17 b(x)p Fp(i)27 b Fs(=)g(0)p Fq(;)44 b Fp(h)p Fq(X)r(;)17 b(g)t Fp(i)27 b Fs(=)h(0)p Fq(;)44 b Fs(and)28 b Fp(h)p Fq(X)r(;)17 b(x)p Fp(i)27 b Fs(=)h(1)p Fq(:)-2731 b Fs(\(4.3\))456 3813 y Fu(Lemma)37 b(4.4.)49 b Fq(D)s Fs(\()p Fq(H)8 b Fs(\))43 b Fg(is)h(gener)-5 b(ate)g(d)43 b(as)h(an)g(algebr)-5 b(a)43 b(by)h Fp(f)p Fq(x;)17 b(g)t(;)g(X)r(;)g(G)p Fp(g)p Fg(.)71 b(The)456 3929 y(r)-5 b(elations)30 b(among)f(these)h(gener)-5 b(ators,)31 b(in)f(addition)g(to)h(the)g(r)-5 b(elations)30 b(in)g Fq(H)38 b Fg(and)456 4046 y Fq(H)545 4009 y Fk(\003)584 4046 y Fg(,)c(ar)-5 b(e)35 b(as)f(fol)5 b(lows:)590 4217 y Fq(g)t(G)27 b Fs(=)h Fq(Gg)t(;)43 b(xG)28 b Fs(=)g Fq(!)1376 4175 y Fk(\000)p Fn(1)1470 4217 y Fq(Gx;)44 b(X)8 b(g)31 b Fs(=)d Fq(!)2009 4175 y Fk(\000)p Fn(1)2102 4217 y Fq(g)t(X)r(;)44 b Fs(and)28 b Fq(xX)j Fp(\000)22 b Fq(X)8 b(x)28 b Fs(=)g Fq(G)22 b Fp(\000)g Fq(g)t(:)456 4399 y Fg(Pr)-5 b(o)g(of.)41 b Fs(It)30 b(is)f(clear)g(that)g(the)h (giv)m(en)g(elemen)m(ts)g(generate)h Fq(D)s Fs(\()p Fq(H)8 b Fs(\).)41 b(T)-8 b(o)30 b(c)m(hec)m(k)i(the)456 4515 y(relations,)41 b(w)m(e)h(use)g(\(4.1\))o(.)68 b(First,)43 b Fq(g)t(G)d Fs(=)i Fp(h)p Fq(G;)17 b(g)t Fp(ih)p Fq(G;)g(g)2536 4479 y Fk(\000)p Fn(1)2627 4515 y Fp(i)p Fq(Gg)45 b Fs(=)d Fp(h)p Fq(G;)17 b Fs(1)p Fp(i)p Fq(Gg)44 b Fs(=)456 4631 y Fq(Gg)t Fs(.)555 4748 y(F)-8 b(or)27 b(the)h(next)h(relation,)e(w)m (e)h(use)h(\001)1867 4763 y Fn(2)1907 4748 y Fs(\()p Fq(x)p Fs(\))f(=)f Fq(x)12 b Fp(\012)g Fs(1)g Fp(\012)g Fs(1)g(+)g Fq(g)k Fp(\012)c Fq(x)g Fp(\012)g Fs(1)g(+)g Fq(g)k Fp(\012)c Fq(g)k Fp(\012)c Fq(x)456 4869 y Fs(and)32 b(the)h(fact)g(that)f Fp(h)p Fq(G;)p 1377 4789 V 17 w(S)5 b(x)p Fp(i)28 b Fs(=)g(0.)43 b(Then)499 5045 y Fq(xG)28 b Fs(=)g Fp(h)p Fq(G;)17 b(x)p Fp(ih)p Fq(G;)g Fs(1)p Fp(i)p Fq(G)p Fs(1)j(+)i Fp(h)p Fq(G;)17 b(g)t Fp(ih)p Fq(G;)g Fs(1)p Fp(i)p Fq(Gx)j Fs(+)i Fp(h)p Fq(G;)17 b(g)t Fp(ih)p Fq(G;)p 2667 4965 V 17 w(S)s(x)p Fp(i)p Fq(Gg)31 b Fs(=)d Fq(!)3148 5004 y Fk(\000)p Fn(1)3242 5045 y Fq(Gx:)456 5216 y Fs(The)33 b(third)f(relation)f(is)h(similar.)p eop %%Page: 14 14 14 13 bop 456 236 a Fl(14)623 b(S.)33 b(MONTGOMER)-6 b(Y)33 b(AND)f(H.-J.)i(SCHNEIDER)555 425 y Fs(F)-8 b(or)31 b(the)i(fourth)f(relation,)e(w)m(e)j(again)e(use)i(\001)2228 440 y Fn(2)2268 425 y Fs(\()p Fq(x)p Fs(\))f(as)g(w)m(ell)f(as)i(\001) 2948 440 y Fn(2)2987 425 y Fs(\()p Fq(X)8 b Fs(\).)43 b(Then)456 541 y(with)32 b Fq(h)c Fs(=)f Fq(x;)17 b(f)39 b Fs(=)27 b Fq(X)41 b Fs(in)31 b(\(4.1\),)h(w)m(e)i(ha)m(v)m(e)687 729 y Fq(xX)i Fs(=)27 b Fp(h)p Fq(";)17 b(x)p Fp(ih)p Fq(X)r(;)g Fs(1)p Fp(i)p Fq(")p Fs(1)j(+)i Fp(h)p Fq(";)17 b(g)t Fp(ih)p Fq(X)r(;)g Fs(1)p Fp(i)p Fq("x)j Fs(+)i Fp(h)p Fq(";)17 b(g)t Fp(ih)p Fq(X)r(;)p 2729 649 66 4 v 17 w(S)t(x)p Fp(i)p Fq("g)929 879 y Fs(+)p Fp(h)p Fq(";)g(x)p Fp(ih)p Fq(G;)g Fs(1)p Fp(i)p Fq(X)8 b Fs(1)20 b(+)i Fp(h)p Fq(";)17 b(g)t Fp(ih)p Fq(G;)g Fs(1)p Fp(i)p Fq(X)8 b(x)19 b Fs(+)j Fp(h)p Fq(";)17 b(g)t Fp(ih)p Fq(G;)p 2839 799 V 17 w(S)s(x)p Fp(i)p Fq(X)8 b(g)929 1029 y Fs(+)p Fp(h)p Fq(X)r(;)17 b(x)p Fp(ih)p Fq(G;)g Fs(1)p Fp(i)p Fq(G)p Fs(1)j(+)i Fp(h)p Fq(X)r(;)17 b(g)t Fp(ih)p Fq(G;)g Fs(1)p Fp(i)p Fq(Gx)k Fs(+)h Fp(h)p Fq(X)r(;)17 b(g)t Fp(ih)p Fq(G;)p 2928 949 V 17 w(S)t(x)p Fp(i)p Fq(Gg)770 1175 y Fs(=)83 b(0)22 b(+)g(0)g Fp(\000)g Fq(g)k Fs(+)c(0)g(+)g Fq(X)8 b(x)22 b Fs(+)g(0)g(+)g Fq(G)g Fs(+)g(0)g(+)g(0)770 1320 y(=)83 b Fq(X)8 b(x)22 b Fs(+)g Fq(G)g Fp(\000)h Fq(g)t(:)p 3374 1502 4 66 v 3378 1440 59 4 v 3378 1502 V 3436 1502 4 66 v 555 1700 a Fs(W)-8 b(e)24 b(no)m(w)h(apply)e(the)i(results)f(of)f(Section)h(2)f(to)h Fq(D)s Fs(\()p Fq(H)8 b Fs(\)-actions)22 b(on)h(the)h(algebra)456 1816 y Fq(A)p Fs(.)54 b(As)37 b(in)e(the)i(last)e(section,)j(w)m(e)f (apply)f(Theorem)g(2.2)g(and)g(Theorem)h(2.5)f(to)456 1932 y(t)m(w)m(o)d(di\013eren)m(t)g(copies)f(of)h(the)g(T)-8 b(aft)32 b(algebra)f(in)h Fq(D)s Fs(\()p Fq(H)8 b Fs(\).)456 2121 y Fu(Theorem)37 b(4.5.)49 b Fg(L)-5 b(et)33 b Fq(A)g Fg(b)-5 b(e)33 b(an)f Fq(n)p Fg(-dimensional)f Fq(k)s Fg(-algebr)-5 b(a)32 b(with)h(no)g(non-zer)-5 b(o)456 2237 y(nilp)g(otent)34 b(elements,)g(and)h(let)g Fq(H)43 b Fg(b)-5 b(e)34 b(the)h(T)-7 b(aft)35 b(algebr)-5 b(a)34 b Fq(T)2619 2254 y Fo(n)2662 2235 y Fm(2)2701 2237 y Fs(\()p Fq(!)t Fs(\))p Fg(.)44 b(Assume)35 b(that)456 2353 y Fq(A)42 b Fg(is)f(a)h Fq(D)s Fs(\()p Fq(H)8 b Fs(\))p Fg(-mo)-5 b(dule)40 b(algebr)-5 b(a)41 b(such)h(that)g Fq(x)28 b Fp(\001)f Fq(A)40 b Fp(6)p Fs(=)h(0)g Fg(or)h(that)g Fq(X)36 b Fp(\001)27 b Fq(A)40 b Fp(6)p Fs(=)h(0)p Fg(.)456 2470 y(Then)34 b(ther)-5 b(e)34 b(exists)h Fq(u)27 b Fp(2)h Fq(A)35 b Fg(and)g Fs(0)27 b Fp(6)p Fs(=)h Fq(\014)6 b(;)17 b(\015)32 b Fp(2)c Fq(k)38 b Fg(such)c(that)456 2586 y(\(a\))g Fq(A)28 b Fs(=)f Fq(k)s Fs(\()p Fq(u)p Fs(\))35 b Fg(and)f Fq(u)1290 2550 y Fo(n)1364 2586 y Fs(=)28 b Fq(\014)33 b Fp(2)28 b Fq(k)s Fg(;)456 2702 y(\(b\))34 b Fq(g)25 b Fp(\001)c Fq(u)28 b Fs(=)f Fq(!)t(u)34 b Fg(and)g Fq(G)22 b Fp(\001)f Fq(u)27 b Fs(=)h Fq(!)1667 2666 y Fk(\000)p Fn(1)1760 2702 y Fq(u)p Fg(,)35 b(and)f(so)g(b)-5 b(oth)35 b Fq(g)j Fg(and)c Fq(G)g 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Fs(\))p Fg(-mo)-5 b(dule)36 b(algebr)-5 b(a)38 b(with)f(the)456 3516 y(given)d(actions.)456 3705 y(Pr)-5 b(o)g(of.)41 b Fs(First)e(assume)i(that)f Fq(x)28 b Fp(\001)f Fq(A)41 b Fp(6)p Fs(=)g(0.)66 b(W)-8 b(e)41 b(apply)f(Theorem)h(2.5)f(to)g Fq(H)48 b Fp(\032)456 3821 y Fq(D)s Fs(\()p Fq(H)8 b Fs(\))47 b(to)g(\014nd)i Fq(u)k Fp(2)h Fq(A)48 b Fs(and)g Fq(\014)6 b(;)17 b(\015)58 b Fp(2)c Fq(k)d Fs(suc)m(h)e(that)f Fq(A)54 b Fs(=)g Fq(k)s Fs(\()p Fq(u)p Fs(\),)d Fq(u)3127 3785 y Fo(n)3227 3821 y Fs(=)j Fq(\014)6 b Fs(,)456 3937 y Fq(x)11 b Fp(\001)g Fq(u)27 b Fs(=)h Fq(!)t(u)p Fs(,)f(and)g Fq(x)11 b Fp(\001)g Fq(u)27 b Fs(=)h Fq(\015)5 b Fs(1.)41 b(This)28 b(pro)m(v)m(es)h(\(a\))d(and)i(the)f(\014rst)h(parts)f(of)g (\(b\))g(and)456 4053 y(\(c\).)42 b(No)m(w)31 b(b)m(y)f(Lemma)f(4.4,)g Fq(g)t Fs(\()p Fq(G)16 b Fp(\001)g Fq(u)p Fs(\))27 b(=)h Fq(G)p Fs(\()p Fq(g)19 b Fp(\001)d Fq(u)p Fs(\))27 b(=)h Fq(G)p Fs(\()p Fq(!)t(u)p Fs(\))e(=)h Fq(!)t Fs(\()p Fq(G)16 b Fp(\001)g Fq(u)p Fs(\).)42 b(Th)m(us)456 4169 y Fq(G)28 b Fp(\001)g Fq(u)43 b Fp(2)g Fq(A)898 4184 y Fn(1)938 4169 y Fs(,)h(whic)m(h)e(is)f(one-dimensional)f(b)m(y)i (Lemma)e(2.4;)46 b(it)41 b(follo)m(ws)g(that)456 4286 y Fq(G)22 b Fp(\001)h Fq(u)29 b Fs(=)g Fq(\013)q(u)p Fs(,)34 b(for)f(some)g Fq(\013)d Fp(2)g Fq(k)s Fs(.)47 b(But)34 b(w)m(e)h(also)d(ha)m(v)m(e)j Fq(x)p Fs(\()p Fq(G)24 b Fp(\001)e Fq(u)p Fs(\))29 b(=)g Fq(!)2986 4250 y Fk(\000)p Fn(1)3080 4286 y Fq(G)p Fs(\()p Fq(x)23 b Fp(\001)g Fq(u)p Fs(\),)456 4402 y(or)32 b Fq(\013)q(x)22 b Fp(\001)g Fq(u)27 b Fs(=)h Fq(\013)q(\015)5 b Fs(1)27 b(=)g Fq(!)1315 4366 y Fk(\000)p Fn(1)1409 4402 y Fq(\015)5 b Fs(1.)43 b(Th)m(us)34 b Fq(\013)29 b Fs(=)e Fq(!)2090 4366 y Fk(\000)p Fn(1)2184 4402 y Fs(,)33 b(\014nishing)e(\(b\).)555 4518 y(No)m(w)43 b(consider)f(the)g(action)f(of)h Fq(H)1865 4482 y Fk(\003)p Fo(cop)2048 4518 y Fp(\032)i Fq(D)s Fs(\()p Fq(H)8 b Fs(\).)70 b(Since)42 b(\001)2860 4482 y Fo(cop)2965 4518 y Fs(\()p Fq(X)8 b Fs(\))43 b(=)h Fq(")28 b Fp(\012)456 4634 y Fq(X)g Fs(+)21 b Fq(X)28 b Fp(\012)22 b Fq(G)p Fs(,)32 b(the)g(elemen)m(t)g Fq(D)e Fs(:=)e Fq(X)8 b(G)1941 4598 y Fk(\000)p Fn(1)2067 4634 y Fs(is)31 b(an)h(\()p Fq(";)17 b(G)2504 4598 y Fk(\000)p Fn(1)2598 4634 y Fs(\)-)31 b(primitiv)m(e)f(elemen)m(t)456 4751 y(and)i(satis\014es)h Fq(D)s(G)1165 4714 y Fk(\000)p Fn(1)1287 4751 y Fs(=)28 b Fq(!)1456 4714 y Fk(\000)p Fn(1)1549 4751 y Fq(G)1626 4714 y Fk(\000)p Fn(1)1720 4751 y Fq(D)s Fs(.)44 b(Th)m(us)34 b Fq(H)2211 4714 y Fk(\003)p Fo(cop)2377 4751 y Fs(=)28 b Fq(k)s Fp(h)p Fq(G)2651 4714 y Fk(\000)p Fn(1)2745 4751 y Fq(;)17 b(D)s Fp(i)2939 4723 y(\030)2940 4755 y Fs(=)3044 4751 y Fq(T)3101 4767 y Fo(n)3144 4748 y Fm(2)3183 4751 y Fs(\()p Fq(!)3286 4714 y Fk(\000)p Fn(1)3379 4751 y Fs(\).)456 4867 y(Apply)38 b(Theorem)g(2.2)g(to)f Fq(H)1547 4831 y Fk(\003)p Fo(cop)1686 4867 y Fs(,)j(replacing)d Fq(\033)42 b Fs(with)37 b Fq(G)2577 4831 y Fk(\000)p Fn(1)2671 4867 y Fs(,)j Fq(\026)e Fs(with)f Fq(!)3127 4831 y Fk(\000)p 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y(w)m(e)d(apply)f(the)h(fourth)g(relation)e(in)g(Lemma)h(4.4)g(to)g Fq(u)p Fs(.)43 b(Th)m(us)1210 723 y Fq(x)p Fs(\()p Fq(X)31 b Fp(\001)22 b Fq(u)p Fs(\))f Fp(\000)i Fq(X)8 b Fs(\()p Fq(x)22 b Fp(\001)g Fq(u)p Fs(\))27 b(=)h Fq(G)22 b Fp(\001)f Fq(u)h Fp(\000)h Fq(g)i Fp(\001)d Fq(u;)456 904 y Fs(or)31 b Fq(x)20 b Fp(\001)g Fs(\()p Fq(\025u)848 868 y Fn(2)887 904 y Fs(\))g Fp(\000)g Fq(\015)5 b(X)28 b Fp(\001)19 b Fs(1)28 b(=)f(\()p Fq(!)1537 868 y Fk(\000)p Fn(1)1651 904 y Fp(\000)20 b Fq(!)t Fs(\))p Fq(u)p Fs(.)42 b(No)m(w)32 b Fq(x)21 b Fp(\001)e Fq(u)2376 868 y Fn(2)2443 904 y Fs(=)28 b Fq(\015)5 b Fs(\(1)19 b(+)h Fq(!)t Fs(\))p Fq(u)30 b Fs(b)m(y)j(Lemma)456 1020 y(2.1,)f(and)h(so)f Fq(\025\015)5 b Fs(\(1)22 b(+)g Fq(!)t Fs(\))p Fq(u)27 b Fs(=)g(\()p Fq(!)1661 984 y Fk(\000)p Fn(1)1777 1020 y Fp(\000)22 b Fq(!)t Fs(\))p Fq(u)p Fs(.)43 b(Th)m(us)34 b Fq(\025\015)e Fs(=)c Fq(!)2661 984 y Fk(\000)p Fn(1)2777 1020 y Fp(\000)22 b Fs(1.)555 1136 y(This)32 b(\014nishes)g(the)f(case) i Fq(x)19 b Fp(\001)g Fq(A)28 b Fp(6)p Fs(=)g(0;)j(note)h(w)m(e)g(ha)m (v)m(e)g(sho)m(wn)h(that)e Fq(X)c Fp(\001)19 b Fq(A)28 b Fp(6)p Fs(=)g(0.)456 1252 y(Assuming)33 b Fq(X)e Fp(\001)23 b Fq(A)31 b Fp(6)p Fs(=)f(0,)k(w)m(e)h(w)m(ould)g(obtain)e(similarly)d (that)k Fq(x)24 b Fp(\001)f Fq(A)30 b Fp(6)p Fs(=)g(0,)35 b(so)f(the)456 1369 y(previous)f(argumen)m(ts)f(w)m(ould)h(apply)-8 b(.)555 1485 y(It)30 b(is)g(clear)f(from)g(the)h(pro)s(of)f(that)h(the) g Fq(D)s Fs(\()p Fq(H)8 b Fs(\)-action)28 b(is)h(determined)h(b)m(y)h (the)456 1601 y Fq(H)8 b Fs(-action.)39 b(The)27 b(con)m(v)m(erse)i (follo)m(ws)c(from)g(Lemma)g(2.1,)i(as)g Fq(A)f Fs(will)e(b)s(e)j(a)f (mo)s(dule)456 1717 y(algebra)j(for)i(b)s(oth)f Fq(H)39 b Fs(and)31 b Fq(H)1568 1681 y Fk(\003)p Fo(cop)1707 1717 y Fs(;)g(b)m(y)h(construction)f(the)g(relations)f(in)g(Lemma)456 1834 y(4.4)i(are)g(satis\014ed,)h(and)g(so)g Fq(A)g Fs(will)d(b)s(e)j (a)f Fq(D)s Fs(\()p Fq(H)8 b Fs(\)-)31 b(mo)s(dule.)p 3374 1834 4 66 v 3378 1771 59 4 v 3378 1834 V 3436 1834 4 66 v 555 2029 a(No)m(w,)43 b(analogously)c(to)i(Corollary)d(3.6,)43 b(w)m(e)e(sho)m(w)h(that)e Fq(A)p Fs(#)p Fq(D)s Fs(\()p Fq(H)8 b Fs(\))40 b(is)g(also)456 2145 y(not)32 b(semisimple.)555 2262 y(First,)e(b)m(y)h(a)f(result)f(of)h(Radford)f([R])h(for)g(an)m(y) g(\014nite-dimensional)d Fq(H)8 b Fs(,)30 b Fq(D)s Fs(\()p Fq(H)8 b Fs(\))456 2378 y(is)36 b(unimo)s(dular)d(with)k(in)m(tegral)d (0)h Fp(6)p Fs(=)f(\003)g(=)g Fq(\025)g(.)-17 b(/)34 b(t)p Fs(,)k(where)g Fq(\025)e Fs(is)g(a)g(left)g(in)m(tegral)456 2494 y(of)c Fq(H)656 2458 y Fk(\003)727 2494 y Fs(and)h Fq(t)g Fs(is)f(a)g(righ)m(t)g(in)m(tegral)f(of)h Fq(H)8 b Fs(.)43 b(Th)m(us)34 b(when)g Fq(H)h Fs(=)27 b Fq(T)2803 2511 y Fo(n)2846 2492 y Fm(2)2918 2494 y Fs(as)32 b(ab)s(o)m(v)m(e,) 1227 2764 y(\003)c(=)f(\()1469 2640 y Fo(n)p Fk(\000)p Fn(1)1464 2669 y Fi(X)1479 2879 y Fo(i)p Fn(=0)1625 2764 y Fq(G)1702 2723 y Fo(i)1730 2764 y Fs(\))p Fq(X)1857 2723 y Fo(n)p Fk(\000)p Fn(1)2021 2764 y Fq(.)-17 b(/)28 b(x)2185 2723 y Fo(n)p Fk(\000)p Fn(1)2322 2764 y Fs(\()2365 2640 y Fo(n)p Fk(\000)p Fn(1)2360 2669 y Fi(X)2371 2879 y Fo(j)t Fn(=0)2521 2764 y Fq(g)2572 2723 y Fo(j)2608 2764 y Fs(\))p Fq(:)456 3057 y Fu(Corollary)37 b(4.6.)49 b Fg(L)-5 b(et)39 b Fq(A)g Fg(b)-5 b(e)38 b(a)g Fq(D)s Fs(\()p Fq(H)8 b Fs(\))p Fg(-mo)-5 b(dule)37 b(algebr)-5 b(a)38 b(as)g(in)h(The)-5 b(or)g(em)37 b(4.5.)456 3173 y(Then)d Fq(A)p Fs(#)p Fq(D)s Fs(\()p Fq(H)8 b Fs(\))34 b Fg(has)g(a)h(non-zer)-5 b(o)34 b(nilp)-5 b(otent)34 b(ide)-5 b(al)34 b Fq(I)i Fs(:=)28 b Fq(A)p Fs(\003)p Fq(A)p Fg(.)456 3362 y(Pr)-5 b(o)g(of.)41 b Fs(First,)c(for)g(an)m(y)g Fq(a)f Fp(2)g Fq(A)p Fs(,)i Fq(w)g Fp(2)e Fq(D)s Fs(\()p Fq(H)8 b Fs(\),)37 b(it)f(is)h(easy)h(to)f(see)h(that)f Fq(w)s(a)p Fs(\003)e(=)456 3478 y(\()p Fq(w)28 b Fp(\001)e Fq(a)p Fs(\)\003)37 b Fp(2)h Fq(A)p Fs(\003;)j(similarly)-8 b(,)36 b(\003)p Fq(aw)j Fp(2)f Fs(\003)p Fq(A)p Fs(.)61 b(It)38 b(follo)m(ws)f(that)h Fq(I)46 b Fs(is)38 b(an)g(ideal)f(of)456 3594 y Fq(A)p Fs(#)p Fq(D)s Fs(\()p Fq(H)8 b Fs(\).)43 b(W)-8 b(e)33 b(claim)d(that)i(\003)22 b Fp(\001)g Fq(A)27 b Fs(=)h(0.)555 3710 y(No)m(w)g(since)f Fq(x)10 b Fp(\001)g Fq(u)28 b Fs(=)g Fq(\015)5 b Fs(1,)28 b(it)d(follo)m(ws)h(from)g(Lemma) f(2.1\(b\))h(that)h Fq(x)2971 3674 y Fo(n)p Fk(\000)p Fn(1)3119 3710 y Fp(\001)10 b Fq(A)28 b Fp(\022)g Fq(k)s Fs(.)456 3827 y(But)h(then)i Fq(X)8 b(x)1010 3790 y Fo(n)p Fk(\000)p Fn(1)1163 3827 y Fp(\001)16 b Fq(A)28 b Fs(=)g(0)h(and)h(so)g (certainly)f(\003)16 b Fp(\001)g Fq(A)27 b Fs(=)h(0.)42 b(It)30 b(no)m(w)g(follo)m(ws)f(that)456 3943 y Fq(I)507 3907 y Fn(2)574 3943 y Fs(=)e Fq(A)p Fs(\003)p Fq(A)p Fs(\003)p Fq(A)h Fs(=)f Fq(A)p Fs(\(\003)22 b Fp(\001)g Fq(A)p Fs(\)\003)p Fq(A)27 b Fs(=)h(0.)p 3374 4059 V 3378 3997 59 4 v 3378 4059 V 3436 4059 4 66 v 456 4247 a Fu(Remark)37 b(4.7.)49 b Fs(An)40 b(alternate)g(approac)m(h)g(to)g (the)h(results)g(in)e(Section)h(3)g(can)456 4363 y(b)s(e)32 b(giv)m(en)h(using)f(the)h(results)g(of)f(this)h(section.)555 4480 y(First,)f(it)g(is)h(w)m(ell-kno)m(wn)g(that)f Fq(u)1791 4495 y Fo(q)1829 4480 y Fs(\()p Fq(sl)1942 4495 y Fn(2)1981 4480 y Fs(\))h(is)g(a)f(Hopf)h(homomorphic)e(image)g(of)456 4596 y Fq(D)s Fs(\()p Fq(H)8 b Fs(\),)25 b(for)f Fq(H)36 b Fs(=)27 b Fq(T)1175 4613 y Fo(n)1218 4594 y Fm(2)1257 4596 y Fs(\()p Fq(!)t Fs(\),)e(b)m(y)h(setting)f Fq(!)31 b Fs(=)c Fq(q)2133 4560 y Fk(\000)p Fn(2)2252 4596 y Fs(and)e(de\014ning)g(\010)j(:)g Fq(D)s Fs(\()p Fq(H)8 b Fs(\))27 b Fp(!)g Fq(u)3407 4611 y Fo(q)456 4712 y Fs(as)32 b(follo)m(ws:)456 4893 y Fq(G)27 b(.)-17 b(/)28 b Fs(1)f Fp(7!)h Fq(K)r(;)44 b(")27 b(.)-17 b(/)28 b(g)j Fp(7!)c Fq(K)1506 4852 y Fk(\000)p Fn(1)1600 4893 y Fq(;)45 b(")27 b(.)-17 b(/)28 b(x)g Fp(7!)f Fq(F)s(;)44 b Fs(and)28 b Fq(X)36 b(.)-17 b(/)27 b Fs(1)h Fp(7!)p 2815 4813 79 4 v 27 w Fq(E)34 b Fs(:=)27 b Fp(\000)p Fs(\()p Fq(q)t Fp(\000)p Fq(q)3337 4852 y Fk(\000)p Fn(1)3432 4893 y Fs(\))p Fq(E)456 5074 y Fs(Moreo)m(v)m(er)37 b(Ker\(\010\))e(=)f Fq(D)s Fs(\()p Fq(H)8 b Fs(\))p Fq(k)s Fp(G)1703 5038 y Fn(+)1761 5074 y Fs(,)38 b(where)f Fp(G)j Fs(=)34 b Fp(f)p Fq(G)2447 5038 y Fo(i)2500 5074 y Fp(\012)25 b Fq(g)2653 5038 y Fo(i)2680 5074 y Fp(j)34 b Fs(0)g Fp(\024)h Fq(i)f(<)g(n)p Fp(g)p Fs(.)54 b(See)456 5191 y([K,)32 b(IX.6].)p eop %%Page: 16 16 16 15 bop 456 236 a Fl(16)623 b(S.)33 b(MONTGOMER)-6 b(Y)33 b(AND)f(H.-J.)i(SCHNEIDER)456 425 y Fu(Corollary)j(4.8.)49 b Fg(Each)33 b(action)f(of)h Fq(D)s Fs(\()p Fq(H)8 b Fs(\))33 b Fg(on)f Fq(A)c Fs(=)g Fq(k)s Fs(\()p Fq(u)p Fs(\))k Fg(as)h(in)g(The)-5 b(or)g(em)32 b(4.5)456 541 y(induc)-5 b(es)34 b(a)g(c)-5 b(orr)g(esp)g(onding)34 b(action)g(of)h Fq(u)1969 556 y Fo(q)2006 541 y Fs(\()p Fq(sl)2119 556 y Fn(2)2159 541 y Fs(\))f Fg(on)h Fq(A)g Fg(via)p 886 638 79 4 v 886 718 a Fq(E)28 b Fp(\001)22 b Fq(u)27 b Fs(:=)h Fq(X)i Fp(\001)21 b Fq(u;)44 b(F)36 b Fp(\001)22 b Fq(u)27 b Fs(:=)h Fq(x)22 b Fp(\001)g Fq(u;)44 b Fs(and)28 b Fq(K)h Fp(\001)22 b Fq(u)27 b Fs(:=)h Fq(g)2766 677 y Fk(\000)p Fn(1)2882 718 y Fp(\001)21 b Fq(u:)456 886 y Fg(Conversely)33 b(any)i(action)g(of)f Fq(u)1606 901 y Fo(q)1679 886 y Fg(on)g Fq(A)h Fg(lifts)g(to)g(an)f (action)h(of)f Fq(D)s Fs(\()p Fq(H)8 b Fs(\))34 b Fg(on)h Fq(A)p Fg(.)456 1067 y(Pr)-5 b(o)g(of.)41 b Fs(W)-8 b(e)53 b(m)m(ust)f(only)g(c)m(hec)m(k)j(that)d(for)g(an)g(action)g(as)g(in)g (Theorem)h(4.5,)456 1183 y(Ker\(\010\))7 b Fp(\001)g Fq(A)27 b Fs(=)g(0.)41 b(Ho)m(w)m(ev)m(er)27 b(for)e(an)m(y)g(suc)m(h)i (action,)e(\()p Fq(G)j(.)-17 b(/)27 b(g)t Fs(\))7 b Fp(\001)g Fq(u)26 b Fs(=)h Fq(G)7 b Fp(\001)g Fs(\()p Fq(!)t(u)p Fs(\))25 b(=)j Fq(u)p Fs(.)456 1299 y(Consequen)m(tly)38 b(\()p Fq(")25 b Fp(\012)g Fs(1)f Fp(\000)h Fq(G)1527 1263 y Fo(i)1580 1299 y Fp(\012)h Fq(g)1734 1263 y Fo(i)1761 1299 y Fs(\))f Fp(\001)f Fq(u)34 b Fs(=)g(0,)j(for)f(all)e Fq(i)p Fs(,)k(and)e(so)h Fq(k)s Fp(G)3015 1263 y Fn(+)3099 1299 y Fp(\001)24 b Fq(A)34 b Fs(=)h(0.)456 1416 y(Th)m(us)f (Ker\(\010\))22 b Fp(\001)g Fq(A)28 b Fs(=)f(0)33 b(and)f(the)h Fq(u)1777 1431 y Fo(q)1815 1416 y Fs(-action)e(is)h(w)m(ell-de\014ned.) p 3374 1416 4 66 v 3378 1353 59 4 v 3378 1416 V 3436 1416 4 66 v 555 1596 a(It)45 b(follo)m(ws)f(that)h(in)f(some)h(sense,) 50 b(Corollary)43 b(3.2)h(and)i(Theorem)f(4.5)f(are)456 1712 y(equiv)-5 b(alen)m(t.)48 b(W)-8 b(e)35 b(use)g(this)f(metho)s(d)g (to)g(describ)s(e)h(the)g(other)f(p)s(ossible)g(actions)456 1828 y(of)e Fq(D)s Fs(\()p Fq(H)8 b Fs(\))31 b(on)i Fq(A)28 b Fs(=)f Fq(k)s Fs(\()p Fq(u)p Fs(\),)32 b(analogously)f(to)h(Theorem)h (3.1.)456 2009 y Fu(Corollary)k(4.9.)49 b Fg(L)-5 b(et)34 b Fq(A)27 b Fs(=)h Fq(k)s Fs(\()p Fq(u)p Fs(\))k Fg(b)-5 b(e)33 b(an)g Fq(n)p Fg(-dimensional)e Fq(k)s Fg(-algebr)-5 b(a.)43 b(Assume)456 2125 y(that)32 b Fq(A)h Fg(is)f(a)g Fq(D)s Fs(\()p Fq(H)8 b Fs(\))p Fg(-mo)-5 b(dule)31 b(algebr)-5 b(a)31 b(such)i(that)f Fq(g)20 b Fp(\001)d Fq(u)27 b Fs(=)g Fq(\013)q(u)p Fg(,)32 b(for)g(some)g Fq(\013)c Fp(2)g Fq(k)s Fg(,)456 2242 y(and)42 b(such)h(that)h Fq(x)29 b Fp(\001)f Fq(A)44 b Fp(6)p Fs(=)f(0)p Fg(.)70 b(Then)42 b(ther)-5 b(e)43 b(exist)g Fs(0)g Fp(6)p Fs(=)h Fq(\014)6 b(;)17 b(\015)5 b(;)17 b(\016)46 b Fp(2)e Fq(k)i Fg(and)d Fs(0)g Fp(\024)456 2358 y Fq(s;)17 b(t;)g(l)36 b Fp(\024)f Fq(n)25 b Fp(\000)g Fs(1)38 b Fg(with)h Fq(t)p Fs(\(1)25 b Fp(\000)g Fq(s)p Fs(\))1657 2330 y Fp(\030)1658 2362 y Fs(=)1769 2358 y(1)38 b Fg(\(mo)-5 b(d)38 b(n\))g(and)g Fq(s)25 b Fs(+)g Fq(l)2672 2330 y Fp(\030)2673 2362 y Fs(=)2784 2358 y(2)39 b Fg(\(mo)-5 b(d)37 b(n\))i(such)456 2474 y(that)555 2590 y(\(a\))i Fq(u)782 2554 y Fo(n)869 2590 y Fs(=)f Fq(\014)6 b Fg(,)43 b Fq(\013)e Fs(=)f Fq(!)1403 2554 y Fo(t)1432 2590 y Fg(,)k Fq(G)27 b Fp(\001)g Fq(u)40 b Fs(=)g Fq(\013)1940 2554 y Fk(\000)p Fn(1)2034 2590 y Fq(u)p Fg(,)j(and)e Fq(g)k Fg(and)c Fq(G)g Fg(have)g(or)-5 b(der)42 b Fq(n)g Fg(in)456 2706 y Fq(Aut)p Fs(\()p Fq(A)p Fs(\))p Fg(;)555 2823 y(\(b\))35 b Fq(x)22 b Fp(\001)g Fq(u)27 b Fs(=)h Fq(\015)5 b(u)1141 2787 y Fo(s)1212 2823 y Fg(and)34 b Fq(X)c Fp(\001)22 b Fq(u)27 b Fs(=)h Fq(\025u)1862 2787 y Fo(l)1887 2823 y Fg(;)555 2939 y(\(c\))36 b Fq(\025)29 b Fs(=)h Fp(\000)p Fs(\()p Fq(q)d Fp(\000)c Fq(q)1240 2903 y Fk(\000)p Fn(1)1334 2939 y Fs(\))p Fq(\016)t Fg(,)36 b(wher)-5 b(e)36 b Fq(!)d Fs(=)c Fq(q)2008 2903 y Fk(\000)p Fn(2)2138 2939 y Fg(and)36 b Fq(\015)5 b(\016)40 b Fg(is)35 b(given)g(as)h(in)g(The)-5 b(or)g(em)456 3055 y(3.1.)555 3171 y(Conversely,)35 b(any)g(choic)-5 b(e)35 b(of)g Fq(\014)6 b(;)17 b(\015)5 b(;)17 b(\016)39 b Fg(and)c Fq(s;)17 b(t;)g(l)37 b Fg(satisfying)e(the)h(ab)-5 b(ove)35 b(r)-5 b(ela-)456 3288 y(tions)34 b(de\014nes)g(a)h Fq(D)s Fs(\()p Fq(H)8 b Fs(\))p Fg(-mo)-5 b(dule)33 b(algebr)-5 b(a)34 b(structur)-5 b(e)36 b(on)e Fq(A)p Fg(.)456 3469 y(Pr)-5 b(o)g(of.)41 b Fs(W)-8 b(e)31 b(use)h(the)g(previous)g (corollary)d(to)i(get)g(the)h(corresp)s(onding)f(induced)456 3585 y(action)g(of)h Fq(u)915 3600 y Fo(q)953 3585 y Fs(\()p Fq(sl)1066 3600 y Fn(2)1105 3585 y Fs(\))h(on)f Fq(A)p Fs(.)44 b(No)m(w)33 b(apply)f(Theorem)h(3.1.)p 3374 3585 V 3378 3522 59 4 v 3378 3585 V 3436 3585 4 66 v 555 3765 a(W)-8 b(e)23 b(can)g(also)f(giv)m(e)h(an)f(alternate)g (pro)s(of)g(of)g(Corollary)f(3.6,)k(as)e(follo)m(ws.)38 b(Giv)m(en)456 3881 y(an)d(action)f(of)h Fq(D)s Fs(\()p Fq(H)8 b Fs(\))34 b(on)h Fq(A)p Fs(,)i(and)e(the)h(corresp)s(onding)f (action)f(of)h Fq(u)3018 3896 y Fo(q)3056 3881 y Fs(\()p Fq(sl)3169 3896 y Fn(2)3208 3881 y Fs(\),)h(the)456 3997 y(homomorphism)22 b(in)j(Remark)g(4.7)g(induces)h(a)g(corresp)s(onding) f(homomorphism)456 4114 y(of)34 b(the)i(smash)f(pro)s(duct)g Fq(A)p Fs(#)p Fq(D)s Fs(\()p Fq(H)8 b Fs(\))35 b(on)m(to)g Fq(A)p Fs(#)p Fq(u)2274 4129 y Fo(q)2312 4114 y Fs(\()p Fq(sl)2425 4129 y Fn(2)2464 4114 y Fs(\).)51 b(In)35 b(particular,)f(if)g(\003)h(is)456 4230 y(the)k(in)m(tegral)f(in)g Fq(D)s Fs(\()p Fq(H)8 b Fs(\),)40 b(then)g(\010\(\003\))f(is)f(the)i (in)m(tegral)d(in)i Fq(u)2727 4245 y Fo(q)2764 4230 y Fs(\()p Fq(sl)2877 4245 y Fn(2)2917 4230 y Fs(\).)62 b(It)40 b(follo)m(ws)456 4346 y(that)c(the)i(nilp)s(oten)m(t)d(ideal)h (of)g Fq(A)p Fs(#)p Fq(D)s Fs(\()p Fq(H)8 b Fs(\))37 b(constructed)h(in)e(Corollary)f(4.6)i(has)456 4462 y(as)28 b(a)g(non-zero)h(image)d(the)j(ideal)e(of)h Fq(A)p Fs(#)p Fq(u)2027 4477 y Fo(q)2065 4462 y Fs(\()p Fq(sl)2178 4477 y Fn(2)2217 4462 y Fs(\))g(considered)h(in)f(Corollary)f(3.6,)456 4578 y(whic)m(h)33 b(m)m(ust)f(therefore)i(b)s(e)e(nilp)s(oten)m(t.)968 4809 y(5.)55 b Fr(Yetter-Drinfel)-7 b('d)37 b(Module)g(Algebras)555 4983 y Fs(In)43 b(this)g(section)g(w)m(e)g(giv)m(e)g(some)g(general)f (results)h(ab)s(out)f(Y)-8 b(etter-Drinfeld)456 5099 y(mo)s(dule)40 b(algebras)h(for)h(an)m(y)g(Hopf)g(algebra)f Fq(H)8 b Fs(,)44 b(and)e(then)h(sp)s(ecialize)d(to)i(the)456 5216 y(T)-8 b(aft)32 b(algebras.)p eop %%Page: 17 17 17 16 bop 3368 233 a Fl(17)555 425 y Fs(Let)46 b Fq(M)56 b Fs(b)s(e)45 b(a)g(left)f Fq(H)8 b Fs(-mo)s(dule)43 b(and)i(a)g(left)g Fq(H)8 b Fs(-como)s(dule)43 b(via)h Fq(\032)50 b Fs(:)f Fq(M)60 b Fp(!)456 541 y Fq(H)29 b Fp(\012)23 b Fq(M)10 b Fs(.)44 b(The)34 b(usual)e(Y)-8 b(etter-Drinfel'd)31 b(condition)g(is)755 714 y Fq(\032)p Fs(\()p Fq(h)22 b Fp(\001)g Fq(m)p Fs(\))28 b(=)1225 619 y Fi(X)1369 714 y Fs(\()p Fq(h)23 b Fp(\001)e Fq(m)p Fs(\))1658 729 y Fk(\000)p Fn(1)1775 714 y Fp(\012)i Fs(\()p Fq(h)f Fp(\001)g Fq(m)p Fs(\))2164 729 y Fn(0)2231 714 y Fs(=)2335 619 y Fi(X)2495 714 y Fq(h)2551 729 y Fn(1)2591 714 y Fq(m)2676 729 y Fk(\000)p Fn(1)2770 714 y Fq(S)6 b(h)2892 729 y Fn(3)2954 714 y Fp(\012)23 b Fq(h)3110 729 y Fn(2)3171 714 y Fp(\001)f Fq(m)3306 729 y Fn(0)456 714 y Fs(\(5.1\))456 897 y(for)33 b(all)f Fq(m)e Fp(2)g Fq(M)5 b(;)17 b(h)30 b Fp(2)g Fq(H)8 b Fs(.)46 b(W)-8 b(e)34 b(will)e(just)i(refer)g(to)g(this)f(as)h(the)g Fq(Y)21 b(D)s Fs(-)p Fq(condition)p Fs(,)456 1013 y(and)32 b(the)h(category)g(of)f(\(left,)g(left\))g Fq(H)8 b Fs(-Y)-8 b(etter-Drinfel'd)30 b(mo)s(dules)h(as)3091 977 y Fo(H)3091 1039 y(H)3158 1013 y Fp(Y)8 b Fq(D)s Fs(.)555 1130 y(The)54 b(left)e Fq(H)8 b Fs(-como)s(dule)50 b(structure)k(dualizes)f(as)g (usual)f(to)g(a)h(righ)m(t)f Fq(H)3373 1093 y Fk(\003)3412 1130 y Fs(-)456 1246 y(mo)s(dule)24 b(structure)j(on)f Fq(M)10 b Fs(,)28 b(via)d Fq(m)8 b Fp(\016)g Fq(f)39 b Fs(=)1986 1171 y Fi(P)2092 1246 y Fp(h)p Fq(f)5 b(;)17 b(m)2313 1261 y Fk(\000)p Fn(1)2407 1246 y Fp(i)p Fq(m)2531 1261 y Fn(0)2570 1246 y Fs(,)28 b(all)23 b Fq(f)39 b Fp(2)28 b Fq(H)3023 1210 y Fk(\003)3062 1246 y Fq(;)17 b(m)27 b Fp(2)i Fq(M)10 b Fs(.)456 1367 y(This)29 b(in)f(turn)i (induces)f(a)g(left)g Fq(H)1677 1331 y Fk(\003)1716 1367 y Fs(-mo)s(dule)e(structure)j(on)f Fq(M)40 b Fs(using)p 3021 1287 66 4 v 29 w Fq(S)6 b Fs(;)30 b(that)f(is,)1034 1550 y Fq(f)k Fp(\001)22 b Fq(m)28 b Fs(=)1381 1456 y Fi(X)1525 1550 y Fp(h)p 1564 1470 V Fq(S)6 b(f)f(;)17 b(m)1812 1565 y Fk(\000)p Fn(1)1906 1550 y Fp(i)p Fq(m)2030 1565 y Fn(0)2125 1550 y Fp(8)p Fq(m)28 b Fp(2)h Fq(M)5 b(;)17 b(f)38 b Fp(2)28 b Fq(H)2800 1509 y Fk(\003)2839 1550 y Fq(:)-2410 b Fs(\(5.2\))456 1734 y(Con)m(v)m(ersely)-8 b(,)45 b(if)40 b Fq(M)52 b Fs(is)40 b(a)h(\014nite-dimensional)d(left)i Fq(H)2482 1698 y Fk(\003)2521 1734 y Fs(-mo)s(dule,)h(then)h Fq(M)52 b Fs(b)s(e-)456 1850 y(comes)36 b(a)f(left)g Fq(H)8 b Fs(-como)s(dule)34 b(using)p 1824 1770 V 35 w Fq(S)6 b Fs(.)53 b(Note,)37 b(ho)m(w)m(ev)m(er,)i(that)p 2857 1770 V 36 w Fq(S)i Fs(induces)c(the)456 1966 y(co)s(opp)s(osite)i (mo)s(dule)f(algebra)h(structure;)44 b(that)c(is,)h(if)e Fq(M)51 b Fs(=)39 b Fq(A)p Fs(,)j(a)e(\014nite-)f(di-)456 2083 y(mensional)32 b Fq(k)s Fs(-algebra,)h(then)i(using)f(\(5.2\),)h Fq(A)f Fs(is)g(a)g(left)f Fq(H)8 b Fs(-como)s(dule)32 b(algebra)456 2199 y Fp(\()-17 b(\))59 b Fq(A)33 b Fs(is)f(a)g(left)g Fq(H)1243 2163 y Fk(\003)p Fo(cop)1382 2199 y Fs(-mo)s(dule)f(algebra.) 555 2315 y(The)d(next)g(lemma)d(is)i(due)g(to)g(S.)g(Ma)5 b(jid,)28 b(see)g([M)q(,)g(10.6.16].)41 b(W)-8 b(e)27 b(giv)m(e)g(a)g(short)456 2431 y(pro)s(of)k(for)h(completeness,)i(and)f 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y(see)i(that)821 3308 y(\()p Fq(hf)11 b Fs(\))22 b Fp(\001)g Fq(m)28 b Fs(=)1300 3213 y Fi(X)1444 3308 y Fp(h)p Fq(f)1531 3323 y Fn(3)1571 3308 y Fq(;)17 b(h)1671 3323 y Fn(1)1710 3308 y Fp(ih)p Fq(f)1836 3323 y Fn(1)1875 3308 y Fq(;)p 1919 3228 V 17 w(S)5 b(h)2040 3323 y Fn(3)2080 3308 y Fp(i)p Fq(f)2167 3323 y Fn(2)2228 3308 y Fp(\001)22 b Fs(\()p Fq(h)2372 3323 y Fn(2)2434 3308 y Fp(\001)g Fq(m)p Fs(\))904 3505 y(=)1063 3410 y Fi(X)1207 3505 y Fp(h)p 1246 3425 V Fq(S)6 b(f)1360 3520 y Fn(3)1399 3505 y Fq(;)17 b(S)6 b(h)1565 3520 y Fn(1)1604 3505 y Fp(ih)p 1682 3425 V Fq(S)f(f)1795 3520 y Fn(1)1835 3505 y Fq(;)17 b(h)1935 3520 y Fn(3)1974 3505 y Fp(ih)p 2052 3425 V Fq(S)6 b(f)2166 3520 y Fn(2)2205 3505 y Fq(;)17 b Fs(\()p Fq(h)2343 3520 y Fn(2)2404 3505 y Fp(\001)22 b Fq(m)p Fs(\))2577 3520 y Fk(\000)p Fn(1)2672 3505 y Fp(i)p Fs(\()p Fq(h)2805 3520 y Fn(2)2866 3505 y Fp(\001)g Fq(m)p Fs(\))3039 3520 y Fn(0)904 3701 y Fs(=)1063 3607 y Fi(X)1207 3701 y Fp(h)p 1246 3621 V Fq(S)6 b(f)f(;)17 b Fs(\()p Fq(S)6 b(h)1569 3716 y Fn(1)1608 3701 y Fs(\)\()p Fq(h)1740 3716 y Fn(2)1802 3701 y Fp(\001)21 b Fq(m)p Fs(\))1974 3716 y Fk(\000)p Fn(1)2069 3701 y Fq(h)2125 3716 y Fn(3)2165 3701 y Fp(i)p Fs(\()p Fq(h)2298 3716 y Fn(2)2359 3701 y Fp(\001)h Fq(m)p Fs(\))2532 3716 y Fn(0)456 3904 y Fs(Also)32 b Fq(h)22 b Fp(\001)g Fs(\()p Fq(f)33 b Fp(\001)21 b Fq(m)p Fs(\))28 b(=)1227 3829 y Fi(P)1349 3904 y Fq(h)22 b Fp(\001)g(h)p 1516 3824 V Fq(S)5 b(f)g(;)17 b(m)1763 3919 y Fk(\000)p Fn(1)1858 3904 y Fp(i)p Fq(m)1982 3919 y Fn(0)2049 3904 y Fs(=)2153 3829 y Fi(P)2258 3904 y Fp(h)p 2297 3824 V Fq(S)5 b(f)g(;)17 b(m)2544 3919 y Fk(\000)p Fn(1)2639 3904 y Fp(i)p Fq(h)22 b Fp(\001)g Fq(m)2891 3919 y Fn(0)2931 3904 y Fs(.)555 4020 y(It)30 b(follo)m(ws)f(that)g Fq(M)41 b Fs(is)29 b(a)h(left)f Fq(D)s Fs(\()p Fq(H)8 b Fs(\)-mo)s(dule)55 b Fp(\()-17 b(\))57 b Fs(\()p Fq(hf)11 b Fs(\))16 b Fp(\001)h Fq(m)28 b Fs(=)f Fq(h)17 b Fp(\001)f Fs(\()p Fq(f)28 b Fp(\001)16 b Fq(m)p Fs(\),)456 4136 y(for)36 b(all)e Fq(h)h Fp(2)g Fq(H)r(;)17 b(f)45 b Fp(2)35 b Fq(H)1350 4100 y Fk(\003)1389 4136 y Fq(;)17 b(m)35 b Fp(2)g Fq(M)82 b Fp(\()-17 b(\))2084 4061 y Fi(P)2205 4136 y Fq(m)2290 4151 y Fk(\000)p Fn(1)2410 4136 y Fp(\012)25 b Fq(h)g Fp(\001)g Fq(m)2731 4151 y Fn(0)2805 4136 y Fs(=)2916 4061 y Fi(P)3021 4136 y Fs(\()p Fq(S)6 b(h)3181 4151 y Fn(1)3220 4136 y Fs(\)\()p Fq(h)3352 4151 y Fn(2)3417 4136 y Fp(\001)456 4252 y Fq(m)p Fs(\))579 4267 y Fk(\000)p Fn(1)673 4252 y Fq(h)729 4267 y Fn(3)791 4252 y Fp(\012)23 b Fs(\()p Fq(h)985 4267 y Fn(2)1046 4252 y Fp(\001)f Fq(m)p Fs(\))1219 4267 y Fn(0)1259 4252 y Fs(,)32 b(for)g(all)f Fq(h)d Fp(2)g Fq(H)r(;)17 b(m)28 b Fp(2)g Fq(M)10 b Fs(.)555 4369 y(This)22 b(last)f(condition)g(is)g(easily)g(seen)i(to)f(b)s(e)g (equiv)-5 b(alen)m(t)22 b(to)f(the)h Fq(Y)g(D)s Fs(-condition)456 4485 y(\(5.1\).)43 b(This)32 b(pro)m(v)m(es)j(the)e(Lemma.)p 3374 4485 4 66 v 3378 4422 59 4 v 3378 4485 V 3436 4485 4 66 v 555 4664 a(W)-8 b(e)48 b(no)m(w)h(sp)s(ecialize)d(to)i Fq(H)61 b Fs(=)53 b Fq(T)1866 4680 y Fo(n)1909 4661 y Fm(2)1948 4664 y Fs(\()p Fq(!)t Fs(\))f(=)i Fq(k)s Fp(h)p Fq(g)t(;)17 b(x)p Fp(i)46 b Fs(and)i(its)f(dual)g Fq(H)3276 4628 y Fk(\003)3369 4664 y Fs(=)456 4780 y Fq(k)s Fp(h)p Fq(G;)17 b(X)8 b Fp(i)p Fs(.)40 b(W)-8 b(e)29 b(\014rst)f(note)g(that)f (\(4.3\))h(extends)h(to)f(all)d(of)j Fq(H)35 b Fs(and)28 b Fq(H)2964 4744 y Fk(\003)3003 4780 y Fs(;)h(the)f(pro)s(of)456 4896 y(is)k(a)g(straigh)m(tforw)m(ard)g(induction,)g(and)g(is)h (presumably)f(w)m(ell-kno)m(wn.)456 5076 y Fu(Lemma)37 b(5.4.)49 b Fp(h)p Fq(X)1204 5040 y Fo(q)1241 5076 y Fq(G)1318 5040 y Fo(r)1356 5076 y Fq(;)17 b(x)1455 5040 y Fo(i)1484 5076 y Fq(y)1536 5040 y Fo(j)1571 5076 y Fp(i)28 b Fs(=)f Fq(\016)1784 5091 y Fo(i;q)1883 5002 y Fi(Q)1977 5028 y Fo(q)1977 5106 y(z)s Fn(=1)2123 4996 y Fi(\000)2169 5032 y Fo(z)2169 5111 y Fn(1)2205 4996 y Fi(\001)2250 5115 y Fo(!)2296 5097 y Ff(\000)p Fm(1)2383 5076 y Fq(!)2448 5040 y Fk(\000)p Fo(j)t(r)456 5210 y Fg(wher)-5 b(e)731 5129 y Fi(\000)776 5166 y Fo(z)777 5244 y Fn(1)812 5129 y Fi(\001)858 5249 y Fo(!)904 5230 y Ff(\000)p Fm(1)1018 5210 y Fs(=)28 b(1)22 b(+)g Fq(!)1356 5174 y Fk(\000)p Fn(1)1472 5210 y Fs(+)g Fp(\001)17 b(\001)g(\001)j Fs(+)i Fq(!)1871 5174 y Fk(\000)p Fn(\()p Fo(z)s Fk(\000)p Fn(1\))2110 5210 y Fg(.)p eop %%Page: 18 18 18 17 bop 456 236 a Fl(18)623 b(S.)33 b(MONTGOMER)-6 b(Y)33 b(AND)f(H.-J.)i(SCHNEIDER)555 425 y Fs(Next)44 b(w)m(e)h(consider)e(our)h(algebra)e Fq(A)h Fs(as)h(in)e(Sections)i (2-4)e(and)i(determine)456 541 y(the)38 b(left)f Fq(H)8 b Fs(-coactions)36 b(corresp)s(onding)i(to)g(the)g(p)s(ossible)f(left)g Fq(H)2921 505 y Fk(\003)2960 541 y Fs(-actions,)h(as)456 658 y(in)30 b(\(5.2\).)43 b(Recall)29 b(from)i(the)g(pro)s(of)g(of)g (Theorem)g(4.5)g(that)g Fq(H)2756 622 y Fk(\003)p Fo(cop)2923 658 y Fs(=)d Fq(k)s Fp(h)p Fq(\033)n(;)17 b(D)s Fp(i)3367 630 y(\030)3368 662 y Fs(=)456 774 y Fq(T)513 791 y Fo(n)556 772 y Fm(2)594 774 y Fs(\()p Fq(!)697 738 y Fk(\000)p Fn(1)791 774 y Fs(\),)32 b(b)m(y)i(setting)e Fq(\033)g Fs(=)27 b Fq(G)1611 738 y Fk(\000)p Fn(1)1738 774 y Fs(and)33 b Fq(D)d Fs(=)e Fq(X)8 b(G)2309 738 y Fk(\000)p Fn(1)2403 774 y Fs(.)456 986 y Fu(Prop)s(osition)35 b(5.5.)49 b Fg(L)-5 b(et)43 b Fq(A)f Fs(=)g Fq(k)s Fs([)p Fq(u)g Fp(j)g Fq(u)1998 949 y Fo(n)2087 986 y Fs(=)f Fq(\014)6 b Fs(])p Fg(,)45 b Fq(\014)i Fp(2)c Fq(k)s Fg(.)68 b(Then)42 b Fq(A)h Fg(is)f(a)g(left)456 1102 y Fq(H)545 1066 y Fk(\003)p Fo(cop)684 1102 y Fg(-mo)-5 b(dule)30 b(algebr)-5 b(a)30 b(via)g Fq(G)13 b Fp(\001)g Fq(u)27 b Fs(=)h Fq(!)1910 1066 y Fo(t)1939 1102 y Fq(u)i Fg(and)g Fq(X)21 b Fp(\001)13 b Fq(u)27 b Fs(=)h Fq(\016)t(u)2643 1066 y Fo(s)2679 1102 y Fg(,)k(wher)-5 b(e)30 b Fs(0)d Fp(6)p Fs(=)h Fq(\016)j Fp(2)d Fq(k)s Fg(,)456 1218 y Fq(t)p Fs(\(1)10 b Fp(\000)g Fq(s)p Fs(\))786 1190 y Fp(\030)786 1222 y Fs(=)891 1218 y(1)29 b Fg(\(mo)-5 b(d)28 b Fq(n)p Fg(\),)i Fs(0)e Fp(\024)g Fq(s;)17 b(t)28 b Fp(\024)g Fq(n)10 b Fp(\000)g Fs(1)p Fg(.)42 b(This)29 b(action)f(determines)g(a)h(unique)456 1334 y(left)34 b Fq(H)8 b Fg(-c)-5 b(omo)g(dule)34 b(algebr)-5 b(a)34 b(structur)-5 b(e)36 b(on)e Fq(A)h Fg(satisfying)g(\(5.2\),)f (given)g(by)1194 1649 y Fq(\032)p Fs(\()p Fq(u)p Fs(\))27 b(=)1570 1524 y Fo(t)1511 1554 y Fi(X)1507 1763 y Fo(m)p Fn(=0)1676 1649 y Fq(a)1727 1664 y Fo(m)1793 1649 y Fq(x)1848 1608 y Fo(m)1915 1649 y Fq(g)1966 1608 y Fk(\000)p Fo(m)p Fn(+)p Fo(t)2190 1649 y Fp(\012)22 b Fq(u)2345 1608 y Fo(m)p Fn(\()p Fo(s)p Fk(\000)p Fn(1\)+1)2679 1649 y Fq(:)456 1957 y Fg(F)-7 b(or)34 b(al)5 b(l)34 b Fq(s)p Fg(,)h Fq(a)939 1972 y Fn(0)1006 1957 y Fs(=)28 b(1)p Fg(.)44 b(When)35 b Fq(s)27 b Fp(\025)h Fs(2)35 b Fg(and)f Fq(m)28 b Fp(\025)h Fs(1)p Fg(,)722 2273 y Fq(a)773 2288 y Fo(m)868 2273 y Fs(=)e(\()p Fp(\000)p Fq(\016)t Fs(\))1171 2232 y Fo(m)1238 2273 y Fq(!)1303 2232 y Fo(m)p Fn(\(1)p Fk(\000)p Fo(t)p Fn(\))1550 2189 y Fs(\()1588 2114 y Fi(P)1693 2140 y Fo(s)p Fk(\000)p Fn(1)1693 2218 y Fo(j)1722 2227 y Fm(1)1756 2218 y Fn(=0)1867 2189 y Fq(!)1932 2153 y Fk(\000)p Fo(j)2016 2162 y Fm(1)2049 2153 y Fo(t)2079 2189 y Fs(\))17 b Fp(\001)g(\001)g(\001)d Fs(\()2304 2114 y Fi(P)2409 2140 y Fn(\()p Fo(m)p Fk(\000)p Fn(1\)\()p Fo(s)p Fk(\000)p Fn(1\))2409 2218 y Fo(j)2438 2227 y Fh(m)p Ff(\000)p Fm(1)2575 2218 y Fn(=0)2815 2189 y Fq(!)2880 2153 y Fk(\000)p Fo(j)2964 2162 y Fh(m)p Ff(\000)p Fm(1)3100 2153 y Fo(t)3130 2189 y Fs(\))p 1550 2250 1618 4 v 1774 2279 a Fi(Q)1868 2305 y Fo(m)p Fk(\000)p Fn(1)1868 2382 y Fo(z)s Fn(=1)2025 2353 y Fs(\(1)21 b(+)h Fq(!)2296 2324 y Fk(\000)p Fn(1)2412 2353 y Fs(+)g Fp(\001)17 b(\001)g(\001)k Fs(+)h Fq(!)2812 2324 y Fk(\000)p Fo(z)2906 2353 y Fs(\))456 2571 y Fg(When)34 b Fq(s)28 b Fs(=)f(0)p Fg(,)35 b(then)g Fq(\032)p Fs(\()p Fq(u)p Fs(\))27 b(=)h Fq(g)d Fp(\012)e Fq(u)e Fp(\000)i Fq(\016)t(x)f Fp(\012)h Fs(1)p Fg(.)456 2783 y(Pr)-5 b(o)g(of.)41 b Fs(W)-8 b(e)33 b(\014rst)g(c)m(hec)m(k)j (that)c(the)i(giv)m(en)f(action)f(of)g Fq(G)h Fs(and)g Fq(X)40 b Fs(determines)33 b(an)456 2899 y Fq(H)545 2863 y Fk(\003)p Fo(cop)684 2899 y Fs(-mo)s(dule)20 b(algebra)h(action.)40 b(First,)23 b Fq(G)q Fp(\001)q Fq(u)k Fs(=)h Fq(!)2324 2863 y Fo(t)2353 2899 y Fq(u)22 b Fs(giv)m(es)g(that)h Fq(\033)5 b Fp(\001)q Fq(u)27 b Fs(=)g Fq(!)3201 2863 y Fk(\000)p Fo(t)3285 2899 y Fq(u)h Fs(=)456 3016 y Fq(\026)515 2979 y Fo(t)544 3016 y Fq(u)p Fs(.)40 b(Th)m(us)27 b(b)m(y)e(Theorem)g (2.2,)h(for)e(an)m(y)i(0)h Fp(6)p Fs(=)h Fq(\016)2161 2979 y Fk(0)2211 3016 y Fp(2)h Fq(k)s Fs(,)d(there)f(is)g(an)f Fq(H)2961 2979 y Fk(\003)p Fo(cop)3100 3016 y Fs(-mo)s(dule)456 3132 y(algebra)33 b(action)g(on)h Fq(A)g Fs(giv)m(en)g(b)m(y)h Fq(D)26 b Fp(\001)d Fq(u)30 b Fs(=)g Fq(\016)2127 3096 y Fk(0)2150 3132 y Fq(u)2206 3096 y Fo(s)2243 3132 y Fs(,)k(where)h Fq(t)p Fs(\(1)24 b Fp(\000)f Fq(s)p Fs(\))2947 3104 y Fp(\030)2948 3136 y Fs(=)3055 3132 y(1)34 b(mo)s(d)f Fq(n)p Fs(.)456 3248 y(Since)26 b Fq(D)31 b Fs(=)c Fq(X)8 b(G)1085 3212 y Fk(\000)p Fn(1)1179 3248 y Fs(,)28 b(it)e(follo)m(ws)f (that)h Fq(X)18 b Fp(\001)10 b Fq(u)27 b Fs(=)g Fq(\016)t(u)2271 3212 y Fo(s)2334 3248 y Fs(do)s(es)g(giv)m(e)g(an)f Fq(H)2961 3212 y Fk(\003)p Fo(cop)3100 3248 y Fs(-mo)s(dule)456 3364 y(algebra)31 b(on)i Fq(A)f Fs(b)m(y)i(setting)e Fq(\016)1541 3328 y Fk(0)1592 3364 y Fs(:=)27 b Fq(!)1787 3328 y Fk(\000)p Fo(t)1871 3364 y Fq(\016)t Fs(.)555 3480 y(>F)-8 b(rom)34 b(\(5.2\),)h(the)h(giv)m(en)g Fq(\032)p Fs(\()p Fq(u)p Fs(\))31 b(=)1874 3406 y Fi(P)1995 3480 y Fq(u)2051 3495 y Fk(\000)p Fn(1)2169 3480 y Fp(\012)25 b Fq(u)2327 3495 y Fn(0)2401 3480 y Fs(is)35 b(the)g(correct)h (coaction)e(if)456 3602 y(and)f(only)f(if,)g(for)g(all)f Fq(f)38 b Fp(2)29 b Fq(H)1532 3565 y Fk(\003)1571 3602 y Fs(,)k Fq(f)g Fp(\001)22 b Fq(u)27 b Fs(=)1949 3527 y Fi(P)2054 3602 y Fp(h)p 2093 3522 66 4 v Fq(S)6 b(f)f(;)17 b(u)2312 3617 y Fk(\000)p Fn(1)2406 3602 y Fp(i)p Fq(u)2501 3617 y Fn(0)2539 3602 y Fs(.)45 b(Since)32 b Fq(S)39 b Fs(is)32 b(bijectiv)m(e,)456 3718 y(w)m(e)h(ma)m(y)g(c)m(hec)m(k)h (instead)f(that)1373 3956 y(\()p Fp(\003)p Fs(\))55 b Fq(S)6 b(f)32 b Fp(\001)22 b Fq(u)27 b Fs(=)1937 3862 y Fi(X)2081 3956 y Fp(h)p Fq(f)5 b(;)17 b(u)2273 3971 y Fk(\000)p Fn(1)2366 3956 y Fp(i)p Fq(u)2461 3971 y Fn(0)2500 3956 y Fq(:)456 4200 y Fs(It)32 b(su\016ces)j(to)d(c)m(hec)m (k)j(this)d(for)g(all)f Fq(f)38 b Fp(2)28 b(f)p Fq(X)2069 4164 y Fo(q)2107 4200 y Fq(G)2184 4164 y Fo(r)2222 4200 y Fp(g)p Fs(,)k(a)g(basis)h(of)f Fq(H)2852 4164 y Fk(\003)2891 4200 y Fs(.)555 4316 y(First)g(assume)h Fq(q)e Fs(=)d(0.)43 b(Then,)34 b(using)e(Lemma)f(5.4,)i(and)f Fq(f)39 b Fs(=)27 b Fq(G)2953 4280 y Fo(r)2991 4316 y Fs(,)456 4555 y Fq(G)533 4513 y Fk(\000)p Fo(r)625 4555 y Fp(\001)p Fq(u)g Fs(=)h Fq(!)905 4513 y Fk(\000)p Fo(tr)1022 4555 y Fq(u)g Fs(=)f Fq(a)1260 4570 y Fn(0)1300 4555 y Fp(h)p Fq(G)1416 4513 y Fo(r)1453 4555 y Fq(;)17 b(g)1548 4513 y Fo(t)1577 4555 y Fp(i)p Fq(u)p Fs(+)1752 4460 y Fi(X)1748 4669 y Fo(m>)p Fn(0)1916 4555 y Fq(a)1967 4570 y Fo(m)2034 4555 y Fp(h)p Fq(G)2150 4513 y Fo(r)2188 4555 y Fq(;)g(x)2287 4513 y Fo(m)2354 4555 y Fq(g)2405 4513 y Fk(\000)p Fo(m)p Fn(+)p Fo(t)2606 4555 y Fp(i)p Fq(u)2701 4513 y Fo(m)p Fn(\()p Fo(s)p Fk(\000)p Fn(1\)+1)3062 4555 y Fs(=)28 b Fq(a)3217 4570 y Fn(0)3256 4555 y Fq(!)3321 4513 y Fk(\000)p Fo(tr)3439 4555 y Fq(u:)456 4867 y Fs(Th)m(us)34 b Fq(a)754 4882 y Fn(0)821 4867 y Fs(=)28 b(1.)43 b(W)-8 b(e)33 b(ma)m(y)f(therefore)h(assume)g(that)g Fq(q)e Fp(\025)d Fs(1.)555 4983 y(No)m(w)e(assume)h Fq(s)g Fp(\025)h Fs(2.)41 b(F)-8 b(or)25 b Fq(f)38 b Fs(=)28 b Fq(X)1843 4947 y Fo(q)1881 4983 y Fq(G)1958 4947 y Fo(r)1996 4983 y Fs(,)f(note)e(that)h Fq(S)6 b(f)38 b Fs(=)28 b Fq(G)2797 4947 y Fk(\000)p Fo(r)2889 4983 y Fs(\()p Fp(\000)p Fq(G)3081 4947 y Fk(\000)p Fn(1)3176 4983 y Fq(X)8 b Fs(\))3303 4947 y Fo(q)3369 4983 y Fs(=)456 5099 y Fq(G)533 5063 y Fk(\000)p Fo(r)625 5099 y Fs(\()p Fp(\000)p Fq(!)t(D)s Fs(\))927 5063 y Fo(q)1016 5099 y Fs(=)52 b(\()p Fp(\000)p Fq(!)t Fs(\))1362 5063 y Fo(q)1400 5099 y Fq(G)1477 5063 y Fk(\000)p Fo(r)1569 5099 y Fq(D)1653 5063 y Fo(q)1691 5099 y Fs(.)86 b(Using)46 b Fq(D)35 b Fp(\001)c Fq(u)51 b Fs(=)h Fq(\016)2549 5063 y Fk(0)2572 5099 y Fq(u)2628 5063 y Fo(s)2716 5099 y Fs(=)f Fq(!)2908 5063 y Fk(\000)p Fo(t)2992 5099 y Fq(\016)t(u)3095 5063 y Fo(s)3131 5099 y Fs(,)g(apply)456 5216 y(Lemma)35 b(2.1\(b\))h(with)g Fq(p)f Fs(=)f(1)i(and)h Fq(\013)e Fs(=)f Fq(!)2070 5179 y Fk(\000)p Fo(t)2191 5216 y Fs(to)i(see)h(that)g(the)g(left-hand)e (side)p eop %%Page: 19 19 19 18 bop 3368 233 a Fl(19)456 425 y Fs(of)32 b(\(*\))g(is)456 586 y Fq(S)6 b(f)33 b Fp(\001)22 b Fq(u)27 b Fs(=)g(\()p Fp(\000)p Fq(!)t Fs(\))1057 544 y Fo(q)1095 586 y Fq(G)1172 544 y Fk(\000)p Fo(r)1265 586 y Fq(D)1349 544 y Fo(q)1409 586 y Fp(\001)22 b Fq(u)539 840 y Fs(=)83 b(\()p Fp(\000)p Fq(!)t Fs(\))916 799 y Fo(q)954 840 y Fs(\()p Fq(!)1057 799 y Fk(\000)p Fo(tr)1174 840 y Fs(\))1212 799 y Fo(q)r Fn(\()p Fo(s)p Fk(\000)p Fn(1\)+1)1518 840 y Fs(\()p Fq(!)1621 799 y Fk(\000)p Fo(t)1705 840 y Fq(\016)t Fs(\))1790 799 y Fo(q)1828 840 y Fs(\()1879 715 y Fo(s)p Fk(\000)p Fn(1)1868 745 y Fi(X)1866 955 y Fo(i)1890 964 y Fm(2)1924 955 y Fn(=0)2031 840 y Fq(!)2096 799 y Fk(\000)p Fo(ti)2200 808 y Fm(2)2239 840 y Fs(\))17 b Fp(\001)g(\001)g(\001)d Fs(\()2464 708 y Fn(\()p Fo(q)r Fk(\000)p Fn(1\)\()p Fo(s)p Fk(\000)p Fn(1\))2570 745 y Fi(X)2568 955 y Fo(i)2592 963 y Fh(q)2626 955 y Fn(=0)2837 840 y Fq(!)2902 799 y Fk(\000)p Fo(ti)3006 807 y Fh(q)3044 840 y Fs(\))p Fq(u)3138 799 y Fn(1+)p Fo(q)r Fn(\()p Fo(s)p Fk(\000)p Fn(1\))539 1213 y Fs(=)83 b(\()p Fp(\000)p Fs(\))851 1171 y Fo(q)889 1213 y Fq(\016)936 1171 y Fo(q)974 1213 y Fq(!)1039 1171 y Fo(q)r Fn(\(1)p Fk(\000)p Fo(t)p Fn(\))1247 1213 y Fq(!)1312 1171 y Fo(r)r Fn(\()p Fo(q)r Fk(\000)p Fo(t)p Fn(\))1519 1213 y Fs(\()1572 1088 y Fo(s)p Fk(\000)p Fn(1)1562 1118 y Fi(X)1557 1328 y Fo(j)1586 1337 y Fm(1)1620 1328 y Fn(=0)1727 1213 y Fq(!)1792 1171 y Fk(\000)p Fo(j)1876 1180 y Fm(1)1909 1171 y Fo(t)1939 1213 y Fs(\))17 b Fp(\001)g(\001)g (\001)d Fs(\()2164 1080 y Fn(\()p Fo(q)r Fk(\000)p Fn(1\)\()p Fo(s)p Fk(\000)p Fn(1\))2270 1118 y Fi(X)2226 1328 y Fo(j)2255 1337 y Fh(q)r Ff(\000)p Fm(1)2368 1328 y Fn(=0)2537 1213 y Fq(!)2602 1171 y Fk(\000)p Fo(j)2686 1180 y Fh(m)p Ff(\000)p Fm(1)2822 1171 y Fo(t)2852 1213 y Fs(\))p Fq(u)2946 1171 y Fo(q)r Fn(\()p Fo(s)p Fk(\000)p Fn(1\)+1)456 1486 y Fs(where)28 b(w)m(e)f(ha)m(v)m(e)h(c)m(hanged)g(the)f(index)g(of)f (summation.)40 b(Moreo)m(v)m(er)28 b Fq(G)3023 1449 y Fk(\000)p Fo(r)3116 1486 y Fq(D)3200 1449 y Fo(q)3248 1486 y Fp(\001)10 b Fq(u)27 b Fp(6)p Fs(=)456 1602 y(0)f(for)g(all)f Fq(q)30 b Fs(suc)m(h)f(that)d(1)i Fp(\024)g Fq(q)j Fp(\024)d Fq(t)p Fs(,)h(and)e(=)g(0)g(for)f(all)e Fq(q)32 b(>)27 b(t)p Fs(.)42 b(F)-8 b(or)26 b(the)h(righ)m(t-hand)456 1718 y(side)32 b(of)g(\(*\),)h(using)f(Lemma)f(5.4,)h(it)g(follo)m(ws)f (that)497 1878 y Fi(X)640 1973 y Fp(h)p Fq(X)768 1932 y Fo(q)806 1973 y Fq(G)883 1932 y Fo(r)921 1973 y Fq(;)17 b(u)1021 1988 y Fk(\000)p Fn(1)1114 1973 y Fp(i)p Fq(u)1209 1988 y Fn(0)1276 1973 y Fs(=)1389 1848 y Fo(n)p Fk(\000)p Fn(1)1384 1878 y Fi(X)1379 2087 y Fo(m)p Fn(=0)1549 1973 y Fq(a)1600 1988 y Fo(m)1666 1973 y Fp(h)p Fq(X)1794 1932 y Fo(q)1832 1973 y Fq(G)1909 1932 y Fo(r)1947 1973 y Fq(;)g(x)2046 1932 y Fo(m)2113 1973 y Fq(g)2164 1932 y Fk(\000)p Fo(m)p Fn(+)p Fo(t)2365 1973 y Fp(i)p Fq(u)2460 1932 y Fo(m)p Fn(\()p Fo(s)p Fk(\000)p Fn(1\)+1)580 2302 y Fs(=)82 b Fq(a)789 2317 y Fo(q)891 2173 y(q)844 2207 y Fi(Y)845 2416 y Fo(z)s Fn(=1)988 2161 y Fi(\022)1062 2234 y Fq(z)1062 2370 y Fs(1)1111 2161 y Fi(\023)1185 2401 y Fo(!)1231 2382 y Ff(\000)p Fm(1)1318 2302 y Fq(!)1383 2261 y Fk(\000)p Fn(\()p Fk(\000)p Fo(q)r Fn(+)p Fo(t)p Fn(\))p Fo(r)1699 2302 y Fq(u)1755 2261 y Fo(q)r Fn(\()p Fo(s)p Fk(\000)p Fn(1\)+1)2088 2302 y Fs(=)27 b Fq(a)2242 2317 y Fo(q)2281 2302 y Fq(!)2346 2261 y Fo(r)r Fn(\()p Fo(q)r Fk(\000)p Fo(t)p Fn(\))2615 2173 y Fo(q)2569 2207 y Fi(Y)2569 2416 y Fo(z)s 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b Fs(=)f(0)32 b(for)g(all)456 3022 y Fq(q)f Fp(\025)e Fs(2,)j(and)h(so,)g(as)g(in)e(the)i(\014rst)h (part)e(of)g(the)h(previous)g(argumen)m(t,)g Fq(a)3077 3037 y Fo(q)3143 3022 y Fs(=)28 b(0)k(for)456 3138 y(all)g Fq(q)i Fp(\025)e Fs(2.)48 b(Th)m(us)36 b Fq(\032)p Fs(\()p Fq(u)p Fs(\))31 b(=)f Fq(a)1522 3153 y Fn(0)1562 3138 y Fq(g)d Fp(\012)c Fq(u)g Fs(+)h Fq(a)1966 3153 y Fn(1)2005 3138 y Fq(x)g Fp(\012)g Fs(1.)48 b(W)-8 b(e)35 b(ha)m(v)m(e)h(already)e (seen)i(that)456 3254 y Fq(a)507 3269 y Fn(0)574 3254 y Fs(=)27 b(1,)33 b(using)f Fq(q)g Fs(=)27 b(0.)43 b(When)34 b Fq(q)d Fs(=)d(1,)456 3414 y Fq(S)6 b(X)20 b Fp(\001)13 b Fq(u)27 b Fs(=)h Fp(\000)p Fq(!)t(D)16 b Fp(\001)d Fq(u)26 b Fs(=)i Fp(\000)p Fq(!)t Fs(\()p Fq(!)1562 3373 y Fk(\000)p Fn(1)1655 3414 y Fq(\016)t Fs(\)1)g(=)f Fq(a)1971 3429 y Fn(0)2011 3414 y Fp(h)p Fq(X)r(;)17 b(g)t Fp(i)p Fq(u)c Fs(+)g Fq(a)2476 3429 y Fn(1)2514 3414 y Fp(h)p Fq(X)r(;)k(x)p Fp(i)p Fs(1)27 b(=)h(0)13 b(+)g Fq(a)3156 3429 y Fn(1)3223 3414 y Fs(=)28 b Fq(a)3378 3429 y Fn(1)3417 3414 y Fq(:)456 3575 y Fs(Th)m(us)34 b Fq(a)754 3590 y Fn(1)821 3575 y Fs(=)28 b Fp(\000)p Fq(\016)t Fs(,)33 b(\014nishing)e(the)i(pro)s(of.)p 3374 3575 4 66 v 3378 3512 59 4 v 3378 3575 V 3436 3575 4 66 v 456 3751 a Fu(Example)j(5.6.) 49 b Fs(Consider)36 b(the)h(sp)s(ecial)d(\(\\generic"\))h(case)i (arising)d(in)h(Theo-)456 3867 y(rem)i(4.5,)j(that)e(is,)h Fq(G)27 b Fp(\001)e Fq(u)37 b Fs(=)h Fq(!)1626 3831 y Fk(\000)p Fn(1)1719 3867 y Fq(u)g Fs(and)h Fq(X)33 b Fp(\001)26 b Fq(u)37 b Fs(=)h Fq(\016)t(u)2487 3831 y Fn(2)2525 3867 y Fs(.)61 b(In)39 b(that)f(case,)j Fq(s)c Fs(=)h(2)456 3984 y(and)d Fq(t)e Fs(=)g Fq(n)24 b Fp(\000)g Fs(1.)52 b(Th)m(us)37 b(the)f(left)f Fq(H)8 b Fs(-coaction)34 b(corresp)s(onding)h(to)g(this)g(action)456 4100 y(is)d(giv)m(en)g(b)m (y)1264 4308 y Fq(\032)p Fs(\()p Fq(u)p Fs(\))27 b(=)1586 4184 y Fo(n)p Fk(\000)p Fn(1)1581 4214 y Fi(X)1577 4423 y Fo(m)p Fn(=0)1746 4308 y Fq(a)1797 4323 y Fo(m)1864 4308 y Fq(x)1919 4267 y Fo(m)1986 4308 y Fq(g)2037 4267 y Fk(\000)p Fn(\()p Fo(m)p Fn(+1\))2324 4308 y Fp(\012)c Fq(u)2480 4267 y Fo(m)p Fn(+1)456 4573 y Fs(where)33 b Fq(a)788 4588 y Fo(m)883 4573 y Fs(=)27 b(\()p Fp(\000)p Fq(\016)t Fs(\))1186 4537 y Fo(m)1253 4573 y Fq(!)1318 4537 y Fo(m)1384 4573 y Fq(!)1459 4502 y Fh(m)p Fm(\()p Fh(m)p Fm(+1\))p 1458 4521 234 3 v 1560 4562 a(2)1707 4573 y Fs(.)555 4689 y(F)-8 b(or,)47 b(in)c(this)h(case,)k(eac)m(h)d (factor)f(in)f(the)i(n)m(umerator)f(of)f Fq(a)2833 4704 y Fo(m)2944 4689 y Fs(of)h(the)h(form)456 4734 y Fi(P)561 4761 y Fo(i)561 4838 y(j)590 4848 y Fh(i)616 4838 y Fn(=0)727 4809 y Fq(!)792 4773 y Fo(j)821 4783 y Fh(i)875 4809 y Fs(is)24 b(paired)g(with)h(the)g(term)f(1)6 b(+)g Fq(!)2057 4773 y Fk(\000)p Fn(1)2157 4809 y Fs(+)g Fq(:)17 b(:)g(:)6 b Fs(+)g Fq(!)2507 4773 y Fk(\000)p Fo(i)2615 4809 y Fs(in)24 b(the)h(denominator,)456 4939 y(for)j Fq(i)g Fs(=)f(1)p Fq(;)17 b(:)g(:)g(:)32 b(;)17 b(m)d Fp(\000)g Fs(1,)30 b(to)e(giv)m(e)g(a)h(quotien)m(t)f(of)g Fq(!)2286 4903 y Fo(i)2314 4939 y Fs(.)42 b(Substituting)28 b(in)m(to)f(the)i (for-)456 5075 y(m)m(ula)i(for)h Fq(a)897 5090 y Fo(m)997 5075 y Fs(in)g(Prop)s(osition)f(5.5,)h(w)m(e)i(see)g(that)f Fq(a)2385 5090 y Fo(m)2480 5075 y Fs(=)28 b(\()p Fp(\000)p Fq(\016)t Fs(\))2784 5039 y Fo(m)2851 5075 y Fq(!)2916 5039 y Fn(2)p Fo(m)3017 5075 y Fq(!)3092 5004 y Fh(m)p Fm(\()p Fh(m)p Ff(\000)p Fm(1\))p 3091 5023 236 3 v 3193 5064 a(2)3369 5075 y Fs(=)456 5216 y(\()p Fp(\000)p Fq(\016)t Fs(\))656 5179 y Fo(m)722 5216 y Fq(!)787 5179 y Fo(m)853 5216 y Fq(!)928 5145 y Fh(m)p Fm(\()p Fh(m)p Fm(+1\))p 928 5164 234 3 v 1029 5205 a(2)1176 5216 y Fs(.)p eop %%Page: 20 20 20 19 bop 456 236 a Fl(20)623 b(S.)33 b(MONTGOMER)-6 b(Y)33 b(AND)f(H.-J.)i(SCHNEIDER)555 425 y Fs(W)-8 b(e)33 b(can)g(no)m(w)g(determine)g(the)g(left)e Fq(Y)21 b(D)s Fs(-structures)34 b(on)f(our)f(ring)f Fq(A)p Fs(.)456 613 y Fu(Theorem)37 b(5.7.)49 b Fg(L)-5 b(et)33 b Fq(A)g Fg(b)-5 b(e)33 b(an)f Fq(n)p Fg(-dimensional)f Fq(k)s Fg(-algebr)-5 b(a)32 b(with)h(no)g(non-zer)-5 b(o)456 729 y(nilp)g(otent)40 b(elements,)i(and)e(let)h Fq(H)47 b Fs(=)39 b Fq(T)1942 746 y Fo(n)1985 727 y Fm(2)2024 729 y Fs(\()p Fq(!)t Fs(\))p Fg(.)62 b(Assume)41 b(that)g Fq(A)g Fg(is)g(a)g(left)g Fq(H)8 b Fg(-)456 845 y(mo)-5 b(dule)38 b(algebr)-5 b(a)38 b(such)g(that)h Fq(x)25 b Fp(\001)g Fq(A)35 b Fp(6)p Fs(=)g(0)p Fg(.)56 b(Then)37 b(for)i(e)-5 b(ach)38 b(such)g(action,)h(ther)-5 b(e)456 962 y(exists)37 b(a)g(unique)h(left)g Fq(H)8 b Fg(-c)-5 b(omo)g(dule)36 b(algebr)-5 b(a)37 b(structur)-5 b(e)38 b(on)g Fq(A)f Fg(such)h(that)g Fq(A)33 b Fp(2)456 1042 y Fo(H)456 1103 y(H)523 1078 y Fp(Y)8 b Fq(D)r Fg(.)555 1194 y(In)45 b(p)-5 b(articular,)47 b(assume)e(the)g Fq(H)8 b Fg(-action)44 b(is)h(as)g(describ)-5 b(e)g(d)44 b(as)h(in)g(The)-5 b(or)g(em)456 1310 y(2.5;)44 b(that)f(is,)g Fq(A)d Fs(=)h Fq(k)s Fs(\()p Fq(u)p Fs(\))g Fg(for)h Fq(u)1686 1274 y Fo(n)1773 1310 y Fs(=)e Fq(\014)46 b Fp(2)41 b Fq(k)s Fg(,)j(and)d Fq(g)31 b Fp(\001)c Fq(u)40 b Fs(=)g Fq(!)t(u)p Fg(,)j Fq(x)27 b Fp(\001)g Fq(u)40 b Fs(=)h Fq(\015)5 b Fs(1)p Fg(.)456 1427 y(Then)28 b(the)i(unique)f (left)h Fq(H)8 b Fg(-c)-5 b(omo)g(dule)27 b(algebr)-5 b(a)29 b(structur)-5 b(e)31 b(on)e Fq(A)p Fg(,)h(such)f(that)h Fq(A)e Fp(2)456 1507 y Fo(H)456 1568 y(H)523 1543 y Fp(Y)8 b Fq(D)r Fg(,)35 b(is)g(given)f(by)870 1818 y Fq(\032)p Fs(\()p Fq(u)p Fs(\))28 b(=)1193 1693 y Fo(n)p Fk(\000)p Fn(1)1188 1723 y Fi(X)1183 1932 y Fo(m)p Fn(=0)1353 1818 y Fq(\015)1409 1776 y Fk(\000)p Fo(m)1530 1818 y Fs(\()p Fq(!)d Fp(\000)e Fs(1\))1841 1776 y Fo(m)1907 1818 y Fq(!)1982 1743 y Fh(m)p Fm(\()p Fh(m)p Fm(+1\))p 1982 1761 234 3 v 2083 1802 a(2)2230 1818 y Fq(x)2285 1776 y Fo(m)2352 1818 y Fq(g)2403 1776 y Fk(\000)p Fn(\()p Fo(m)p Fn(+1\))2691 1818 y Fp(\012)f Fq(u)2846 1776 y Fo(m)p Fn(+1)3003 1818 y Fq(:)456 2097 y Fg(Pr)-5 b(o)g(of.)41 b Fs(Apply)c(Theorem)h(4.5)f(to)g(get)h(a)f(unique)h(left)f Fq(D)s Fs(\()p Fq(H)8 b Fs(\)-mo)s(dule)34 b(algebra)456 2213 y(structure)29 b(on)f Fq(A)p Fs(;)h(b)m(y)g(Prop)s(osition)e(5.5)g (the)i(left)e Fq(H)2352 2177 y Fk(\003)p Fo(cop)2491 2213 y Fs(-mo)s(dule)f(algebra)h(struc-)456 2329 y(ture)32 b(giv)m(es)h(a)g(unique)g(left)e Fq(H)8 b Fs(-coaction)31 b(on)h Fq(A)p Fs(.)44 b(That)33 b(this)f(implies)e Fq(A)j Fp(2)3191 2293 y Fo(H)3191 2354 y(H)3286 2329 y Fp(Y)8 b Fq(D)456 2445 y Fs(follo)m(ws)31 b(from)g(Lemma)g(5.3.)555 2561 y(No)m(w)g(consider)g(the)g(particular)e(\\generic")i(case)g(in)f (the)h(theorem.)43 b(By)31 b(The-)456 2678 y(orem)45 b(4.5,)50 b(w)m(e)d(m)m(ust)f(ha)m(v)m(e)i Fq(G)31 b Fp(\001)h Fq(u)50 b Fs(=)h Fq(!)2032 2642 y Fk(\000)p Fn(1)2126 2678 y Fq(u)45 b Fs(and)i Fq(X)39 b Fp(\001)31 b Fq(u)51 b Fs(=)g Fq(\016)t(u)2947 2642 y Fn(2)3032 2678 y Fs(suc)m(h)d(that)456 2794 y Fq(\016)t(\015)f Fs(=)c Fq(!)785 2758 y Fk(\000)p Fn(1)906 2794 y Fp(\000)29 b Fs(1.)69 b(Th)m(us)43 b(\()p Fp(\000)p Fq(\016)t Fs(\))1613 2758 y Fo(m)1723 2794 y Fs(=)f Fq(\015)1897 2758 y Fk(\000)p Fo(m)2019 2794 y Fs(\(1)28 b Fp(\000)g Fq(!)2304 2758 y Fk(\000)p Fn(1)2398 2794 y Fs(\))2436 2758 y Fo(m)2502 2794 y Fs(.)70 b(Since)42 b(w)m(e)g(are)f(in)g(the)456 2910 y(situation)29 b(of)h(Example)h(5.6,)f(w)m(e)i(use)g(the)f (simpli\014ed)e(form)h(of)g Fq(a)2858 2925 y Fo(m)2956 2910 y Fs(found)h(there)456 3026 y(to)h(see)i(that)1256 3170 y Fq(a)1307 3185 y Fo(m)1402 3170 y Fs(=)27 b Fq(\015)1561 3129 y Fk(\000)p Fo(m)1683 3170 y Fs(\(\(1)21 b Fp(\000)i Fq(!)1994 3129 y Fk(\000)p Fn(1)2088 3170 y Fs(\))p Fq(!)t Fs(\))2229 3129 y Fo(m)2294 3170 y Fq(!)2369 3095 y Fh(m)p Fm(\()p Fh(m)p Fm(+1\))p 2369 3114 V 2470 3155 a(2)2617 3170 y Fq(;)456 3330 y Fs(whic)m(h)44 b(is)g(the)g(desired)h(co)s (e\016cien)m(t.)79 b(The)45 b(form)d(of)i Fq(\032)p Fs(\()p Fq(u)p Fs(\))g(also)f(follo)m(ws)g(from)456 3446 y(Example)32 b(5.6.)p 3374 3446 4 66 v 3378 3384 59 4 v 3378 3446 V 3436 3446 4 66 v 555 3641 a(The)38 b(theorem)f(giv)m(es)h(an)f(answ)m (er)h(to)f(the)h(question)f(raised)g(in)g([CFM,)i(2.2];)456 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b(then)h(one)g(m)m(ust)456 4455 y(still)j(c)m(hec)m(k)35 b(that)733 4580 y Fo(n)p Fk(\000)p Fn(1)727 4609 y Fi(X)737 4819 y Fo(p)p Fn(=0)888 4704 y Fq(a)939 4719 y Fo(p)979 4704 y Fq(x)1034 4663 y Fo(p)1074 4704 y Fq(g)1125 4663 y Fk(\000)p Fn(\()p Fo(p)p Fn(+1\))1386 4704 y Fp(\012)23 b Fq(\032)p Fs(\()p Fq(u)p Fs(\))1668 4663 y Fo(p)p Fn(+1)1825 4704 y Fs(=)1938 4580 y Fo(n)p Fk(\000)p Fn(1)1933 4609 y Fi(X)1928 4819 y Fo(m)p Fn(=0)2098 4704 y Fq(a)2149 4719 y Fo(m)2215 4704 y Fs(\001\()p Fq(x)2389 4663 y Fo(m)2457 4704 y Fq(g)2508 4663 y Fk(\000)p Fn(\()p Fo(m)p Fn(+1\))2773 4704 y Fs(\))g Fp(\012)f Fq(u)2989 4663 y Fo(m)p Fn(+1)3145 4704 y Fq(:)456 4983 y Fs(This)28 b(is)f(the)h(problem.)41 b(The)29 b(indirect)e(argumen)m(t)g(via)h (passing)f(to)h Fq(H)3023 4947 y Fk(\003)3062 4983 y Fs(-mo)s(dules)456 5099 y(as)45 b(in)g(Theorem)h(5.7)f(ab)s(o)m(v)m(e)i (a)m(v)m(oids)f(ha)m(ving)f(to)g(c)m(hec)m(k)j(this)d(iden)m(tit)m(y)-8 b(.)82 b(The)456 5216 y(iden)m(tit)m(y)32 b(ma)m(y)g(b)s(e)h(of)f (indep)s(enden)m(t)i(in)m(terest.)p eop %%Page: 21 21 21 20 bop 3368 233 a Fl(21)1660 425 y Fr(References)456 583 y Ft([CFM])41 b(M.)21 b(Cohen,)g(D.)g(Fisc)n(hman,)g(and)f(S.)g (Mon)n(tgomery)-7 b(,)20 b(On)g(Y)-7 b(etter-)20 b(Drinfeld)h (categories)675 683 y(and)27 b Fe(H)7 b Ft(-comm)n(utativit)n(y)-7 b(,)27 b(Comm.)g(Alg)h(27)f(\(1999\),)f(1321-1345.)456 782 y([D])110 b(V.)26 b(G.)g(Drinfel'd,)h(Quan)n(tum)f(groups,)f(Pro)r (c.)g(In)n(t.)h(Cong.)f(Math.)i(Berk)n(eley)d(1)i(\(1996\),)675 882 y(798-218.)456 981 y([J])130 b(N.)34 b(Jacobson,)f(Classes)g(of)g (restricted)g(Lie)h(algebras)e(I)r(I,)i(Duk)n(e)g(Math)f(J)h(10)f (\(1943\),)675 1081 y(107-121.)456 1181 y([Ji])107 b(M.)39 b(Jim)n(b)r(o,)j(A)d(q-di\013erence)f(anlogue)g(of)h(U\(g\))h(and)e (the)i(Y)-7 b(ang-Baxter)37 b(equation,)675 1280 y(Lett.)28 b(Math.)g(Ph)n(ysics)e(10)h(\(1985\),)f(63-69.)456 1380 y([K])108 b(C.)27 b(Kassel,)g(Quan)n(tum)g(Groups,)g(GTM)h(155,)e (Springer-V)-7 b(erlag,)25 b(1995.)456 1480 y([M])97 b(S.)27 b(Mon)n(tgomery)-7 b(,)25 b(Hopf)i(Algebras)e(and)i(their)g (Actions)f(on)h(Rings,)f(CBMS)h(Lectures,)675 1579 y(V)-7 b(ol.)27 b(82,)g(AMS,)h(Pro)n(vidence,)e(RI,)i(1993.)456 1679 y([MSm])42 b(S.)30 b(Mon)n(tgomery)f(and)h(S.)h(P)-7 b(.)30 b(Smith,)j(Sk)n(ew)d(deriv)-5 b(ations)29 b(and)i Fe(U)2825 1691 y Fd(q)2861 1679 y Ft(\()p Fe(sl)r Ft(\(2\)\),)g(Israel) f(J.)675 1778 y(Math)d(72)g(\(1990\),)f(158-166.)456 1878 y([R])112 b(D.)48 b(E.)f(Radford,)53 b(Minimal)48 b(quasi-triangular)d(Hopf)j(algebras,)j(J.)c(Algebra)g(157)675 1978 y(\(1993\),)26 b(285-315.)456 2077 y([T])113 b(E.)22 b(T)-7 b(aft,)25 b(The)e(order)f(of)h(the)g(an)n(tip)r(o)r(de)g(of)g(a) g(\014nite-dimensional)g(Hopf)g(algebra,)g(Pro)r(c.)675 2177 y(Nat.)k(Acad.)h(Sci.)g(USA)g(68)f(\(1971\),)f(2631-2633.)456 2282 y([Z])122 b(H.)21 b(Zassenhaus,)1244 2261 y(\177)1233 2282 y(Ub)r(er)g(Liesc)n(he)f(Ringe)g(mit)i(Primzahlc)n(harakteristk,)c (Abhand.)j(Mat.)675 2382 y(Sem.)28 b(Hansisc)n(hen)f(Univ.)h(13)e (\(1940\),)h(1-100.)555 2569 y Fc(University)k(of)h(Southern)f(Calif)n (ornia,)g(Los)g(Angeles,)g(CA)g(90089-1113)555 2669 y Fb(E-mail)g(addr)l(ess)7 b Ft(:)38 b Fa(smontgom@math.us)o(c.)o(edu)555 2838 y Fc(Ma)-6 b(thema)g(tisches)24 b(Institut,)f(Universit)2057 2831 y(\177)2054 2838 y(at)e(M)2260 2831 y(\177)2257 2838 y(unchen,)i(D-80333)f(Munich,)h(Ger-)456 2937 y(many)555 3037 y Fb(E-mail)31 b(addr)l(ess)7 b Ft(:)38 b Fa(hanssch@rz.mathm)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