%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: SelfDualPrep5.dvi %%Pages: 11 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips SelfDualPrep5.dvi %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 2001.07.09:1744 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N 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2310149013A33803C1A014E0380180C0A319107F8F1C>I<38FF0F80383C0700EA1C0613 04A26C5AA26C5AA3EA03A0A2EA01C0A36C5AA248C7FCA212E112E212E4127811177F8F14 >121 DI E /Fo 7 117 df<1303497EA2497EA3EB1BE0A2EB3BF013 31A2EB60F8A2EBE0FCEBC07CA248487EEBFFFE487FEB001F4814800006130FA248EB07C0 39FF803FFCA21E1A7F9921>65 D97 D<12FCA2123CA713FE383F878038 3E01C0003C13E0EB00F0A214F8A514F0A2EB01E0003E13C0383B07803830FE00151A7E99 19>II114 DI<1206A4120EA2121EEA3FF012FFEA1E00 A81318A5EA0F30EA03E00D187F9711>I E /Fp 22 128 df69 D75 D<3803F020380C0C60EA1802383001E0EA70000060136012E0A21420A36C1300A2127812 7FEA3FF0EA1FFE6C7E0003138038003FC0EB07E01301EB00F0A214707EA46C1360A26C13 C07E38C8018038C60700EA81FC14247DA21B>83 D89 D<387FFFFE387E003E0078133C007013781260004013F012C0EB01E0388003C0A2EB0780 1200EB0F005B131E5BA25BA25B1201EBE001EA03C0A2EA07801403EA0F00001E1302A248 1306140E48131E00F8137EB512FE18227DA11E>I97 D99 D<14E0130F13011300ABEA01F8 EA0704EA0C02EA1C01EA38001278127012F0A7127012781238EA1801EA0C0238070CF038 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EC23F01421A2EC40F8A3EC807CA2903801007E153EA20102133F81A2496D7EA3496D7EA2 011880011FB5FCA29039200003F01501A2496D7EA349147CA20001157E90C8123EA24815 3F825AD81F80EC3F80D8FFE0903801FFFCA22E327EB132>65 D72 D77 D<90387F80203801FFE03907C07860380F001C001EEB06E04813030038130100 7813001270156012F0A21520A37E1500127C127E7E13C0EA1FF86CB47E6C13F06C13FCC6 13FF010F1380010013C0EC1FE01407EC03F01401140015F8A26C1478A57E15706C14F015 E07E6CEB01C000ECEB038000C7EB070038C1F01E38807FFCEB0FF01D337CB125>83 D<13FE380303C0380C00E00010137080003C133C003E131C141EA21208C7FCA3EB0FFEEB FC1EEA03E0EA0F80EA1F00123E123C127C481404A3143EA21278007C135E6CEB8F08390F 0307F03903FC03E01E1F7D9E21>97 DI<15F0141FA2140114 00AFEB0FC0EB70303801C00C3803800238070001120E001E13005AA2127C1278A212F8A7 1278A2127C123CA27E000E13016C1302380380046C6C487E3A00F030FF80EB1FC021327E B125>100 DII<15F090387F03083901C1C41C 380380E8390700700848EB7800001E7FA2003E133EA6001E133CA26C5B6C13706D5A3809 C1C0D8087FC7FC0018C8FCA5121C7E380FFFF86C13FF6C1480390E000FC00018EB01E048 EB00F000701470481438A500701470A26C14E06CEB01C00007EB07003801C01C38003FE0 1E2F7E9F21>I<120FEA1F80A4EA0F00C7FCABEA0780127FA2120F1207B3A6EA0FC0EAFF F8A20D307EAF12>105 D 108 D<260780FEEB1FC03BFF83078060F0903A8C03C180783B0F9001E2003CD807A013E4 DA00F47F01C013F8A2495BB3A2486C486C133F3CFFFC1FFF83FFF0A2341F7E9E38>I111 D<380781FC39FF86078090388801C0390F9000E0D807 A0137001C01378497F153E151E151FA2811680A716005DA2151E153E153C6D5B01A01370 5D90389803C0D9860FC7FCEB81F80180C8FCAB487EEAFFFCA2212D7E9E25>I<380783E0 38FF8418EB887CEA0F90EA07A01438EBC000A35BB3487EEAFFFEA2161F7E9E19>114 D<3801FC10380E0330381800F048137048133012E01410A37E6C1300127EEA3FF06CB4FC 6C13C0000313E038003FF0EB01F813006C133CA2141C7EA27E14186C1338143000CC1360 38C301C03880FE00161F7E9E1A>I117 D E end %%EndProlog %%BeginSetup %%Feature: *Resolution 300dpi TeXDict begin %%PaperSize: a4 %%EndSetup %%Page: 1 1 1 0 bop 444 369 a Fq(Self-dual)20 b(Mo)r(dules)h(of)g(Semisimple)704 461 y(Hopf)g(Algebras)453 622 y Fp(Y)l(evgenia)16 b(Kashina)65 b(Y)l(orc)o(k)15 b(Sommerh\177)-24 b(auser)744 715 y(Y)l(ongc)o(hang)17 b(Zh)o(u)748 854 y(Second)f(Edition)825 1036 y Fo(Abstract)327 1134 y Fn(W)m(e)11 b(pro)o(v)o(e)i(that,)e(o)o(v)o(er)h(an)g (algebraical)q(ly)j(closed)e(\014eld)g(of)e(c)o(haracteristic)j(zero,) 328 1180 y(a)e(semisimple)j(Hopf)d(algebra)h(that)f(has)h(a)f(non)o (trivial)j(self-dual)f(simple)f(mo)q(d-)328 1226 y(ule)k(m)o(ust)f(ha)o (v)o(e)h(ev)o(en)f(dimension.)28 b(This)17 b(generalizes)h(a)e (classical)j(result)e(of)328 1271 y(W.)g(Burnside.)29 b(As)17 b(an)f(applicatio)q(n,)k(w)o(e)c(sho)o(w)h(under)g(the)g(same)g (assump-)328 1317 y(tions)j(that)e(a)h(semisimple)i(Hopf)d(algebra)i (that)f(has)g(a)g(simple)h(mo)q(dule)g(of)328 1363 y(ev)o(en)14 b(dimension)h(m)o(ust)e(itself)h(ha)o(v)o(e)g(ev)o(en)f(dimension.)224 1503 y Fm(1)48 b Fl(Supp)q(ose)14 b(that)e Fk(H)k Fl(is)c(a)h (\014nite-dimensional)d(Hopf)i(algebra)g(that)h(is)g(de\014ned)g(o)o(v) o(er)224 1553 y(the)21 b(\014eld)g Fk(K)s Fl(.)37 b(W)m(e)20 b(denote)i(its)e(com)o(ultiplication)d(b)o(y)j(\001,)h(its)f(counit)h (b)o(y)f Fk(")p Fl(,)i(and)224 1602 y(its)17 b(an)o(tip)q(o)q(de)g(b)o (y)f Fk(S)r Fl(.)27 b(F)m(or)17 b(the)g(com)o(ultiplication,)d(w)o(e)j (use)g(the)h(sigma)c(notation)i(of)224 1652 y(R.)d(G.)g(Heyneman)h(and) f(M.)h(E.)f(Sw)o(eedler)i(in)f(the)g(follo)o(wing)d(v)n(arian)o(t:)743 1748 y(\001\()p Fk(h)p Fl(\))g(=)h Fk(h)913 1755 y Fj(\(1\))966 1748 y Fi(\012)e Fk(h)1032 1755 y Fj(\(2\))224 1844 y Fl(W)m(e)20 b(view)g(the)h(dual)f(space)h Fk(H)736 1829 y Fh(\003)775 1844 y Fl(as)f(a)g(Hopf)g(algebra)g(whose)h(unit)f(is)g (the)g(counit)224 1894 y(of)15 b Fk(H)s Fl(,)g(whose)h(counit)g(is)f (the)h(ev)n(aluation)e(at)h(1,)g(whose)h(an)o(tip)q(o)q(de)g(is)f(the)h (transp)q(ose)224 1944 y(of)j(the)h(an)o(tip)q(o)q(de)g(of)f Fk(H)s Fl(,)h(and)f(whose)h(m)o(ultiplication)c(and)k(com)o (ultiplicatio)o(n)d(are)224 1993 y(determined)d(b)o(y)g(the)g(form)o (ulas)408 2089 y(\()p Fk('')478 2072 y Fh(0)491 2089 y Fl(\)\()p Fk(h)p Fl(\))d(=)h Fk(')p Fl(\()p Fk(h)685 2096 y Fj(\(1\))730 2089 y Fl(\))p Fk(')773 2072 y Fh(0)785 2089 y Fl(\()p Fk(h)825 2096 y Fj(\(2\))870 2089 y Fl(\))83 b Fk(')996 2096 y Fj(\(1\))1040 2089 y Fl(\()p Fk(h)p Fl(\))p Fk(')1123 2096 y Fj(\(2\))1168 2089 y Fl(\()p Fk(h)1208 2072 y Fh(0)1220 2089 y Fl(\))12 b(=)g Fk(')p Fl(\()p Fk(hh)1383 2072 y Fh(0)1394 2089 y Fl(\))224 2185 y(for)i Fk(h;)7 b(h)355 2170 y Fh(0)377 2185 y Fi(2)12 b Fk(H)k Fl(and)e Fk(';)7 b(')622 2170 y Fh(0)645 2185 y Fi(2)k Fk(H)722 2170 y Fh(\003)741 2185 y Fl(.)899 2310 y(1)p eop %%Page: 2 2 2 1 bop 177 195 a Fl(With)11 b Fk(H)s Fl(,)h(w)o(e)g(can)g(asso)q (ciate)g(its)g(Drinfel'd)e(double)i Fk(D)q Fl(\()p Fk(H)s Fl(\))g(\(cf.)f([18)o(],)h Fi(x)g Fl(10.3,)e(p.)h(187\).)177 245 y(This)16 b(is)f(a)h(Hopf)f(algebra)h(whose)g(underlying)f(v)o (ector)i(space)g(is)e Fk(D)q Fl(\()p 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Fk(e)797 1353 y Fh(\000)p Fj(1)855 1368 y Fl(is)g(also)f(a)h(righ)o(t)f(in)o(v)o(erse)i(of)e Fk(e)p Fl(.)177 1455 y(The)h(ev)n(aluation)e(form)g(w)o(as)i (considered)h(b)o(y)e(T.)g(Kerler,)i(who)e(pro)o(v)o(ed)h(the)g(follo)o (wing)177 1505 y(prop)q(ert)o(y)h(\(cf.)f([12)o(],)e(Prop.)i(7,)f(p.)h (366\):)177 1608 y Fm(Prop)q(osition)e(1)21 b Fl(The)e(ev)n(aluation)d (form)h(is)h(a)g(symmetric)e(F)m(rob)q(enius)j(homom)o(or-)177 1658 y(phism.)177 1762 y Fm(Pro)q(of.)36 b Fl(W)m(e)13 b(giv)o(e)h(a)f(di\013eren)o(t)i(pro)q(of.)i(By)d(the)h(de\014nition)e (of)g(a)h(F)m(rob)q(enius)g(algebra)177 1812 y(\(cf.)k([10)o(],)g(Kap.) f(13,)h(Def.)f(13.5.4,)f(p.)i(306\),)g(w)o(e)g(ha)o(v)o(e)f(to)h(sho)o (w)g(that)g(the)g(bilinear)177 1861 y(form)12 b(asso)q(ciated)j(with)e Fk(e)i Fl(is)e(symmetric)g(and)g(nondegenerate.)20 b(Since)14 b(w)o(e)g(ha)o(v)o(e)281 1954 y Fk(e)p Fl(\(\()p Fk(')c Fi(\012)f Fk(h)p Fl(\))q(\()p Fk(')494 1937 y Fh(0)515 1954 y Fi(\012)g Fk(h)580 1937 y Fh(0)592 1954 y Fl(\)\))j(=)f Fk(')706 1937 y Fh(0)706 1966 y Fj(\(1\))751 1954 y Fl(\()p Fk(S)794 1937 y Fh(\000)p Fj(1)840 1954 y Fl(\()p Fk(h)880 1961 y Fj(\(3\))924 1954 y Fl(\)\))p Fk(')983 1937 y Fh(0)983 1966 y Fj(\(3\))1028 1954 y Fl(\()p Fk(h)1068 1961 y Fj(\(1\))1113 1954 y Fl(\))h Fk(e)p Fl(\()p Fk('')1230 1937 y Fh(0)1230 1966 y Fj(\(2\))1284 1954 y Fi(\012)e Fk(h)1350 1961 y Fj(\(2\))1394 1954 y Fk(h)1418 1937 y Fh(0)1430 1954 y Fl(\))462 2025 y(=)i Fk(')533 2008 y Fh(0)533 2037 y Fj(\(1\))578 2025 y Fl(\()p Fk(S)621 2008 y Fh(\000)p Fj(1)666 2025 y Fl(\()p Fk(h)706 2032 y Fj(\(4\))751 2025 y Fl(\)\))p Fk(')810 2008 y Fh(0)810 2037 y Fj(\(3\))855 2025 y Fl(\()p Fk(h)895 2032 y Fj(\(1\))939 2025 y Fl(\))p Fk(')p Fl(\()p Fk(h)1022 2032 y Fj(\(2\))1067 2025 y Fk(h)1091 2008 y Fh(0)1091 2037 y Fj(\(1\))1136 2025 y Fl(\))p Fk(')1179 2008 y Fh(0)1179 2037 y Fj(\(2\))1224 2025 y Fl(\()p Fk(h)1264 2032 y Fj(\(3\))1308 2025 y Fk(h)1332 2008 y Fh(0)1332 2037 y Fj(\(2\))1377 2025 y Fl(\))462 2096 y(=)g Fk(')533 2079 y Fh(0)545 2096 y Fl(\()p Fk(S)588 2079 y Fh(\000)p Fj(1)633 2096 y Fl(\()p Fk(h)673 2103 y Fj(\(4\))718 2096 y Fl(\))p Fk(h)758 2103 y Fj(\(3\))802 2096 y Fk(h)826 2079 y Fh(0)826 2108 y Fj(\(2\))871 2096 y Fk(h)895 2103 y Fj(\(1\))939 2096 y Fl(\))p Fk(')p Fl(\()p Fk(h)1022 2103 y Fj(\(2\))1067 2096 y Fk(h)1091 2079 y Fh(0)1091 2108 y Fj(\(1\))1136 2096 y Fl(\))462 2166 y(=)g Fk(')533 2149 y Fh(0)545 2166 y Fl(\()p Fk(h)585 2149 y Fh(0)585 2178 y Fj(\(2\))629 2166 y Fk(h)653 2173 y Fj(\(1\))698 2166 y Fl(\))p Fk(')p Fl(\()p Fk(h)781 2173 y Fj(\(2\))826 2166 y Fk(h)850 2149 y Fh(0)850 2178 y Fj(\(1\))894 2166 y Fl(\))852 2310 y(2)p eop %%Page: 3 3 3 2 bop 224 195 a Fl(w)o(e)14 b(see)h(that)e(this)h(bilinear)e(form)g (is)h(symmetric.)j(T)m(o)d(see)h(that)g(it)f(is)g(also)g(nondegen-)224 245 y(erate,)h(consider)g(the)f(righ)o(t)f(m)o(ultiplication)e Fk(R)967 251 y Fg(e)997 245 y Fl(b)o(y)i Fk(e)h Fl(in)g Fk(D)q Fl(\()p Fk(H)s Fl(\))1238 230 y Fh(\003)1257 245 y 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1521 y Fl(\))224 1615 y(This)14 b(sho)o(ws)g(that)g(the)h(order)f(of)g Fk(e)g Fl(is)g(related)g(to)g(the)g(exp)q(onen)o(t)h(of)e Fk(H)s Fl(:)224 1720 y Fm(Prop)q(osition)g(2)20 b Fl(Supp)q(ose)13 b(that)e Fk(H)k Fl(is)c(semisimple)e(and)i(that)h(the)g(base)g(\014eld) g Fk(K)i Fl(has)224 1770 y(c)o(haracteristic)k(zero.)25 b(Then)16 b(the)h(order)f(of)g Fk(e)g Fl(is)g(equal)f(to)h(the)g(exp)q (onen)o(t)h(of)e Fk(H)s Fl(.)24 b(In)224 1820 y(particular,)14 b(the)g(order)h(of)e Fk(e)h Fl(divides)g(\(dim)n(\()p Fk(H)s Fl(\)\))999 1805 y Fj(3)1018 1820 y Fl(.)224 1926 y Fm(Pro)q(of.)36 b Fl(In)18 b(this)f(situation,)g(w)o(e)h(kno)o(w)e (from)g([15)o(],)h(Thm.)f(3.3,)g(p.)h(276,)g(and)g([16)o(],)224 1975 y(Thm.)11 b(3,)g(p.)h(194)f(that)h Fk(H)j Fl(is)d(also)g (cosemisimple)e(and)h(that)i(the)f(an)o(tip)q(o)q(de)g(of)g Fk(H)j Fl(is)d(an)224 2025 y(in)o(v)o(olution.)22 b(It)16 b(therefore)h(follo)o(ws)d(from)g(the)i(de\014nition)g(of)f(the)h(exp)q (onen)o(t)h(\(cf.)e([5)o(],)224 2075 y(Def.)e(2.1,)f(p.)h(132\))g(that) g(the)h(order)g(of)f Fk(e)h Fl(is)f(the)h(exp)q(onen)o(t)g(of)f Fk(H)1253 2060 y Fj(op)1290 2075 y Fl(,)g(whic)o(h)g(coincides)224 2125 y(with)h(the)h(exp)q(onen)o(t)f(of)g Fk(H)i Fl(b)o(y)e([5)o(],)f (Cor.)h(2.6,)e(p.)i(134.)j(The)d(divisibilit)o(y)e(prop)q(ert)o(y)j(is) 224 2175 y(pro)o(v)o(ed)f(in)g([5)o(],)f(Thm.)f(4.3,)g(p.)i(136.)j Fd(2)899 2310 y Fl(3)p eop %%Page: 4 4 4 3 bop 177 195 a Fm(3)48 b Fl(Let)18 b(us)g(consider)g(no)o(w)g(the)g (case)h(that)e Fk(H)k Fl(is)c(semisimple)e(and)j(that)f(the)i(base)177 245 y(\014eld)14 b Fk(K)i Fl(is)d(algebraically)f(closed)i(of)f(c)o (haracteristic)i(zero.)j(Note)c(that)g(a)f(semisimple)177 295 y(Hopf)j(algebra)h(is)f(necessarily)i(\014nite-dimensional)d(\(cf.) h([22)o(],)h(Cor.)f(2.7,)g(p.)g(330,)g(or)177 344 y([23)o(],)f(Chap.)g (V,)g(Ex.)h(4,)f(p.)g(108\).)23 b(By)16 b(Masc)o(hk)o(e's)g(theorem)f (\(cf.)h([18)o(],)f(Thm.)e(2.2.1,)177 394 y(p.)i(20\),)g(there)i(is)f (a)f(unique)h(t)o(w)o(o-sided)f(in)o(tegral)g(\003)h(that)f (satis\014es)i Fk(")p Fl(\(\003\))e(=)f(1.)23 b(Sup-)177 444 y(p)q(ose)e(that)f Fk(V)29 b Fl(is)20 b(a)f(simple)f Fk(H)s Fl(-mo)q(dule)h(with)g(c)o(haracter)i Fk(\037)p Fl(.)36 b(W)m(e)20 b(sa)o(y)f(that)h Fk(V)29 b Fl(is)177 494 y(self-dual)16 b(if)g Fk(V)436 483 y Fi(\030)436 496 y Fl(=)485 494 y Fk(V)518 479 y Fh(\003)537 494 y Fl(.)27 b(This)16 b(is)h(equiv)n(alen)o(t)f(to)h(the)g(requiremen)o(t)g (that)g(there)h(is)e(a)177 544 y(nondegenerate)j(in)o(v)n(arian)o(t)d (bilinear)g(form)g(on)h Fk(V)10 b Fl(,)17 b(i.e.,)g(a)g(nondegenerate)i (bilinear)177 594 y(form)698 646 y Fi(h\001)p Fk(;)7 b Fi(\001i)j Fl(:)h Fk(V)19 b Fi(\002)9 b Fk(V)21 b Fi(!)11 b Fk(K)177 723 y Fl(that)j(satis\014es)611 776 y Fi(h)p Fk(h)651 783 y Fj(\(1\))696 776 y Fk(:v)q(;)7 b(h)772 783 y Fj(\(2\))816 776 y Fk(:v)849 758 y Fh(0)861 776 y Fi(i)k Fl(=)h Fk(")p Fl(\()p Fk(h)p Fl(\))p Fi(h)p Fk(v)q(;)7 b(v)1084 758 y Fh(0)1097 776 y Fi(i)177 853 y Fl(for)k(all)f Fk(h)i Fi(2)f Fk(H)j Fl(and)e(all)e Fk(v)q(;)d(v)612 838 y Fh(0)636 853 y Fi(2)k Fk(V)e Fl(.)18 b(F)m(ollo)o(wing)8 b([17)o(],)j(w)o(e)h(de\014ne)g(the)g(F)m(rob)q (enius-Sc)o(h)o(ur)177 903 y(indicator,)20 b(also)f(brie\015y)h(called) f(the)h(Sc)o(h)o(ur)g(indicator,)g Fk(\027)1130 909 y Fj(2)1148 903 y Fl(\()p Fk(\037)p Fl(\))g(of)f(the)h(irreducible)177 952 y(c)o(haracter)15 b Fk(\037)f Fl(corresp)q(onding)h(to)f(the)g (simple)f(mo)q(dule)f Fk(V)e Fl(:)678 1045 y Fk(\027)699 1051 y Fj(2)717 1045 y Fl(\()p Fk(\037)p Fl(\))h(:=)h Fk(\037)p Fl(\(\003)913 1052 y Fj(\(1\))957 1045 y Fl(\003)986 1052 y Fj(\(2\))1031 1045 y Fl(\))177 1138 y(The)j(F)m(rob)q(enius-Sc)o (h)o(ur)h(theorem)e(for)g(Hopf)h(algebras)f(\(cf.)h([17)o(],)e(Thm.)g (3.1,)h(p.)g(349\))177 1188 y(then)h(asserts,)g(among)d(other)i (things,)f(the)i(follo)o(wing:)177 1292 y Fm(Theorem)36 b Fl(The)11 b(Sc)o(h)o(ur)h(indicator)f Fk(\027)790 1298 y Fj(2)808 1292 y Fl(\()p Fk(\037)p Fl(\))g(can)h(only)e(tak)o(e)i(the) g(v)n(alues)f(1,)g Fi(\000)p Fl(1,)g(and)g(0:)228 1414 y(1.)20 b(W)m(e)12 b(ha)o(v)o(e)g Fk(\027)466 1420 y Fj(2)484 1414 y Fl(\()p Fk(\037)p Fl(\))g(=)g(1)g(if)f(and)h(only)g(if) f Fk(V)22 b Fl(admits)11 b(a)h(symmetric)f(nondegenerate)281 1464 y(in)o(v)n(arian)o(t)h(bilinear)h(form.)228 1551 y(2.)20 b(W)m(e)12 b(ha)o(v)o(e)h Fk(\027)467 1557 y Fj(2)485 1551 y Fl(\()p Fk(\037)p Fl(\))f(=)g Fi(\000)p Fl(1)h(if)f(and)h(only)f(if)g Fk(V)22 b Fl(admits)12 b(a)g(sk)o(ew-symmetric)g(nonde-)281 1600 y(generate)j(in)o(v)n(arian)o (t)d(bilinear)h(form.)228 1687 y(3.)20 b(W)m(e)13 b(ha)o(v)o(e)h Fk(\027)469 1693 y Fj(2)487 1687 y Fl(\()p Fk(\037)p Fl(\))e(=)g(0)h(if)h(and)f(only)g(if)g Fk(V)24 b Fl(is)13 b(not)h(self-dual.)177 1824 y Fm(4)48 b Fl(Using)12 b(these)h (preparations,)f(w)o(e)g(can)h(pro)o(v)o(e)f(the)g(main)e(theorem.)17 b(It)12 b(generalizes)177 1874 y(a)g(classical)f(result)i(of)e(W.)g (Burnside)i(in)e(the)h(theory)h(of)e(\014nite)h(groups)g(\(cf.)f([3)o (],)h(P)o(ar.)f(2,)177 1924 y(p.)k(167;)g([4)o(],)g Fi(x)g Fl(222,)g(Thm.)e(I)q(I,)i(p.)g(294\).)21 b(W)m(e)15 b(note)h(that)f (this)h(theorem)e(w)o(as)i(kno)o(wn)177 1973 y(in)e(the)g(case)h(of)e (co)q(cen)o(tral)i(ab)q(elian)e(extensions)i(\(cf.)f([11)o(],)f(Cor.)g (3.2,)f(p.)i(5\).)177 2077 y Fm(Theorem)36 b Fl(Supp)q(ose)22 b(that)g Fk(H)i Fl(is)e(a)f(semisimple)e(Hopf)i(algebra)g(o)o(v)o(er)h (an)f(alge-)177 2127 y(braically)16 b(closed)h(\014eld)g(of)g(c)o (haracteristic)h(zero.)28 b(If)16 b Fk(H)k Fl(has)d(a)g(non)o(trivial)e (self-dual)177 2177 y(simple)e(mo)q(dule,)f(then)i(the)h(dimension)d (of)h Fk(H)k Fl(is)d(ev)o(en.)852 2310 y(4)p eop %%Page: 5 5 5 4 bop 224 195 a Fm(Pro)q(of.)36 b Fl(Supp)q(ose)18 b(that)f Fk(V)26 b Fl(is)16 b(an)h Fk(H)s Fl(-mo)q(dule)e(with)h(c)o (haracter)i Fk(\037)f Fl(and)f(that)h Fk(W)23 b Fl(is)224 245 y(an)d Fk(H)326 230 y Fh(\003)345 245 y Fl(-mo)q(dule)e(with)i(c)o (haracter)h Fk(\021)q Fl(.)36 b(As)20 b(an)f(algebra,)i(the)f(dual)f Fk(D)q Fl(\()p Fk(H)s Fl(\))1444 230 y Fh(\003)1484 245 y Fl(of)g(the)224 295 y(Drinfel'd)13 b(double)h(is)g(isomorphic)f(to)h Fk(H)874 280 y Fj(op)920 295 y Fi(\012)c Fk(H)1000 280 y Fh(\003)1019 295 y Fl(.)19 b(W)m(e)13 b(can)i(therefore)g(turn)g Fk(V)k Fi(\012)10 b Fk(W)224 344 y Fl(in)o(to)k(a)f Fk(D)q Fl(\()p Fk(H)s Fl(\))448 329 y Fh(\003)468 344 y Fl(-mo)q(dule)f(b)o(y) i(de\014ning)623 426 y(\()p Fk(h)9 b Fi(\012)g Fk(')p Fl(\))p Fk(:)p Fl(\()p Fk(v)i Fi(\012)f Fk(w)q Fl(\))h(=)h Fk(S)r Fl(\()p Fk(h)p Fl(\))p Fk(:v)f Fi(\012)f Fk(':w)224 507 y Fl(If)k(w)o(e)h(iden)o(tify)e Fk(H)517 492 y Fh(\003\003)568 507 y Fl(and)h Fk(H)s Fl(,)f(w)o(e)i(can)f(consider)i Fk(\021)f Fl(as)f(an)g(elemen)o(t)g(of)g Fk(H)s Fl(.)19 b(Denoting)224 556 y(the)13 b(c)o(haracter)h(of)d Fk(V)553 541 y Fh(\003)584 556 y Fl(b)o(y)17 b(\026)-26 b Fk(\037)p Fl(,)12 b(the)h(trace)g(of)e(the)i(action)f(of)f Fk(e)i Fl(on)f Fk(V)j Fi(\012)6 b Fk(W)18 b Fl(is)12 b(then)h(giv)o(en)224 606 y(b)o(y)h(the)h(form)o(ula)647 656 y(\()5 b(\026)-26 b Fk(\037)10 b Fi(\012)f Fk(\021)q Fl(\)\()p Fk(e)p Fl(\))k(=)e Fk(e)p Fl(\()5 b(\026)-26 b Fk(\037)10 b Fi(\012)g Fk(\021)q Fl(\))h(=)17 b(\026)-26 b Fk(\037)p Fl(\()p Fk(\021)q Fl(\))224 725 y(Similarly)m(,)10 b(the)15 b(trace)f(of)g Fk(e)651 710 y Fj(2)683 725 y Fl(is)g(giv)o(en)g(b)o(y)f(the)i(form)o (ula)575 806 y(\()5 b(\026)-26 b Fk(\037)9 b Fi(\012)g Fk(\021)q Fl(\)\()p Fk(e)740 789 y Fj(2)760 806 y Fl(\))i(=)h Fk(e)850 789 y Fj(2)869 806 y Fl(\()5 b(\026)-26 b Fk(\037)10 b Fi(\012)f Fk(\021)q Fl(\))j(=)17 b(\026)-26 b Fk(\037)p Fl(\()p Fk(\021)1119 813 y Fj(\(2\))1163 806 y Fk(\021)1184 813 y Fj(\(1\))1228 806 y Fl(\))224 887 y(W)m(e)12 b(no)o(w)f(assume)g (that)h Fk(V)21 b Fl(is)11 b(simple,)g(non)o(trivial,)f(and)h (self-dual)g(and)g(that)h Fk(W)18 b Fl(=)12 b Fk(H)1576 872 y Fh(\003)224 937 y Fl(is)17 b(the)h(regular)e(represen)o(tation.) 29 b(W)m(e)16 b(then)i(kno)o(w)e(that,)h(if)f(\003)h(is)g(an)f(in)o (tegral)g(that)224 986 y(satis\014es)f Fk(")p Fl(\(\003\))d(=)g(1,)h (the)i(c)o(haracter)g(of)e(the)i(regular)f(represen)o(tation)h(is)f (giv)o(en)f(b)o(y)727 1068 y Fk(\021)q Fl(\()p Fk(')p Fl(\))f(=)g(dim)n(\()p Fk(H)s Fl(\))p Fk(')p Fl(\(\003\))224 1149 y(i.e.,)17 b(up)f(to)h(the)h(iden)o(ti\014cation)e(of)g Fk(H)839 1134 y Fh(\003\003)892 1149 y Fl(and)h Fk(H)s Fl(,)g(w)o(e)g(ha)o(v)o(e)f Fk(\021)i Fl(=)f(dim)n(\()p Fk(H)s Fl(\)\003.)27 b(Since)224 1198 y Fk(V)f Fl(is)17 b(non)o(trivial,)e Fk(\037)i Fl(v)n(anishes)g(on)f(the)h(in)o(tegral,)f (and)h(since)g(the)h(self-dualit)o(y)d(of)h Fk(V)224 1248 y Fl(implies)11 b(that)17 b(\026)-26 b Fk(\037)12 b Fl(=)f Fk(\037)p Fl(,)h(w)o(e)h(get)f(from)f(the)i(ab)q(o)o(v)o(e)f (and)g(the)h(F)m(rob)q(enius-Sc)o(h)o(ur)g(theorem)224 1298 y(that)521 1348 y(\()5 b(\026)-26 b Fk(\037)9 b Fi(\012)h Fk(\021)q Fl(\)\()p Fk(e)p Fl(\))i(=)g(0)83 b(\()5 b(\026)-26 b Fk(\037)9 b Fi(\012)h Fk(\021)q Fl(\)\()p Fk(e)1029 1331 y Fj(2)1048 1348 y Fl(\))i(=)f Fi(\006)c Fl(dim)o(\()p Fk(H)s Fl(\))224 1448 y(No)o(w)12 b(supp)q(ose)i(that)e Fk(n)g Fl(is)g(the)h(exp)q(onen)o(t)g(of)f Fk(H)j Fl(and)d(that)g Fk(\020)j Fl(is)d(a)g(primitiv)o(e)e Fk(n)p Fl(-th)i(ro)q(ot)224 1498 y(of)j(unit)o(y)m(.)23 b(Since)16 b Fk(e)g Fl(has)g(order)g Fk(n)g Fl(b)o(y)f(Prop)q(osition)g(2.2,)g Fk(V)20 b Fi(\012)11 b Fk(W)21 b Fl(is)16 b(the)g(direct)g(sum)224 1547 y(of)g(the)h (eigenspaces)h(corresp)q(onding)g(to)e(the)h(p)q(o)o(w)o(ers)g(of)f Fk(\020)s Fl(,)h(whose)g(dimensions)e(w)o(e)224 1597 y(denote)g(b)o(y)580 1647 y Fk(a)602 1653 y Fg(k)634 1647 y Fl(:=)d(dim)n(\()p Fi(f)p Fk(z)h Fi(2)f Fk(V)18 b Fi(\012)10 b Fk(W)17 b Fi(j)11 b Fk(e:z)j Fl(=)e Fk(\020)1160 1630 y Fg(k)1180 1647 y Fk(z)r Fi(g)p Fl(\))224 1716 y(If)i(w)o(e)g(in)o(tro)q(duce)h(the)f(p)q(olynomial)697 1834 y Fk(p)p Fl(\()p Fk(x)p Fl(\))d(:=)841 1782 y Fg(n)p Fh(\000)p Fj(1)842 1794 y Fc(X)842 1884 y Fg(k)q Fj(=0)910 1834 y Fk(a)932 1840 y Fg(k)953 1834 y Fk(x)977 1817 y Fg(k)1008 1834 y Fi(2)h Fb(Z)-14 b Fl([)p Fk(x)p Fl(])224 1955 y(w)o(e)18 b(see)h(that)e Fk(p)p Fl(\()p Fk(\020)s Fl(\))h(=)g(\()5 b(\026)-26 b Fk(\037)12 b Fi(\012)g Fk(\021)q Fl(\)\()p Fk(e)p Fl(\))18 b(=)g(0.)29 b(Therefore,)19 b(if)d Fk(q)1176 1961 y Fg(n)1216 1955 y Fl(denotes)j(the)f Fk(n)p Fl(-th)f(cy-)224 2004 y(clotomic)d(p)q(olynomial,)e(w)o(e)k(see) h(that)f Fk(q)863 2010 y Fg(n)900 2004 y Fl(divides)g Fk(p)p Fl(.)23 b(On)15 b(the)i(other)f(hand,)f Fk(e)1488 1989 y Fj(2)1523 2004 y Fl(acts)224 2054 y(on)g(the)h(eigenspace)g(of)e Fk(e)h Fl(corresp)q(onding)h(to)f(the)g(eigen)o(v)n(alue)f Fk(\020)1254 2039 y Fg(i)1283 2054 y Fl(b)o(y)h(m)o(ultiplicatio)o(n) 224 2104 y(with)f Fk(\020)340 2089 y Fj(2)p Fg(i)371 2104 y Fl(.)k(Therefore,)c(w)o(e)g(get)580 2185 y Fk(p)p Fl(\()p Fk(\020)638 2168 y Fj(2)656 2185 y Fl(\))e(=)g(\()5 b(\026)-26 b Fk(\037)9 b Fi(\012)h Fk(\021)q Fl(\)\()p Fk(e)894 2168 y Fj(2)913 2185 y Fl(\))i(=)g Fi(\006)7 b Fl(dim)n(\()p Fk(H)s Fl(\))12 b Fi(6)p Fl(=)g(0)899 2310 y(5)p eop %%Page: 6 6 6 5 bop 177 195 a Fl(whic)o(h)17 b(implies)f(that)h(also)g Fk(q)643 201 y Fg(n)665 195 y Fl(\()p Fk(\020)702 180 y Fj(2)721 195 y Fl(\))h Fi(6)p Fl(=)g(0.)28 b(Therefore,)19 b Fk(\020)1091 180 y Fj(2)1127 195 y Fl(is)f(not)f(a)g(primitiv)o(e)e Fk(n)p Fl(-th)177 245 y(ro)q(ot)g(of)f(unit)o(y)m(,)g(whic)o(h)h (implies)e(that)h(2)h(and)g Fk(n)f Fl(are)i(not)e(relativ)o(ely)h (prime,)e(i.e.,)h Fk(n)g Fl(is)177 295 y(ev)o(en.)21 b(Since)15 b Fk(n)f Fl(divides)h(\(dim)n(\()p Fk(H)s Fl(\)\))750 280 y Fj(3)784 295 y Fl(b)o(y)f(Prop)q(osition)g(2.2,)f(w)o (e)i(see)h(that)e(dim)o(\()p Fk(H)s Fl(\))g(is)177 344 y(also)f(ev)o(en.)19 b Fd(2)177 483 y Fl(W)m(e)14 b(note)g(that)g(the)h (con)o(v)o(erse)g(of)e(the)i(ab)q(o)o(v)o(e)f(theorem)f(also)h(holds:)j (If)d(a)g(semisimple)177 533 y(Hopf)d(algebra)f(has)h(ev)o(en)h (dimension,)e(it)g(has)h(a)g(non)o(trivial)e(self-dual)i(simple)e(mo)q (dule.)177 583 y(T)m(o)g(see)i(this,)g(lo)q(ok)e(at)g(the)i(action)f (of)f(the)h(an)o(tip)q(o)q(de)g(on)g(the)g(minima)o(l)d(t)o(w)o (o-sided)j(ideals)177 633 y(that)i(app)q(ear)h(in)e(the)h(W)m (edderburn)h(decomp)q(osition.)j(A)c(simple)f(mo)q(dule)f(is)i (self-dual)177 683 y(if)k(and)h(only)f(if)g(the)i(an)o(tip)q(o)q(de)e (preserv)o(es)k(the)d(corresp)q(onding)h(minim)o(al)13 b(t)o(w)o(o-sided)177 733 y(ideal.)26 b(If)17 b(this)g(happ)q(ens)h (only)e(for)g(the)i(one-dimensional)c(ideal)i(that)h(corresp)q(onds)177 782 y(to)c(the)h(trivial)f(represen)o(tation,)h(the)g(remaining)e (minim)o(al)e(t)o(w)o(o-sided)j(ideals)g(can)h(b)q(e)177 832 y(group)q(ed)19 b(in)o(to)e(pairs)h(of)f(ideals)h(of)f(equal)h (dimension.)29 b(As)18 b(the)h(dimension)d(of)i(the)177 882 y(Hopf)d(algebra)f(is)h(the)g(sum)f(of)h(the)g(dimensions)f(of)g (the)i(minim)o(al)11 b(t)o(w)o(o-sided)k(ideals,)177 932 y(this)f(m)o(ust)f(then)i(b)q(e)f(an)g(o)q(dd)g(n)o(um)o(b)q(er.) 177 1026 y(The)k(argumen)o(ts)f(that)g(w)o(e)h(ha)o(v)o(e)f(giv)o(en)g (so)g(far)g(also)g(pro)o(v)o(e)g(t)o(w)o(o)g(facts)h(that)f(are)h(of) 177 1076 y(indep)q(enden)o(t)d(in)o(terest:)177 1195 y Fm(Corollary)35 b Fl(Supp)q(ose)22 b(that)e Fk(H)k Fl(is)c(a)h(semisimple)d(Hopf)i(algebra)g(o)o(v)o(er)h(an)f(alge-)177 1245 y(braically)13 b(closed)h(\014eld)g Fk(K)j Fl(of)d(c)o (haracteristic)h(zero.)228 1380 y(1.)20 b(If)14 b Fk(\037)h Fl(is)g(an)f(irreducible)h(c)o(haracter)h(of)f Fk(H)i Fl(and)e Fk(\021)g Fl(is)g(an)f(irreducible)i(c)o(haracter)281 1430 y(of)d Fk(H)366 1415 y Fh(\003)385 1430 y Fl(,)h(then)g Fk(\021)q Fl(\()p Fk(\037)p Fl(\))g(is)g(con)o(tained)g(in)f(the)i Fk(n)p Fl(-th)f(cyclotomic)e(\014eld)i Fb(Q)p Fl(\()p Fk(\020)1406 1436 y Fg(n)1426 1430 y Fl(\))e Fi(\032)f Fk(K)s Fl(,)281 1479 y(where)k Fk(n)f Fl(is)g(the)h(exp)q(onen)o(t)g (of)f Fk(H)j Fl(and)d Fk(\020)932 1485 y Fg(n)969 1479 y Fl(is)g(a)g(primitiv)o(e)e Fk(n)p Fl(-th)i(ro)q(ot)g(of)g(unit)o(y) 281 1529 y(of)f Fk(K)s Fl(.)228 1628 y(2.)20 b(If)13 b(the)i(dimension)d(of)h Fk(H)k Fl(is)d(ev)o(en,)g(then)h(the)f(exp)q (onen)o(t)h(of)e Fk(H)k Fl(is)d(also)f(ev)o(en.)177 1747 y Fm(Pro)q(of.)36 b Fl(The)18 b(\014rst)g(statemen)o(t)f(follo)o(ws)e (from)h(the)i(considerations)f(at)h(the)f(b)q(egin-)177 1797 y(ning)f(of)g(the)h(pro)q(of)f(of)g(the)i(theorem.)25 b(The)17 b(second)h(statemen)o(t)e(hold)g(since,)i(if)e(the)177 1847 y(dimension)d(of)h Fk(H)j Fl(is)d(ev)o(en,)h(w)o(e)f(ha)o(v)o(e)h (just)f(seen)i(that)e Fk(H)k Fl(has)c(a)g(non)o(trivial)f(self-dual)177 1897 y(simple)c(mo)q(dule,)g(and)h(w)o(e)g(ha)o(v)o(e)g(seen)i(in)d (the)i(pro)q(of)f(of)f(the)i(theorem)f(that)g(this)g(implies)177 1946 y(that)k(the)h(exp)q(onen)o(t)f(of)g Fk(H)i Fl(is)e(ev)o(en.)19 b Fd(2)177 2085 y Fl(The)d(second)g(statemen)o(t)g(can)f(b)q(e)h(seen)h (as)e(a)g(\014rst)h(partial)f(answ)o(er)h(to)f(the)h(question)177 2135 y(whether)e(the)g(exp)q(onen)o(t)f(and)g(the)g(dimension)e(of)h Fk(H)k Fl(ha)o(v)o(e)d(the)g(same)f(prime)f(divisors)177 2185 y(\(cf.)j([5)o(],)f(Qu.)h(5.1,)e(p.)i(138\).)852 2310 y(6)p eop %%Page: 7 7 7 6 bop 224 195 a Fm(5)48 b Fl(An)16 b(imp)q(ortan)o(t)d(op)q(en)j (problem)e(in)h(the)h(theory)g(of)f(semisimple)e(Hopf)i(algebras)224 245 y(is)g(to)f(pro)o(v)o(e)g(that,)g(o)o(v)o(er)g(an)g(algebraically)f (closed)h(\014eld)h(of)e(c)o(haracteristic)j(zero,)f(the)224 295 y(dimension)i(of)g(a)h(simple)e(mo)q(dule)h(divides)h(the)g (dimension)f(of)g(the)i(Hopf)e(algebra.)224 344 y(This)g(w)o(as)g(the)g (sixth)g(out)g(of)f(a)h(list)f(of)g(ten)i(problems)e(p)q(osed)h(b)o(y)g (I.)f(Kaplansky)h(in)224 394 y(1975)c(\(cf.)h([9)o(],)f([21)o(]\).)18 b(The)c(ab)q(o)o(v)o(e)g(theorem)g(can)g(b)q(e)g(used)h(to)f(giv)o(e)f (a)h(partial)f(answ)o(er)224 444 y(to)h(this)g(question:)224 539 y Fm(Theorem)36 b Fl(Supp)q(ose)22 b(that)g Fk(H)i Fl(is)e(a)f(semisimple)e(Hopf)i(algebra)g(o)o(v)o(er)h(an)f(alge-)224 589 y(braically)d(closed)i(\014eld)f(of)f(c)o(haracteristic)i(zero.)34 b(If)19 b Fk(H)i Fl(has)f(a)e(simple)g(mo)q(dule)f(of)224 639 y(ev)o(en)e(dimension,)d(then)i(the)h(dimension)d(of)h Fk(H)k Fl(is)d(ev)o(en.)224 734 y Fm(Pro)q(of.)36 b Fl(Assume)15 b(on)f(the)h(con)o(trary)g(that)g(the)g(dimension)e(of)h Fk(H)k Fl(is)c(o)q(dd.)21 b(Supp)q(ose)224 784 y(that)e Fk(\037)345 790 y Fj(1)364 784 y Fk(;)7 b(:)g(:)g(:)k(;)c(\037)489 790 y Fg(k)528 784 y Fl(are)19 b(the)g(irreducible)g(c)o(haracters)i (of)d Fk(H)s Fl(,)h(where)h Fk(\037)1363 790 y Fj(1)1401 784 y Fl(=)f Fk(")g Fl(is)g(the)224 833 y(c)o(haracter)e(of)d(the)h (trivial)f(mo)q(dule.)19 b(F)m(rom)13 b(Theorem)h(4,)g(w)o(e)h(kno)o(w) g(that)f(the)i(trivial)224 883 y(mo)q(dule)f(is)g(the)i(only)e (self-dual)g(simple)f(mo)q(dule.)22 b(Therefore)17 b Fk(k)f Fl(=)e(2)p Fk(l)e Fl(+)f(1)k(m)o(ust)g(b)q(e)224 933 y(o)q(dd,)g(and)g(w)o(e)h(can)f(n)o(um)o(b)q(er)f(the)i(c)o (haracters)h(in)e(suc)o(h)h(a)e(w)o(a)o(y)h(that)g(no)g(pair)g(of)f (dual)224 983 y(c)o(haracters)20 b(is)e(con)o(tained)g(in)f Fk(\037)742 989 y Fj(2)761 983 y Fk(;)7 b(:)g(:)g(:)12 b(;)7 b(\037)887 989 y Fg(l)p Fj(+1)959 983 y Fl(and)17 b(that)h(the)h(remaining)d(c)o(haracters)224 1033 y Fk(\037)250 1039 y Fg(l)p Fj(+2)305 1033 y Fk(;)7 b(:)g(:)g(:)12 b(;)7 b(\037)431 1039 y Fg(k)466 1033 y Fl(are)16 b(the)g(duals)g(of)f (the)h(\014rst,)g(so)g(that)g(w)o(e)g(ha)o(v)o(e)f(for)g Fk(i)g Fl(=)g(2)p Fk(;)7 b(:)g(:)g(:)k(;)c(l)k Fl(+)g(1)224 1082 y(that)824 1132 y Fk(\037)850 1138 y Fg(i)p Fj(+)p Fg(l)911 1132 y Fl(=)17 b(\026)-26 b Fk(\037)981 1138 y Fg(i)224 1235 y Fl(No)o(w)14 b(assume)f(that)g Fk(\037)h Fl(is)f(an)g(irreducible)i(c)o(haracter)f(of)f(ev)o(en)i(degree)f Fk(n)p Fl(.)k(By)c(Sc)o(h)o(ur's)224 1284 y(lemma,)7 b(the)j(trivial)f(c)o(haracter)h(app)q(ears)h(exactly)e(once)i(in)e (the)h(decomp)q(osition)e(of)h Fk(\037)c Fl(\026)-26 b Fk(\037)p Fl(.)224 1334 y(This)14 b(decomp)q(osition)f(therefore)i (has)f(the)h(form)611 1457 y Fk(\037)5 b Fl(\026)-26 b Fk(\037)11 b Fl(=)h Fk(\037)744 1463 y Fj(1)772 1457 y Fl(+)817 1405 y Fg(l)p Fj(+1)813 1417 y Fc(X)817 1505 y Fg(i)p Fj(=2)880 1457 y Fk(m)916 1463 y Fg(i)931 1457 y Fk(\037)957 1463 y Fg(i)980 1457 y Fl(+)1057 1405 y Fg(k)1036 1417 y Fc(X)1021 1506 y Fg(i)p Fj(=)p Fg(l)p Fj(+2)1118 1457 y Fk(m)1154 1463 y Fg(i)1168 1457 y Fk(\037)1194 1463 y Fg(i)224 1581 y Fl(where)i Fk(m)379 1587 y Fg(i)404 1581 y Fi(2)e Fb(N)476 1587 y Fj(0)504 1581 y Fl(is)g(the)h(m)o (ultiplicit)o(y)c(of)i Fk(\037)906 1587 y Fg(i)932 1581 y Fl(in)h Fk(\037)5 b Fl(\026)-26 b Fk(\037)p Fl(.)17 b(Since)c Fk(\037)5 b Fl(\026)-26 b Fk(\037)12 b Fl(is)g(self-dual,)f (w)o(e)h(m)o(ust)224 1631 y(ha)o(v)o(e)17 b Fk(m)359 1637 y Fg(i)p Fj(+)p Fg(l)424 1631 y Fl(=)f Fk(m)508 1637 y Fg(i)539 1631 y Fl(for)g Fk(i)f Fl(=)h(2)p Fk(;)7 b(:)g(:)g(:)12 b(;)7 b(l)k Fl(+)g(1.)25 b(Denoting)16 b(the)h(degree)g(of)f Fk(\037)1390 1637 y Fg(i)1420 1631 y Fl(b)o(y)g Fk(n)1505 1637 y Fg(i)1519 1631 y Fl(,)g(w)o(e)224 1680 y(can)f(compare)e(degrees)i(in)f(the)g(ab)q(o)o(v)o(e)g(equation)f (to)h(get)732 1803 y Fk(n)757 1785 y Fj(2)788 1803 y Fl(=)d(1)e(+)h(2)934 1751 y Fg(l)p Fj(+1)931 1763 y Fc(X)934 1852 y Fg(i)p Fj(=2)997 1803 y Fk(m)1033 1809 y Fg(i)1048 1803 y Fk(n)1073 1809 y Fg(i)224 1922 y Fl(Since)19 b(the)h(left)e (hand)g(side)h(is)g(ev)o(en)g(and)f(the)i(righ)o(t)e(hand)g(side)h(is)f (o)q(dd,)i(this)e(is)h(a)224 1972 y(con)o(tradiction.)f Fd(2)224 2085 y Fl(This)h(corollary)e(generalizes,)j(o)o(v)o(er)e (algebraically)f(closed)i(\014elds)f(of)g(c)o(haracteristic)224 2135 y(zero,)f(a)e(result)h(of)f(W.)f(D.)h(Nic)o(hols)g(and)g(M.)g(B.)h (Ric)o(hmond,)d(who)i(pro)o(v)o(ed)h(it)f(in)g(the)224 2185 y(case)g(where)g(the)g(simple)d(mo)q(dule)h(has)h(dimension)e(2)h (\(cf.)h([19)o(],)f(Cor.)g(12,)g(p.)h(306\).)899 2310 y(7)p eop %%Page: 8 8 8 7 bop 177 195 a Fl(It)13 b(is)g(clear)g(from)e(the)i(ab)q(o)o(v)o(e)g (pro)q(of)f(that)h(the)g(dimension)e(of)i Fk(H)i Fl(m)o(ust)d(b)q(e)h (ev)o(en)h(if)e(the)177 245 y(dimension)h(of)g(the)i(c)o(haracter)h (ring)e(Ch\()p Fk(H)s Fl(\))g(is)g(ev)o(en.)20 b(Ho)o(w)o(ev)o(er,)14 b(this)g(result)h(can)g(b)q(e)177 295 y(substan)o(tially)e(sharp)q (ened:)177 404 y Fm(Corollary)35 b Fl(Supp)q(ose)22 b(that)e Fk(H)k Fl(is)c(a)h(semisimple)d(Hopf)i(algebra)g(o)o(v)o(er)h(an)f (alge-)177 453 y(braically)12 b(closed)i(\014eld)f(of)f(c)o (haracteristic)j(zero.)k(If)12 b(the)i(dimension)e(of)g Fk(H)k Fl(is)d(o)q(dd,)g(w)o(e)177 503 y(ha)o(v)o(e)548 559 y(dim)n(\()p Fk(H)s Fl(\))f Fi(\021)g Fl(dim)n(\(Ch\()p Fk(H)s Fl(\)\))42 b(\(mo)q(d)13 b(16\))177 667 y Fm(Pro)q(of.)20 b Fl(The)10 b(pro)q(of)f(from)e(group)j(theory)m(,)f(whic)o(h)h(w)o(as) f(originally)e(giv)o(en)h(b)o(y)h(W.)g(Burn-)177 717 y(side)j(\(cf.)f([3)o(],)g(P)o(ar.)g(3,)g(p.)g(169;)g([4)o(],)g Fi(x)h Fl(222,)e(p.)h(295\),)g(no)o(w)g(carries)h(o)o(v)o(er)g (directly:)17 b(With)177 767 y(the)e(notation)e(from)g(ab)q(o)o(v)o(e,) g(the)i(degree)h Fk(n)870 773 y Fg(i)898 767 y Fl(of)d Fk(\037)971 773 y Fg(i)999 767 y Fl(is)h(an)g(o)q(dd)h(n)o(um)o(b)q(er) e Fk(n)1358 773 y Fg(i)1384 767 y Fl(=)f(2)p Fk(l)1461 773 y Fg(i)1485 767 y Fi(\000)d Fl(1)177 817 y(for)14 b(some)f(natural)g(n)o(um)o(b)q(er)g Fk(l)653 823 y Fg(i)679 817 y Fi(2)e Fb(N)p Fl(,)h(where)j Fk(l)906 823 y Fj(1)936 817 y Fl(=)d(1.)18 b(W)m(e)13 b(then)i(ha)o(v)o(e)281 951 y(dim)n(\()p Fk(H)s Fl(\))d(=)496 899 y Fg(k)476 911 y Fc(X)479 1000 y Fg(i)p Fj(=1)543 951 y Fk(n)568 933 y Fj(2)568 961 y Fg(i)598 951 y Fl(=)f(1)e(+)h(2)744 899 y Fg(l)p Fj(+1)741 911 y Fc(X)744 1000 y Fg(i)p Fj(=2)801 951 y Fl(\(2)p Fk(l)850 957 y Fg(i)873 951 y Fi(\000)f Fl(1\))951 933 y Fj(2)432 1097 y Fl(=)j(1)d(+)g(2)578 1045 y Fg(l)p Fj(+1)575 1057 y Fc(X)578 1145 y Fg(i)p Fj(=2)635 1097 y Fl(\(4)p Fk(l)685 1079 y Fj(2)684 1107 y Fg(i)713 1097 y Fi(\000)g Fl(4)p Fk(l)787 1103 y Fg(i)810 1097 y Fl(+)h(1\))h(=)h Fk(k)e Fl(+)g(8)1049 1045 y Fg(l)p Fj(+1)1045 1057 y Fc(X)1049 1145 y Fg(i)p Fj(=2)1112 1097 y Fk(l)1124 1103 y Fg(i)1139 1097 y Fl(\()p Fk(l)1167 1103 y Fg(i)1190 1097 y Fi(\000)g Fl(1\))177 1230 y(Since)15 b Fk(l)298 1236 y Fg(i)312 1230 y Fl(\()p Fk(l)340 1236 y Fg(i)363 1230 y Fi(\000)10 b Fl(1\))k(is)f(ev)o(en,)h(the)h(result)g (follo)o(ws.)h Fd(2)177 1369 y Fm(6)48 b Fl(By)15 b(the)h(lifting)d (theorems)i(of)g(P)m(.)f(Etingof)h(and)g(S.)f(Gelaki,)g(results)i(of)f (the)h(t)o(yp)q(e)177 1419 y(of)d(the)h(ab)q(o)o(v)o(e)g(theorems)f (can)h(b)q(e)g(carried)h(o)o(v)o(er)e(to)h(\014elds)g(of)f(p)q(ositiv)o (e)g(c)o(haracteristic)177 1469 y(under)g(the)g(additional)e (assumption)g(that)h(the)h(Hopf)f(algebra)g(is)g(also)g(cosemisimple.) 177 1519 y(W)m(e)i(therefore)h(ha)o(v)o(e)f(the)g(follo)o(wing)d (consequence:)177 1628 y Fm(Corollary)35 b Fl(Supp)q(ose)12 b(that)f Fk(H)j Fl(is)d(a)g(semisimple)d(cosemisimple)h(Hopf)i(algebra) f(o)o(v)o(er)177 1678 y(an)g(algebraically)f(closed)i(\014eld)g Fk(K)s Fl(.)17 b(If)10 b Fk(H)j Fl(has)e(a)f(non)o(trivial)e(self-dual) i(simple)f(mo)q(dule,)177 1727 y(then)15 b(the)f(dimension)e(of)i Fk(H)i Fl(is)e(ev)o(en.)177 1836 y Fm(Pro)q(of.)36 b Fl(Let)19 b(us)f(explain)g(in)g(detail)f(ho)o(w)h(the)h(lifting)d (theorems)j(can)f(b)q(e)h(applied)177 1886 y(to)e(our)f(situation:)23 b(Ob)o(viously)m(,)15 b(w)o(e)i(can)g(assume)f(that)h(the)g(c)o (haracteristic)h Fk(p)e Fl(of)g Fk(K)177 1936 y Fl(is)i(p)q(ositiv)o (e.)31 b(Since)18 b Fk(K)j Fl(is)d(p)q(erfect,)j(there)e(exists)g(a)f (complete)f(discrete)j(v)n(aluation)177 1986 y(ring)e Fk(R)g Fl(of)g(c)o(haracteristic)h(zero)g(with)f(residue)i(\014eld)e Fk(K)j Fl(whose)e(maxim)o(al)c(ideal)i(is)177 2036 y(generated)j(b)o(y) f Fk(p)g Fl(\(cf.)f([20)o(],)h Fi(x)h Fl(I)q(I.5,)f(Thm.)e(3,)i(p.)f (36\),)i(namely)d(the)i(ring)g(of)f(Witt)177 2085 y(v)o(ectors)f(of)f Fk(K)s Fl(.)25 b(Suc)o(h)17 b(a)f(discrete)h(v)n(aluation)e(ring)g(is)h (unique)h(up)f(to)g(isomorphism,)177 2135 y(and)i(w)o(e)g(denote)g(its) g(quotien)o(t)f(\014eld)h(b)o(y)f Fk(F)6 b Fl(.)29 b(By)18 b([6)o(],)g(Thm.)e(2.1,)h(p.)g(855,)g(there)i(is)177 2185 y(an)c Fk(R)p Fl(-Hopf)f(algebra)h Fk(A)g Fl(with)f(the)i(prop)q (erties)g(that)f Fk(A)c Fi(\012)1106 2191 y Fg(R)1143 2185 y Fk(F)20 b Fl(is)15 b(a)g(semisimple)d(and)852 2310 y(8)p eop %%Page: 9 9 9 8 bop 224 195 a Fl(cosemisimple)15 b(Hopf)h(algebra)g(o)o(v)o(er)h (the)g(\014eld)f Fk(F)22 b Fl(of)16 b(c)o(haracteristic)i(zero)g(and)e (that)224 245 y Fk(A=pA)e Fl(is)g(isomorphic)e(to)i Fk(H)s Fl(.)k(If)697 364 y Fk(H)746 353 y Fi(\030)746 366 y Fl(=)812 312 y Fg(k)790 324 y Fc(M)795 413 y Fg(i)p Fj(=1)860 364 y Fk(M)5 b Fl(\()p Fk(n)946 370 y Fg(i)969 364 y Fi(\002)k Fk(n)1035 370 y Fg(i)1049 364 y Fk(;)e(K)s Fl(\))224 478 y(is)17 b(the)g(W)m(edderburn)h(decomp)q(osition)d(of)h Fk(H)s Fl(,)h(w)o(e)g(can)g(construct)h(an)f(isomorphism)224 528 y(b)q(et)o(w)o(een)f Fk(A)e Fl(and)510 497 y Fc(L)556 507 y Fg(k)556 540 y(i)p Fj(=1)619 528 y Fk(M)5 b Fl(\()p Fk(n)705 534 y Fg(i)728 528 y Fi(\002)10 b Fk(n)795 534 y Fg(i)808 528 y Fk(;)d(R)p Fl(\))14 b(in)f(suc)o(h)i(a)e(w)o(a)o(y)h (that)f(the)i(diagram)527 802 y Fk(H)987 762 y Fg(k)973 771 y Fc(L)969 838 y Fg(i)p Fj(=1)1030 802 y Fk(M)5 b Fl(\()p Fk(n)1116 808 y Fg(i)1139 802 y Fi(\002)k Fk(n)1205 808 y Fg(i)1219 802 y Fk(;)e(K)s Fl(\))p 577 791 377 2 v 912 790 a Fa(-)530 636 y Fk(A)990 596 y Fg(k)976 605 y Fc(L)972 672 y Fg(i)p Fj(=1)1033 636 y Fk(M)e Fl(\()p Fk(n)1119 642 y Fg(i)1142 636 y Fi(\002)k Fk(n)1208 642 y Fg(i)1222 636 y Fk(;)e(R)p Fl(\))p 574 625 383 2 v 915 624 a Fa(-)p 545 761 2 108 v 546 761 a(?)p 1126 761 V 539 w(?)224 903 y Fl(comm)o(utes,)k(where)j(the)g(v)o(ertical)e(arro) o(ws)h(arise)g(from)e(the)j(quotien)o(t)e(mappings)f(from)224 953 y Fk(A)j Fl(to)g Fk(H)i Fl(resp.)f(from)d Fk(R)h Fl(to)h Fk(K)j Fl(\(cf.)c([13)o(],)g Fi(x)h Fl(I)q(I)q(I.5,)f(Lem.)f (\(5.1.16\),)g(p.)h(142;)g([14)o(],)f Fi(x)i Fl(22,)224 1003 y(Thm.)e(\(22.11\),)h(p.)g(342\).)224 1083 y(Supp)q(ose)20 b(no)o(w)f(that)g Fk(V)28 b Fl(is)19 b(a)f(non)o(trivial)g(self-dual)g 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b(the)f(an)o(tip)q(o)q (de)g(of)g Fk(A)g Fl(lifts)f(the)i(an)o(tip)q(o)q(de)224 1481 y(of)c Fk(H)s Fl(,)f(and)h(there)h(is)f(a)g(one-to-one)g(corresp)q (ondence)j(b)q(et)o(w)o(een)e(the)g(cen)o(trally)f(primi-)224 1531 y(tiv)o(e)f(idemp)q(oten)o(ts)g(of)f Fk(A)h Fl(and)g(the)h(cen)o (trally)f(primitiv)o(e)e(idemp)q(oten)o(ts)h(of)h Fk(H)j Fl(\(cf.)c([14)o(],)224 1581 y(lo)q(c.)i(cit.\),)f(the)i(an)o(tip)q(o)q (de)f(of)f Fk(A)h Fl(also)f(preserv)o(es)k(the)d Fk(j)r Fl(-th)g(cen)o(trally)g(primitiv)o(e)e(idem-)224 1631 y(p)q(oten)o(t)17 b Fk(e)377 1616 y Fh(0)377 1641 y Fg(j)410 1631 y Fl(of)e Fk(A)p Fl(.)24 b(Therefore,)17 b(the)f(an)o(tip)q(o)q (de)g(of)f Fk(A)c Fi(\012)1098 1637 y Fg(R)1136 1631 y Fk(F)21 b Fl(preserv)o(es)d(the)f(cen)o(trally)224 1680 y(primitiv)o(e)10 b(idemp)q(oten)o(t)h Fk(e)636 1665 y Fh(0)636 1691 y Fg(j)659 1680 y Fi(\012)5 b Fl(1,)12 b(whic)o(h)g(implies)e(that)i Fk(A)5 b Fi(\012)1153 1686 y Fg(R)1186 1680 y Fk(F)17 b Fl(has)12 b(a)f(non)o(trivial)g(self-)224 1730 y(dual)h(simple)f(mo)q(dule.)16 b(By)d(Theorem)f(4,)g(w)o(e)h(see) h(that)f(dim)1172 1736 y Fg(K)1204 1730 y Fl(\()p Fk(H)s Fl(\))e(=)h(dim)1398 1736 y Fg(F)1426 1730 y Fl(\()p Fk(A)7 b Fi(\012)1512 1736 y Fg(R)1546 1730 y Fk(F)f Fl(\))224 1780 y(is)14 b(ev)o(en.)19 b Fd(2)224 1889 y Fl(By)c(a)e(similar)f(argumen)o(t,)g(one)i(can)g(sho)o(w)g(the)g (follo)o(wing:)224 1981 y Fm(Corollary)36 b Fl(Supp)q(ose)11 b(that)h Fk(H)h Fl(is)e(a)g(semisimple)e(cosemisimple)f(Hopf)j(algebra) f(o)o(v)o(er)224 2030 y(an)16 b(algebraically)e(closed)j(\014eld.)24 b(If)16 b Fk(H)j Fl(has)d(a)g(simple)e(mo)q(dule)h(of)g(ev)o(en)i (dimension,)224 2080 y(then)e(the)f(dimension)f(of)g Fk(H)k Fl(is)c(ev)o(en.)224 2189 y(Of)h(course,)h(it)e(is)h(also)f(p)q (ossible)i(to)e(use)i(the)g(pro)q(of)e(of)g(Theorem)h(5)f(directly)m(.) 899 2310 y(9)p eop %%Page: 10 10 10 9 bop 177 195 a Fm(References)198 313 y Fl([1])19 b(P)m(.)g(Ban)o(ta)o(y:)30 b(The)20 b(F)m(rob)q(enius-Sc)o(h)o(ur)g 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b(\014nite-dimensional)e(semisimple)f(and)j (cosemi-)262 1025 y(simple)12 b(Hopf)h(algebras)h(in)f(p)q(ositiv)o(e)g (c)o(haracteristic,)h(In)o(t.)f(Math.)g(Res.)h(Not.)f(16)262 1075 y(\(1998\),)g(851-864)198 1158 y([7])19 b(J.)e(F)m(uc)o(hs/A.)g (Ch.)f(Ganc)o(hev/K.)g(Szlac)o(h\023)-21 b(an)o(yi/P)m(.)15 b(V)m(ecsern)o(y)o(\023)-20 b(es:)26 b Fk(S)1334 1164 y Fj(4)1353 1158 y Fl(-symmetry)262 1208 y(of)21 b(6)p Fk(j)r Fl(-sym)o(b)q(ols)e(and)i(F)m(rob)q(enius-Sc)o(h)o(ur)g (indicators)g(in)f(rigid)g(monoidal)e Fk(C)1515 1193 y Fh(\003)1533 1208 y Fl(-)262 1257 y(categories,)d(J.)e(Math.)h(Ph)o (ys.)g(40)f(\(1999\),)g(408-426)198 1340 y([8])19 b(J.)c(F)m(uc)o (hs/C.)f(Sc)o(h)o(w)o(eigert:)21 b(Category)14 b(theory)i(for)e (conformal)e(b)q(oundary)j(con-)262 1390 y(ditions,)e(Preprin)o(t,)i (math.CT/0106)o(050,)10 b(2001)198 1473 y([9])19 b(I.)f(Kaplansky:)26 b(Bialgebras,)19 b(Lecture)h(Notes,)g(Univ.)d(of)g(Chicago,)h(Chicago,) 262 1522 y(1975)177 1605 y([10])h(F.)g(Kasc)o(h:)30 b(Mo)q(duln)19 b(und)g(Ringe,)g(Math.)g(Leitf\177)-21 b(aden,)20 b(T)m(eubner,)h (Stuttgart,)262 1655 y(1977)177 1738 y([11])e(Y.)14 b(Kashina/G.)e (Mason/S.)g(Mon)o(tgomery)m(,)f(Computing)h(the)i(Sc)o(h)o(ur)g (indicator)262 1787 y(for)g(ab)q(elian)f(extensions)i(of)e(Hopf)h (algebras,)f(Preprin)o(t,)h(2001)177 1870 y([12])19 b(T.)h(Kerler:)31 b(Mapping)20 b(class)g(group)g(actions)g(on)f(quan)o(tum)g(doubles,)i (Com-)262 1920 y(m)o(un.)12 b(Math.)i(Ph)o(ys.)g(168)f(\(1995\),)f (353-388)177 2003 y([13])19 b(M.-A.)e(Kn)o(us:)25 b(Quadratic)17 b(and)g(hermitian)e(forms)h(o)o(v)o(er)h(rings,)g(Grundlehren)262 2053 y(Math.)d(Wiss.,)f(V)m(ol.)f(294,)h(Springer,)g(Berlin,)h(1991)177 2135 y([14])19 b(T.)11 b(Y.)g(Lam:)16 b(A)11 b(\014rst)h(course)h(in)e (noncomm)o(utativ)o(e)d(rings,)k(Grad.)e(T)m(exts)i(Math.,)262 2185 y(V)m(ol.)h(131,)f(Springer,)i(Berlin,)g(1991)841 2310 y(10)p eop %%Page: 11 11 11 10 bop 224 195 a Fl([15])20 b(R.)e(G.)h(Larson/D.)g(E.)g(Radford:)29 b(Finite)19 b(dimensional)f(cosemisimple)f(Hopf)310 245 y(algebras)c(in)h(c)o(haracteristic)h(0)e(are)h(semisimple,)d(J.)i (Algebra)h(117)f(\(1988\),)f(267-)310 295 y(289)224 378 y([16])20 b(R.)14 b(G.)g(Larson/D.)g(E.)h(Radford:)20 b(Semisimple)12 b(cosemisimple)h(Hopf)i(algebras,)310 428 y(Am.)d(J.)i(Math.)f(109)g(\(1987\),)g(187-195)224 511 y([17])20 b(V.)g(Linc)o(henk)o(o/S.)f(Mon)o(tgomery:)30 b(A)20 b(F)m(rob)q(enius-Sc)o(h)o(ur)h(theorem)f(for)g(Hopf)310 560 y(algebras,)13 b(Algebr.)h(Represen)o(t.)h(Theory)f(3)f(\(2000\),)g (347-355)224 643 y([18])20 b(S.)14 b(Mon)o(tgomery:)20 b(Hopf)14 b(algebras)i(and)f(their)g(actions)g(on)g(rings,)g(2nd)g (revised)310 693 y(prin)o(ting,)f(Reg.)g(Conf.)g(Ser.)h(Math.,)g(V)m (ol.)e(82,)h(Am.)g(Math.)g(So)q(c.,)h(Pro)o(vidence,)310 743 y(1997)224 826 y([19])20 b(W.)d(D.)g(Nic)o(hols/M.)g(B.)h(Ric)o (hmond:)25 b(The)18 b(Grothendiec)o(k)h(group)f(of)f(a)h(Hopf)310 876 y(algebra,)13 b(J.)g(Pure)i(Appl.)e(Algebra)h(106)f(\(1996\),)g (297-306)224 959 y([20])20 b(J.)15 b(P)m(.)f(Serre:)22 b(Lo)q(cal)15 b(\014elds,)h(Grad.)e(T)m(exts)i(Math.,)e(V)m(ol.)g(67,)g (Springer,)i(Berlin,)310 1009 y(1979)224 1092 y([21])k(Y.)c (Sommerh\177)-21 b(auser:)22 b(On)16 b(Kaplansky's)g(conjectures.)i (In:)24 b(F.)16 b(v.)g(Oystaey)o(en/)310 1142 y(M.)f(Saorin)g(\(ed.\):) 22 b(In)o(teractions)17 b(b)q(et)o(w)o(een)g(ring)f(theory)g(and)g (represen)o(tations)310 1191 y(of)11 b(algebras,)g(Lect.)h(Notes)h (Pure)f(Appl.)f(Math.,)g(V)m(ol.)f(210,)h(Dekk)o(er,)h(New)g(Y)m(ork,) 310 1241 y(2000,)g(393-412)224 1324 y([22])20 b(M.)c(E.)h(Sw)o(eedler:) 25 b(In)o(tegrals)18 b(for)e(Hopf)h(algebras,)g(Ann.)g(Math.,)g(I)q(I.) f(Ser.,)i(89)310 1374 y(\(1969\),)12 b(323-335)224 1457 y([23])20 b(M.)13 b(E.)h(Sw)o(eedler:)19 b(Hopf)13 b(algebras,)h (Benjamin,)e(New)i(Y)m(ork,)f(1969)224 1576 y(T)o(yp)q(eset)j(using)d Fi(A)517 1585 y(M)562 1576 y(S)h Fl(-)g(L)640 1567 y Fj(A)658 1576 y Fl(T)681 1585 y(E)704 1576 y(X)889 2310 y(11)p eop %%Trailer end userdict 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