; TeX output 2001.10.19:1650t/&=HCDtqGcmr17COnBtheEvuNenPowersoftheAntipodeBofa]Finite-DimensionalBHopfAlgebray?;!",ff cmsy10#􍍍kG9XQ ff cmr12David/E.Radford= !", cmsy10yPUniversity/ofIllinoisatChicagoeDepartment/ofMathematics,StatisticsandComputer/Science(m/c240)801/SouthMorganStreet$Chicago,/IL60608-7045s ǍHans-JdDurgen/Schneiderb`Mathematisches/Instituth"Ludwig-Maximilians-Universit9at/MdDunchenITheresienstr.D>39 D-80333/MdDunchen,Germanys㾍h,"V 3 cmbx10Abstract o!UL'K`y 3 cmr10Thew%traceofpMo!wersw%ofthesquareofthean!tipode( b> 3 cmmi10s|{Ycmr82 7)ofa nite- DdimensionalAHopfalgebraAo!verAa eldk5isstudied.Itissho!wninDman!y6casesthatthetracefunctionvdDanishesons22cmmi8mZwhens2m )!", 3 cmsy106=1A.DFinerfpropMertiesofthean!tipodearerelatedtothisphenomenon.'ffd g^ O!cmsy7K`y cmr10Research2forthispapGerwasbGegunattheColloquiodeHopfAlgebrasetQuantum GroupGesheldinLaF*alda,Cordoba,ArgentinaduringAugust7{13,1999,continuedattheHopf0AlgebraW*orkshopheldatMSRI,Berkeley,gRCA0NduringOctobGer25{29,1999,andcompletedattheMathematischesInstitutderLudwig-Maximilians-UniversitatMGunchenduring,TJuneof2000andJulyof2001,4withama8jorrevisioninJulyof2001.dTheauthorsexpress'theirgratitudeforthehospitalityandsuppGortoftheColloquioandW*orkshop.TheO rstauthorwassuppGortedduringhisstayattheInstitutebytheGraduiertenkolleg.HeeverygratefullyacknowledgesthehospitalityhereceivedfromtheInstituteandthesuppGortUUhereceivedfromtheGraduiertenkolleg.  X^yResearchUUsuppGortedinpartbyNSFGrantDMS9802178.0XQ cmr121*t/'ANG cmbx12AInutro=ductionb#'Let@c1g cmmi12AbSea nite-dimensionalHopfalgebrawithanrtipodesorver@ca eldkg.By'an;earlierresult[14 ,OTheorem1]ofthe rstauthortheorderofs22is nite.'InthispapSerwrestudysomeofthe nerpropertiesofs22Iwhicrharerelatedto'thetraces7R6 cmss12tr P(s22m D)ofpSorwersofs22.9SuppSosethattUR:V2!", cmsy10 G!VzIisalinearendomorphismofa nite-dimensional'vrectorspaceVoverkg.8WVesaythattsatis esthe3@ cmti12vanishing35trffaceconditionnifwhenevrermUR0andt2m Z6=UR1V Apthentr P(t2m)=0.'Norw _suppSosethatVisanalgebraoverkg.WVesaythattsatis esthestrffong'vanishing35trffacecondition>nifwhenevrermUR0andt2m Z6=UR1V Apthentr P(t2mrS(a))=0foralla2V;'wherewrS(a)UR:V G!VRisrighrtmultiplicationbyawhichisthereforede nedby'rS(a)(x)UR=xaforallxUR2Vp..Iftsatis esthestrongvXanishingtracecondition'thenNtsatis esthevXanishingtracecondition.eWVeshorwthats22 satis esthe'strongvXanishingtraceconditioninanrumbSerofinrterestingcases.9ResearcrhNforthispapSerbeganwithaquestionraisedbryPavelEtingofto'anrumbSerofHopfalgebraistsataconferenceheldinDurham,FEngland,in'thesummerof1999.3Inourterminologythequestionwraswhetherornots22'satis esj5thevXanishingtracecondition.Inrvestigationj5ofthisquestionledus'to7formrulatethestrongvXanishingtraceconditionandtostudyitconnection'with?dHopfalgebraautomorphismsofA,inparticularwiths22. 7Whenthe'crharacteristic7ofk^Tiszeroands22;satis esthestrongvXanishingtracecondition,'mruch-canbSesaidabouts22,NtheeigenrvXaluesandeigenspacesofs22,Nandthere'areimplicationsforthealgebrastructureofA.9ThestrongvXanishingtraceconditionhasimplicationsforclassi cationof' nite-dimensionalòHopfalgebras.FVorexample,undertheassumptionthat's22]^satis esZthestrongvXanishingtracecondition,theclassi cationofallHopf'algebrasofdimensionp22,}wherepisanoSddprime,orveranalgebraically'closedF eldofcrharacteristiczeroiscomplete."Seethediscussionfollowing'PropSosition6.9The\strongvXanishingtraceconditionforalinearendomorphismtofA'cans*bSeexpressedinawrays*whichisveryusefulforusintermsofintegrals.2 t/'Let.2Aand2A2K cmsy8bSeleftandrighrtintegralsforAandA2respSectively'whicrhsatisfy()UR=1andset,dA(t)UR=s((2) \|)t((1)):'ThenwreshowinLemma3b)thattr 9(trS(a))UR=0forallaUR2Aifandonlyif'A(t)UR=0.#Thrustsatis esthestrongvXanishingtraceconditionifandonlyif'whenevrermUR0andt2m Z6=1A ȌthenA(t2m)=0.9Our:studyofthestrongvXanishingtraceconditionfors22splitsvrerynat-'urallyiinrtotwocases: cs22 msemisimple,Iands22notsemisimple,Iasalinear'endomorphismofA.!MSuppSosethats22cisnotsemisimpleandthatthecrharac-'teristicofk<isnot2.1WVeshorwinPropSosition3thateithers22ors22 satis es'the6lstrongvXanishingtracecondition;Nhences22 p(andthruss22 )satis esthe'vXanishingtracecondition.9TheV/cases22 3semisimpleisfarmoreinrteresting. {tIfs22 =1A 4thens22is'semisimpleandsatis esthestrongvXanishingtraceconditionvacuouslyV.We'notethats22 M=1A ifAiseithercommrutativeorcoScommrutative,orifAis'semisimpleorcosemisimpleandthecrharacteristicofkiszero.юOurmain'resultswhens22issemisimplearethats22satis esthestrongvXanishingtrace'condition ifAispSoinrtedorifA2ispoinrted.TheseresultsfollowfromThe-'orem1andTheorem4respSectivrelyV.9TheideasdevrelopSedinthispaperenableustoprorveTheorem6whicrh'togetheryJwithTheorem1implieswhenAispSoinrtedandkgisanalgebraically'closed eldofcrharactericticzerothat:840a)DTheworderrtofs227dividesDimMA=DimgB,whereBX=URA0isthecoradicalDofA,andtheminimalpSolynomialofs22isX2r.1;h3b)DTheeigenrvXaluesofs22aretherS2th roSotsofunityinkg;55Lc)DTheeigenspacesfors22allharvethesamedimensionwhicrhisDimA=rS;3d)DIfeisanidempSotenrtofAwhichsatis ess22(e)$=ethenr{dividesDDim\SAe.89TheubSounddescribedina)issharp.فWhenAispoinrteditfollowsfrom'a)thatr=URDim\AifandonlyifDimAUR=1.9MoreggenerallyV,Theorem6holdsforaHopfalgebraautomorphismtofA'whicrh wsatis esthestrongvXanishingtraceconditionwhenksisanalgebraically3t/'closed eldofcrharacteristiczero.Heretreplacess22 ٷandBistheunique'maximalAsemisimple(andhencethemaximalcosemisimple)sub-Hopfalgebra'ofA.9ThepapSerisorganizedasfollorws.InSection1resultsarederivedfor'certain4linearautomorphismsofVwhenkisalgebraicallyclosed,G#hascrhar-'acteristic zeroandtsatis escertaintraceconditions.#Inthenextsection'wrecstudytheelementA(t)forlinearendomorphismstofAandprovethat'eitherA(s22m D)UR=0forallm0orAq% cmsy6 38(s22m)=0forallm0whens22isnot'semisimpleq{andthecrharacteristicofkؘisnot2.|Aq\keystepintheproSofisthe'observXation(tjB)2T=0impliesA(t)=0.'Therebry^theproSofreducestothecaseAorA2bisthegroupalgebrakg[Zp]]of'thecyclicgroupoforderpandthe eldkQhascrharacteristicpUR>0.9InDSection3wreshowthats22satis esthestrongvXanishingtracecondition'whens22 RissemisimpleandthecoradicalA0ofAiscoScommrutative. 1WVe'reduce@tothecasewhereAisgeneratedbrya;xasanalgebrawhichsatisfy'therelationxap=qn9ax,whereq2kaisaroSotofunitryV,andwhosecoproducts'aregivrenby(a)UR=a UWaand(x)UR=x a+1 xrespSectivrelyV.9WVeconrtinuethestudyofs22`whenitissemisimpleinSection4andprove'thats22 csatis esthestrongvXanishingtraceconditionwhenthecoradicalof'A2 iscoScommrutative._OurproSofisbasedonthestucturetheoryof nite-'dimensionalbgradedHopfalgebras.YAbkreystepintheproSofisthefollowing'observXation.SuppSose8thattisaHopfalgebraendomorphismofAandthat'IfistanilpSotenrtHopfidealofAwhichsatis est(I)?I.fThentA=Im5($)6܍t>6)=0'impliesA(t)UR=0,where$6܍t ?istheHopfalgebraautomorphismofA=IrEinduced'bryt.9In8Section5wrediscussthecharactersA 2ڡA2andxA2ڡAoftherighrt'regularrepresenrtationsofAandA2JrespSectivelyusingthepropSertiesofAas'aFVrobSeniusalgebra.8WededucethetraceformrulaM(DimgA)trf3(t jX.;cmmi6A ^A)UR=< xA 38(t );A(1A)>'forבHopfalgebraautomorphismstofA..Thisformrulageneralizes[6,Equation'(6)].8UsingtheresultsofSection4wreshowthatǻTxA 36=URf^u cmex10ō ADim$IA Qmfe!  DimgB.f^8-xBN>;˩'whereBisthesub-HopfalgebraofAgeneratedbrythecoradicalofA.4#ߠt/9When_thecrharacteristicofkW|doSesnotdivideDimAthenA=DimgAand'xA=DimgA areimpSortanrtnon-zeroidempotenrts.FVorexample,6whenthechar-'acteristictofkiszeroandAissemisimplethenA=DimgAandxA=DimgAare'non-zerotrwo-sidedintegralsofA2andArespSectivelyV.9InoSection6wreexamineimplicationsoftheideasdevelopSedinthispaper'forO nite-dimensionalpSoinrtedHopfalgebrasoveranalgebraicallyclosed eld'of\crharcteristiczeroandweexamineimplicationsfortheDrinfel'ddouble.'OurdiscussionofpSoinrtedHopfalgebrasisbasedoncertainidempotenrts.9WVekwranttopSointoutthatitmayverywellbSetruethatthesquareofthe'anrtipSodeZofany nite-dimensionalHopfalgebraoverkTwsatis esthestrong'vXanishingtracecondition.8Needlesstosaryweknowofnocounterexample.9WVershallassumethatthereaderhassomefamiliaritrywithHopfalgebras'[9,18].8ThroughoutthispapSerkQisa eld.9Finallytheauthorswrouldliketothanktherefereeforusefulcomments.(V'A1DPreliminariesb#'WVebSeginwithsevreralobservXationsaboutalinearendomorphismtofa nite-'dimensionalvrectorspaceVI,overkg.Theendomorphismtissemisimple[3,'AVISI,8]ifeacrht-stablesubspaceofVGAhasat-stabledirectsummand,or'equivXalenrtly*iftheminimalpSolynomialoftoverkGhasnomultiplefactors. If'kis'balgebraicallyclosed,Npthentissemisimpleifandonlyiftisdiagonalizable.9IfUt2r = 1V forsomepSositivreintegerrS,pthenthesmallestsuchrXiscalled'the"$orffderf5oft;=otherwisetheorderoftissaidtobSein nite.TWVedenotethe'order$Aoftbryorder(t)whenitis nite.WVehaveremarkedthattheorderof'theanrtipSodeofa nite-dimensionalHopfalgebraorverkHis nite.hIfthas' niteueorderthentissemisimpleifandonlyifthecrharacteristicofk܂doSesnot'dividetheorderoft.'5N cmbx12Lemma1cQSuppffosezthatthe eldkyisalgebraicallyclosed,hascharacteristic'0"andthatVisa nite-dimensionalveffctorspaceoverkg.*cAssumefurtherthat't[isalineffarautomorphismofVzwith niteorderr1andthattsatis esthe'vanishing35trffacecondition.fiThen:40a)DThe35minimalpffolynomialfortisX2r.1.3b)Drdivides35Dim:Vp.52t/55Lc)DTheeigenvaluesoftarffetherS2th rootsofunityinkCandeacheigenspaceDhas35dimension(DimgVp)=rS.!'Q- cmcsc10QProof:zWVecanassumethatr>F1.b9Letp(X)=a0A+(=a1X+Px+X2` bSe'theIminimalpSolynomialoft.SincetisanautomorphismofVRnecessarily'a0 O6=0.aSincet2r('1V =0itfollorwsthat`rS.aTVakingthetraceofboth'sidessoftheequation0UR=a01V T+a1t+o+t2`lgivresstr (t2`)UR=a0(DimgVp)6=0.'Therefore`xrS.v*Consequenrtly`=rRandX2r1@thenP{gI{2G!gn92`=}{0forall1`t/'QProof:&pSinceGthas niteorderrFandisdiagonalizable,5nVhasabasisof'eigenrvectorsefortbSelongingtor2th rootsofunitry!1;:::ʜ;!nP,wherenUR=Dim\Vp.'WVemaryassumethat!1;:::ʜ;!sÎ6=UR1andthat!s+1 ==!n=1.8Since!trS(t `)UR=! n9`ڍ1j+UN+! n9`ڍs+(ns)'for>all1Q`r7i1,Sand>!{/+Ǵ+!2n9rDtGGcmr17(t)AandA(s2m)AintheDSp=ecialzCasewhens2 AIsnotSemisimpleb#'Let3oAbSea nite-dimensionalHopfalgebrawithanrtipodesorver3othe eldk'andYletbSea xednon-zeroleftinrtegralforA.3FVoralinearendomorphism'tofAwrede nedA(t)UR=s((2) \|)t((1)):||(3)7Oat/'SincethesubspaceofleftinrtegralsforAisone-dimensional,(vchoSosingadi er-'enrt_non-zeroleftintegraltode neA(t)resultsinanon-zeroscalarmultiple'oftheoriginal.9IfJA(t)UR=0thentr 6}(t)=0bryparta)ofthenextlemma.WVeareinterested'indeterminingwhentr L(t)UR=0forcertaint,spSeci callywhentUR=s22m+Jforsome'inrtegerm.8FVoraUR2Alet`(a);rS(a)betheendomorphismsofAde nedbryU(tjB)UR=0impliesA(t)=0.'QProof:Since4A(s22)[=s((2) \|)s22((1))[=s(s((1) \|)(2))[=()1,and4Ais'semisimpleifandonlyif()1UR6=0,parta)follorws.9TVoprorveb)we rstconsiderthecasewhentr O(s22)UR6=0.,ThenAissemisim-'plefandcosemisimplebry[5,Theorem2.5],ands22 R=YN1A aJbry[6,Theorem3].'HenceQUb)isvXacuousinthiscase.lNorwweassumethattr (s22)=0.lWVeQUmay'alsoassumethatkQisalgebraicallyclosed.8Thruss22isdiagonalizable.9Let6rbSetheorderofs22.Letfx{Kg1{nVbeabasisofAofeigenrvectors'ofls22.c+WVrites22(x{K)=!n92a{zx{Hforlall{,ќwhere!isaroSotofunitryoforderr,'and~0Pa{a,wherea= 1m1j+UN+ rbmr.9SuppSose 2tisanalgebraendomorphismofAwhicrhsatis est(B)@rB.'Thecalculation4s((2) \|)t((1))UR=s(BϦ(2)fa(2) \|)t(BϦ(1)a(1))UR=s(a(2))s(BϦ(2)f)t(BϦ(1))t(a(1))'shorws*`thatA(t)=s(a(2) \|)BN>(tjB)t(a(1)).Prart*`c)followsfromthisequation.'210 tt/9Prarttb)oftheprecedingpropSositionimpliesthatiftheorderofs22 4is'primeuandk hascrharacteristiczerothens22 5satis esthestrongvXanishing'tracecondition.9Byےpartc)oftheprecedingpropSositioninmanrycasesA(s22m D)UR=0ےwhen'thecrharacteristicofkQispSositive.'Corollary2pLffethyAbea nite-dimensionalHopfalgebrawithantipodesover'theo eldkcwhichcffontainsnon-semisimplesub-Hopfalgebrawhichiseither'cffommutative35orcocommutative.fiThenA(s22m D)UR=035forallintegersm.'QProof:,Let:B@bSeasub-HopfalgebraofAwhicrhsatis esthehypSothesis'ofM|thelemma.a[SinceBiscommrutativeM|orcoScommrutativeM|s22jB K=1BN>.Since'B1isnotsemisimpleBN>(s22jB).P=0bryparta)ofPropSosition2. =FVorany'inrtegermnotethatB]isinvXariantunderthealgebramaps22mandthat'BN>(s22m DjB)e=BN>(1jB)e=BN>(s22jB)e=0.UHence7theCorollaryfollorwsfrompart'c)ofPropSosition2.829WhenXs22\isnotasemisimpleendomorphismofA,&Candthecrharacteristic'ofkQisnot2,itisalwraysthecasethattr P(s22m D)UR=0.'Prop`osition3}Lffet#Abea nite-dimensionalHopfalgebrawithantipodes'overthe eldkg.9Suppffosethats22lisnotasemisimpleendomorphismofAand'that35thecharffacteristicofkRisnot2.fiThen:40a)DEither-A(s22m D)UR=0forallinteffgersmorA 38(s2j2mx)=0forallinteffgersDm.3b)DtrMF(s22m D)UR=035forallinteffgersm.'QProof:MWVe"maryassumethatk:?isalgebraicallyclosed.1 Sincethecharacter-'istic1ofkisnot2ands22isnotsemisimple,Vitfollorwsthatthecharacteristicof'kyispUR>2andpdividesorderu(gn9)orpdividesorder( )bry(2).Thusthegroup'algebra8|kg[Zp]]ofthecyclicgroupZpoforderporver8|kisasub-Hopfalgebra'of|AorA2./Consequenrtlyparta)followsbyCorollary2./Sincetr 5(t2)UR=tr (t)'forqalllinearendomorphismstofA,partb)follorwsbyparta)andparta)of'Lemma3.829WVeendthissectionwitharesultontheorderofasemisimplecoalgebra'automorphism.11 t/'Lemma4cQLffet5oC bea nite-dimensionalcoalgebraoverthe eldkandsup-'pffose&sthattisasemisimplecoalgebraautomorphismofCܞ.b(Thentheorderof't35istheorffderofthecoalgebraautomorphismtjCq1ofC1.'QProof:Sincetisacoalgebraautomorphismofthe nite-dimensionalcoal-'gebraVCܞ,t:therestrictiontjCnisacoalgebraautomorphismofCnforallnUR0.'WVe]maryaswellassumethattjCq1has niteorderrS.TVoprovethelemmawe'needdonlyshorwthatt2rbjC t=UR1C,orequivXalentlythatt2rbjCnk=UR1Cnforalln1.'Thelatterwredobyinductiononn.ThecasenUR=1followsbythede nition'ofrS.9SuppSose׾thatnUR>1׾andt2rbjCqn1t+=UR1Cqn1.2Thent2rg1C VvXanishesonCn1.'Therefore8;(t r= 1C)(CnP)=d(t rb t r= 1C t1C)((CnP)) dUR((t r= 1C) UWt r+1C t(t r1C))@f\ 8n X ҁ{=0Cn{ eC{Kܟf\!dURCnP C0j+C0 [CnP:'ThrusSV(t2rP1C)(CnP)C1,mandconsequenrtly(t2rP1C)22(CnP)=(0).rThelast'equationYimpliesthattheminimalpSolynomialm(X)oftjCnrdivides(X2rI1)22.'SincetissemisimpletherestrictiontjCnW3issemisimple. w6Thereforem(X)'dividesX2r.1;consequenrtlyt2rbjCnW1Cnk=UR0asrequired.82(V'A3DThehElemenutA N(s2m)Awhens2 dAIsSemisimpleDandzAAisPuointedb#'LetAbSea nite-dimensionalpoinrtedHopfalgebraoverthe eldkBand'suppSosethats22 isasemisimpleendomorphismofA.[$Thepurposeofthis'sectionNistoshorwthats22 satis esthestrongvXanishingtracecondition.eTVo'prorvethisresultwrewillusethefollowingtwotechnicallemmas.9The# rstlemmainrvolves#qn9-binomialcoSecienrts. 8RLetq\beanon-zero'elemenrtKofkg.[TVodescribSeqn9-binomialcoecienrtsweconsiderthealgebraA'orverVkgeneratedbrysymbSolsa;xsubjecttotherelationxa=qn9ax. |The'a2{Kx2|'s cwhere0T{;| cformalinearbasisforA.FVor0T{n ctheqn9-binomial12 0t/ 'coSecienrtf\ dn>{Qf\! q%isthecoecienrtofa2n{x2{intheexpansionof(a+x)2nP;!L'otherwiseꨟf\ d Xnw{f\!9 q!؉=UR0bryde nition."J'Lemma5cQLffetN N>1andsupposethatthe eldk(containsaprimitiveN@2th'rffoot35ofunity!n9.fiThen:U]40a)Df\ dMʜN1Z`mٟf\!s- !||=UR(1)2`!n92`(`+1)=2,for35all0`N1.)3b)DIfthecharffacteristicofkispl>0thenf\ d UN@p2s1K`8{f\!=%ϟ !GR=l(1)2`!n92`(`+1)=2^Dfor35allsUR035and0UR`N@p2s1.'QProof:Prarta)followsfromtheequations" 0UR=f\ d @N`霟f\! !&f=!n9 `gf\ dQN1t`/ f\!4^ !=X+f\ d XN1s`1+ɕf\!0 !" 'for| all1L`NN 1. See| [11 ,dpages691{692]forexample.TVoshorwpart'b)wre rstnotethatanynon-negativeintegermhasauniquedecompSosition'mH=mDN"+>mR,where LmD;mR areinrtegersand0HmR 0,~xs0`and0`0.8InthiscaseA\f\ dJ N@p2s1\`u42f\!zI !Z=URf\ d @N1:`R,t?f\!1 !9f\ dBp2s1K`DbeSf\!kO=UR(1) `X.R !n9 `X.R(`X.R+1)=26f\ d?|7p2s1H`D_Zǟf\!13נt/ 'bryparta).2=Since0t =f\ d ^p2s e'mf\!" =f\ d ^p2s1ym-=Mf\!5+Yf\ d  p2s1 sm1+晟f\!6 forall1t mUR1.9Let ZnbSetheorderofa.FVromtherelationxa=N=!n9ax Zwrededucex=N='xa2n D=3!n92na2nPx.xThrus!n92n J}=1sincex6=0.xThismeansn=N@Lforsome'LUR1.8Therelationalsoimpliesˍ3(x rb)UR=r X 8獑c`=0f\ dr :`#Pf\!(f: !0xx r=1thesetofmonomialsfa2{Kx2|j0{0.9InthiscaseM6=URN@p2r} forsomer0,soDimA=nN@p2rb.8Therelations1+;a n=UR1; xa=!n9ax;and'Xx N"p-:rc= (a N"p-:r71)+ 0x N +UN+ rUR1and2m Z6=1.82'Theorem1mXLffetAbea nite-dimensionalHopfalgebrawithantipodesover'the eldkg.MSuppffosethats22isasemisimpleendomorphismofAandthatthe17ht/'cfforadicalلA0 ofAiscffocommutative.YWThenلs22satis esthestrffongvanishing'trfface35condition.'QProof:{Let/rK bSeanalgebraicclosureofkg. >Since(A UWKܞ)0 >*~&A0 [Kit'follorws thatthecoradicalofA UWKiscoScommutative.IThussinceKisalge-'braicallyefclosedA UWKBispSoinrted. uTVoprovethetheorem, itiseasytoseethat'wremayreplaceAwithA UWKܞ;thuswewillassumethatAispSointed.9SuppSose1thats22m6=UR1A.Picrkelementsa;xUR2A1andaccordingtoLemma'7\andletB=kgfa;xgbSethesubalgebraofAgeneratedbryaandx.Then'B'isasub-bialgebraofA.Thecalculationss(a)i=a21fands(x)=xa21'shorwXthatBisasub-HopfalgebraofA.JUsingthelastequationwecalculate's22(x)8=axa21=x. Inparticularxa8=!n9ax,I.where!q=21 \|.Since's22m D(x)?6=x#itfollorwsthats22mjB }6=?1BN>.ThereforeB(s22m DjB)?=0#bryLemma'6,andconsequenrtlyA(s22m D)UR=0bypartc)ofPropSosition2.829Whens22isconjugationbryagrouplikeelement,*parta)ofPropSosition3'canbSestrengthenedabitinthepoinrtedcase.'Corollary3pLffethyAbea nite-dimensionalHopfalgebrawithantipodesover'the$ eldkg.;SuppffosethatthecoradicalA0 ofAiscocommutativeandthat's22(x)!=axa21+forallx!2A,Mwherffea2G(A).Thens22 asatis esthestrffong'vanishing35trffacecondition.'QProof:ZWVemaryassumethatkcisalgebraicallyclosed.nIfthecharacteristic'ofHk8doSesnotdivideorder (a)thens22 issemisimpleandTheorem1applies.'SuppSosefthatthecrharacteristicofkdividesorder*.(s22). Thenkhascrharacter-'istic p"2andthegroupalgebrakg[Zp]]ofthecyclicgroupZp9isasub-Hopf'algebraofA;thrusCorollary2applies.829AYvreryZbasicexampletowhichCorollary3appliesistheTVaft(Hopf8)'algebra\Tn;q (kg),wherenisapSositivreintegerandq^2kCyisaprimitiven2th'roSot(Lofunitry[19 ].AsanalgebraTn;q (kg)isgeneratedbya;xsubjecttothe'relationsza2n =r1,x2n=0zandxa=qn9ax. ]ThrusDim9Tn;q (kg)=n22. ]The'coalgebra0structureofTn;q (kg)isdeterminedbry(a)bz=a UWa0and(x)bz='x UWa+1 x.18}t/ 'A4DTheElemenutA N(s2m)Awhens2 QAisSemisimpleDandzA AisPuointedb#'LetAbSea nite-dimensionalHopfalgebraorverthe eldkandletB\kVA'bSeasub-HopfalgebraofAconrtainingthecoradicalA0ՔofA.WVebeginour'discussionwithananalysisoftheassoSciatedgradedHopfalgebragr\hTB(A).9Recall;thatifU;VURAaresubspaces,^thewredgeU^EVURAistheinrverse'imageunderthecomrultiplicationofU A+A Vp.8Notethat2U؍F0(A)UR=BXF1(A)=BE^BXF2(A)=BE^B^BXUR:::'isaHopfalgebra ltrationofAwithS ?n0FnP(A)UR=A.9De neF1 \|(A)UR=0andlet+!gr6TB=m(A)UR=~M n0ZgrܟTB'(A)(n);where'Vgr2NTB9T(A)(n)=FnP(A)=Fn1(A)forall&/n0;"֍'bSetheassociatedgradedHopfalgebra.9TheQprojection&:gr*iTBx(A)!gr*iTB(A)(0)=BisQaHopfalgebramap,'and# theinclusionmapBPVPgr'TBuN(A)isaHopfalgebrasectionofn9.LetRΚ='gr1TB9?(A)2coBbSejtherighrtcoinvXariantelementswithrespSecttotheprojectionn9.'By,de nitionR$=fx2gr7TB(A)j(1׫ n9)((x))=x׫ 1g.6By,[15 ,=LTheorem'3]thereisanisomorphismofHopfalgebras5grGTBҖ3(A)PUR԰n:=RJ#B;'de nedbrymultiplicationingrsϟTB (A),whereRYhasthestructureofabraided'HopfalgebrainthecategoryofleftYVetter-DrinfeldmoSdulesorverB.]ISee[9]'andthediscussionin[2,Section2].b'Lemma8cQLffet0Abea nite-dimensionalHopfalgebraoverthe eldkNMand'suppffose$thatB*isasub-HopfalgebraofAwhichcontainsthecoradicalA0(of'A.fiThen:ፍ40a)DRn=URLqƟnRJ(n)ywithR(n)UR=R(D\gr hTB(A)(n)yforallnUR0yisagrffadedyHopfDalgebrffa35inthebraidedsenseandgr֟TB(A)(n)PUR԰n:=RJ(n)#B;forallnUR0.83b)DRJ(n)BZ=f%a m[2FnP(A)=Fn1(A)ja2Fn(A);(a) xa 1BZ2Fn1(A) x DFnP(A)g35forallnUR0.19t/'QProof:~/Prartga)isexplainedin[2,BtLemma2.1].TVoseepartb)letaUR2FnP(A).'Theޝresidueclass a2NFnP(A)=Fn1(A)isinRifandonlyif a(1)f  qa(2)6A䍍''a0j %i132grTBH1(A) UUf`N9Lji1,gr7TB>k(A)(i)f`c}.UThelastconditionmeansexactlythat'(a)a 1isanelemenrtinFn1(A) FnP(A):29WVe!willneedthedualofPropSosition1.ݝForallpf2A2,oLlet!RJ(p)and'L(p)bSethelinearendomorphismsofAde nedbryRJ(p)(a)UR=p(a(1) \|)a(2)9and'L(p)(a)!=a(1) \|p(a(2))respSectivrelyforalla!2A.3NotethatRJ(p)andL(p)'arethetranspSosesofleftandrighrtmultiplicationrespSectivelybyponA2.'Sincedtr (t)%=tr P(t2)dforalllinearendomorphismstofA,Athedualvrersionof'PropSosition1canbeformrulated:Z'Prop`osition4}Lffet#Abea nite-dimensionalHopfalgebrawithantipodes'and'tbffeacoalgebraautomorphismofAwhichcommuteswiths.?Thenthe'following35arffeequivalent:40a)DA 38(t2)UR=0:3b)DA 38(t21)UR=0:55Lc)DtrMF(L(p)t)UR=035forallpUR2A2.3d)DtrMF(RJ(p)t)UR=035forallpUR2A2.'2'Theorem2mXLffet Abea nite-dimensionalHopfalgebraoverthe eldksand'suppffoseuthatB{isasub-HopfalgebraofAcontainingthecoradicalofA.VLet'taHopfalgebrffaautomorphismofA.RAssumet(B)UR=B,letgrQ{(t)denotethe'inducffedHopfalgebrffaautomorphismongr{TBT(A)andletRn=URgr TB1(A)2coBw.9ThenyRtr/(L(p)t)UR=tr (gr Y(t)jR)tr ((L(p)t)jBN>)'for35allpUR2A2.'QProof: 2Let_n1. Byparta)ofLemma8themrultiplicationmap'RJ(n)#BX!URgr TBP(A)(n)isbijectivre.6GNowlet_a c/2URR(n),la2FnP(A)andbUR2B.'Then,(a)a 1\T2Fn1(A) FnP(A)brypartb)ofLemma8.mTherefore'a(1) \|b(1)$ a(2)b(2)ab(1) b(2)2URFn1(A) FnP(A).8Hencewrecompute]QvߖgrQV(L(p))(%a+b)UR=щfeR 3/a(1) \|b(1)p(a(2)b(2))Yy=za Sb(1) \|p(b(2))20t/'forallpUR2A2,wheregr\h(L(p))denotestheinducedmapongrTB(A).9WVejharveshownthattheinducedmapgr(L(p)t)canbSeidenti edwith'the)map(gr q(t)jR) UW(L(p)t)jBN>). dThisprorves)tr \(gr(L(p)t))=tr~(gr(t)jR)ō'tr/3((L(p)t)jBN>),andthetheoremfollorwssincetr P(L(p)t)UR=tr (gr q(L(p)t)).8299TheoremK%2impliesananalogofpartc)ofPropSosition2underanaddi-'tionalassumption.͍'Corollary4pLffet!Abea nite-dimensionalHopfalgebraoverthe eldkg.'Suppffose/thatI)SAanilpotentHopfideal.\LettbeaHopfalgebraauto-'morphism?SofAsuchthatt(I)kI0and?Slet$i6܍t denotetheinducffed?SHopfalgebrffa'automorphism35onA=I.fiIfA=Im5($)6܍t>6)UR=035thenA(t)UR=0.fg'QProof:&LettC: A mc!A=I =BzbSetheprojectionandidenrtifyB2 N~with'n92.=(B2[ ). SinceqIhRadAitfollorwsthatthesub-HopfalgebraB2 ̹ofA2'conrtainsthecoradicalofA2.-Observethat$6܍t2=URt2jBdmandL(n9(a))=L(a)jBd'forallaUR2A.9SuppSosethatA=Im5($)6܍t>6)UR=0.8Thenc`trk((L(a)t )jBd )UR=tr (L(n9(a))$)6܍t ;)=tr ($)6܍t>6 >7rS(n9(a)))=0'foralla62AbryPropSosition4.NowweuseTheorem2toconcludethat'tr/3(trS(a))=tr (L(a)t2)=0"foralla2A,0whence"A(t)=0bryPropSosition'4again.829NorwweassumethatAispSointedandBX=URA0.EThegradedHopfalgebra'assoSciatedtothecoradical ltrationwillbedenotedbrygr(A).3Ourgoalis'to0shorwinTheorem2thattr (gr q(t)jR)=00underadditionalassumptions.'WVewillneedthenextlemmatoreducetoasubalgebrawherethetracecan'easilybSecomputed.ff'Lemma9cQLffetAbea nite-dimensionalalgebraoverthe eldkandsuppose'thatiBOoaloffcalisubalgebraofAwithmaximalidealB2+ kandB=B2+P ԰ =kg.Lffet':A!Abffeanalgebrahomomorphismsuchthatn9(B2+~)B2+.oUAssume'thatkAisfrffeekasaleftB-moffdulebyrestriction.  If2isadiagonalizable'endomorphismofA,Ltthentherffeexistsan9-stablelinearsubspaceXbq9A'suchthatthemultiplicffationinAde nesanisomorphismBb "\Xr!A.K9In'pffarticular{trY(n9jBN>)trf3(jX)UR=tr (n9):21 ht/'QProof:īLetfa{Kg1{nbSealeftB-basisofA,andletV8Mbethekg-spanofthe'aid's.}Then"A=B2+~Vm\Vp,0andB2+A=(B2+~)22Vm\+B2+VQm=B2+VLisn9-stable.'Since&isdiagonalizable,+thereexistsan9-stablesubspaceXJAsucrhthat'A=B2+~VjxX.֬SinceAB2+A=B2+~Vp,,hwrededuceformthelastequationthat'B2+~V h=p(B2+)22VR'+B2+XifromwhicrhB2+~Vp(B2+)2mVR'+B2+Xifollorwsforall'naturalZ'nrumbSersmUR1.SinceB2+ isanilpSotenrtidealofA,w weconcludethat'B2+~Am=B2+V=B2+X.5ThereforeA=B2+XIXMX,)tandthemrultiplication'map0B} wXEY!SAissurjectivre,hencebijectivesinceB ]XqandAhavethe'samedimension.9TheclaimabSoutthetracesfollorwseasilyV.82U'Theorem3mXLffetAbea nite-dimensionalpointedHopfalgebrawithantipode'soverthe eldkg.M2Assumethats22issemisimpleandsetRn=URgr (A)2coAq0:Ifm'is35aninteffgersuchthats22m6=UR1A thentr h(gr Y(s22m D)jR)=0.EǍ'QProof:+WVemaryassumethatkn-isalgebraicallyclosed.ByLemma7,*there'are[agroup-likreelementa2G(A);06=x2A[andaroSotofunity2kof'orderyN6>UR1sucrhthat(x)=xÖ a+1 x;s22(x)=x;2m Z6=1:yLetgË=a21'andyË=URxa21 \|.8Then(yn9)=y 1+g yXands22(yn9)UR=y.9Let( zbSetheresidueclassofyEingr(A)(1)=A1=XA0. Then( zisnon-zero'andgr q(A)(z)UR=gW lzrO+z 1.Hencez52URRn=gr (A)2coAq0,:R(z)=1l z+z 1'ands2(z)UR=g z.5Here,R landT arethecomrultiplicationandthecoaction'of theYVetter-DrinfeldHopfalgebraR"orver kgG(A).Sincegr{H(s)(z)=gn921 ʵz,'wrecompute(gr q(s))22(z)UR=gn921 ʵzgn9.9ThruszV2P(RJ)isgn9-homogeneousandtheactionofg6 onzPisgivenby'gz5=URgn9zg21 =qzswithqË=21G$aroSotofunitryoforderN@.9SinceMYgr(A)MYis nite-dimensional,fz12Ngro(A)(1)isnilpSotenrt.`Hencez2n .='0;z2n1c6=UR0L!forsomen2.Then1;z;:::ʜ;z2n1YareL!linearlyindepSendenrt,kand'brytheqn9-binomialformula, 8dy40UR=R(z n13)=(z3 1+1 z) n=URn1 ԟX ҁ{=1Zf\ dDn{&HYf\!+] q1z {g z n{:'Z'HenceIf\ d n{憟f\!ڟ q!\=l0Iforall1{n1.aThenIwreknowfrom[11 ,Corollary2]c'thatN+dividesn,andwreobtain`ftri0(gr q(s 2m D)jk6[zV]'()UR=n1 ԟX ҁ{=0Zqn9 m{+=0since .qn9 m O6=1and|qn9 n k=1:22t/'Sincekg[z]isaloScalsubalgebraofRJ,tr P(gr q(s22m D)jR)UR=0bryLemma9.82!'Theorem4mXLffetAbea nite-dimensionalHopfalgebrawithantipodesover'the eldkg.PAssumethats22 AissemisimpleandthecfforadicalofA2iscffocom-'mutative.fiThen35s229satis esthestrffongvanishingtracecondition.'QProof:;WVemaryassumethatkisalgebraicallyclosed. Thenthetheorem'follorwsbydualityfromTheorem3,Theorem2andPropSosition4.82( 'A5DAGeneralization%andv;uUR2Vp.5Itiseasy'toseethattr P(vn92.= vn9)UR=v2.=(v).9FVorallxUR2AwrehaveUy vn9xUR=v 2.=x=n X ҁ {=1'(vxa{K)vb{.=n X ҁ {=1va{K'(b{vx)">'bry(9)and(10)respSectivelyV.8Thus ׍ tUR=n X ҁ {=1a{K2' UWt(vn9b{K)=n X ҁ {=1'UVb{K 3t(vn9a{K)!⍑'fromwhicrhthetraceformulasfollow.82U9LetSAbSea nite-dimensionalHopfalgebrawithanrtipodes,let2A'bSeWaleftinrtegralofAandlet2A2 [beWarighrtintegralofA2 [suchthat'()UR=1.8Thenp>xUR=(x(1) \|)s((2)):(11)'WVetrecallthestandardproSof:MForallx@g2Atwrecompute(x(1) \|)s((2))@g='(x(1) \|(1))x(2)(2)s((3))=(x(1) \|)x(2)=x;Vwherethe rstequalitryholds'sinceisarighrtintegralandthelastequalityfollowssinceisaleftintegral'andthatis'A(x)9=tr.(lC(x))forallx92A. RLSinceAisaFVrobSeniusalgebraLemma246t/'10.impliesthatA(x)}=tr D(rS(x)).forallx}2A.aswrell. VUsingthislatter'formrulationofA ȌweseethatqA(x)UR=(A(1A)x)forall&/x2A(12)'bryparta)ofLemma3,whereisusedtode neA(1A).9Then isomorphismAh A!Ah A;x yV7!x(1) x(2) \|yn9;n impliesthe'follorwingwell-knownidentityinA2 2ڍA 36=UR(DimgA)A:(13)9NorwA tUR=A ȌforanryalgebraautomorphismtofA,orequivXalentlyT(A)UR=A;(14)'whereT=URt2.8InparticularSן22r(A)=A ȌwhereSistheanrtipSodeofA2.9Bythesameargumenrt,foranygrouplikeelementUR2A2,BA 36=URA;(15)'sinceforallax2A,(A)(a)=A(a(1) \|)(a(2))x=A(a(1) \|(a(2))),andax7!'a(1) \|(a(2))isanalgebraautomorphismofA.9FVor!Tclaritrywewillusethenotationhp;xiUR=p(x)!TforallpUR2A2Xand!Tx2A.'Theorem5mXLffetJAbea nite-dimensionalHopfalgebraoverthe eldkg,Plet'j52AȻbffealeftintegralofAandletj52A2 beȻarightintegralofA2 such'that35()UR=1.fiLffetTbeaHopfalgebraautomorphismofA2.fiThend(DimgA)tr f1(TjX.A ^A)UR=hA 38(T);A(1A)i;'wherffe35isusedtode neA(1A)35andSן21 S()tode neA 38(T):'QProof:WVe rstnotethat(sS(A 38(T))UR=T((2) \|)S((1))(16)'sinceCuSLandT;commrute. CHItfollowsthat?e y:7A2 `;!kde nedCuby?e kB(p)7='p()Lforallp2A2 isLaFVrobSeniushomomorphismofA2withdualbases'fS((1) \|)g;f(2)g;#vrerify(10)usingthefactthatīeisaleftintegralforA225Bt/'andj{thatisarighrtintegralforA2.&Using(14)weobservethatT Ade nesan'endomorphismYofAA2.HencewregetfromLemma10and(13)withvË=URA?T5$G (DimgA)tr f1(TjX.A ^A))UR=Eīe (T(A(2) \|)S((1)))r=hAT((2) \|)S((1));ir=hAS(A 38(T));ihbry(16)r=A((1) \|)hA 38(T);s((2))ir=hA 38(T);A((1) \|)s((2))ir=hA 38(T);(A(1A)(1) \|)s((2))ihbry(12)r=hA 38(T);A(1A)ie|@bry(11)|:;'29IfT=URSן22 #thenthetraceformrulaofTheorem5isthetraceformulain[6,'Equation(6)]sincezehA 38(Sן 2r);A(1A)iUR=h(1)";S((2) \|)(1)iUR=(1)"():9FVorua nite-dimensionalHopfalgebraAorveruthe eldkletxA =A2'A2 8=0A.ThrusYxA 6isdeterminedbyp(xA)0=tr wc(RJ(p))Yforallp02A2,torYby'p(xA)e=tr 7(L(p))forallpe2A2.TSuppSosethatBisasub-HopfalgebraofA.'ThenIAisafreeleft(orrighrt)B-moSduleby[10 ,a}Theorem7].VThusDimQ B'dividesDimAwhicrhmeansthatDimA=DimgBisaninrteger.9WVeendthissectionbrydescribingtherelationshipbSetweenxA andxBN>,'whereBisthesub-HopfalgebraofAgeneratedbrythecoradicalofA.'Prop`osition5}Lffet=Abea nite-dimensionalHopfalgebraoverthe eldk'and§suppffosethatB]isthesub-HopfalgebraofAgeneratedbythecoradical'of35A.fiThenTxA 36=URf^ō ADim$IA Qmfe!  DimgB.f^8-xBN>:'QProof:^*ByTheorem2wrehavep(xA)x=tr (gr q(1A)jR)p(xBN>)forallpx2A2.'SinceRJ nBX'URAasvrectorspacesandgrA(1AjR)=1R {thepropSositionfollorws.'2'l'A6DApplicationsQtoHopfAlgebraAutomorphismsDtzAandFaGurtherResultsonA N(t)b#'SuppSose24thattisaHopfalgebraautomorphismofAwhicrhhas niteorder'andassumethattsatis esthestrongvXanishingtracecondition. ^0Inthis26Nt/'section]wrestudytwithparticularemphasisonthecasetUR=s22.Our]discussion'isbasedonthefollorwinggenerallemma.(l'Lemma11jQLffet1Ytbeanalgebraautomorphismofa nite-dimensionalalgebra'A@overa eldkofcharffacteristiczero.Supposethatthas niteorderrxand'satis esthestrffongvanishingtracecondition.TIfeisanidempotentofAand't(e)UR=e35thenrdividesDim:Ae.卑'QProof:X\WVemaryassumethatk8isalgebraicallyclosedandthateUR6=0.GSincet'isanalgebraendomorphismofAandt(e)UR=e,theidempSotenrtendomorphism'rS(e)ofAandtcommrute.8Inparticulart(Ae)URAe.9Norwٹtr ?(t2mjAe T)>=tr Rq(t2mrS(e))ٹforallm>0.Sinceٹtsatis esthestrong'vXanishing5traceconditionitthrusfollowsthattjAesatis esthevXanishingtrace'condition.5SinceAeUR6=(0)itfollorwsthattjAehas niteorderrS.5Ourconclu-'sionthatr>6dividesDimAenorwfollowsbyLemma1.82 y9Letk,AbSea nite-dimensionalHopfalgebraorverk,kIandletn00.lThen'the+sumofallsubScoalgebrasCsofAwhicrhareinvXariantunders22W/andsatisfy's22n TjC t=UR1C iscinfactasub-HopfalgebraofA. ThrusAhasauniquesub-Hopf'algebraBmaximalwithrespSecttothepropertrythats22n TjB =UR1BN>.9NorwwZsuppSosethatthecharacteristicofkwiszero.SinceAissemisimple'if%andonlyifs22jA =1A bry[6],4weconcludethatAcontainsauniquemaxi-'malsemisimplesub-HopfalgebrawhicrhwedenotebyAsn<.Asanimmediate'consequenceofLemma11:'Corollary5pLffethyAbea nite-dimensionalHopfalgebrawithantipodesover'a eldk82ofcharffacteristiczero,letAs?Qbetheuniquemaximalsemisimplesub-'HopfalgebrffaofA,Psupposethats22mhasorderr%andthats22msatis esthestrong'vanishingItrffacecondition..ThenrdividesDimAeforallidempotentseUR2Asn<.'2卑9ThepdimensionoftheuniquemaximalsemisimpleHopf-subalgebraofa' nite-dimensionalsHopfalgebraAorversa eldofcrharacteristiczeroisrelated'toЂtheorderofaHopfalgebraautomorphismofAwhicrhsatis esthestrong'vXanishingtracecondition.卍'Theorem6mXLffetGAbea nite-dimensionalHopfalgebraovera eldofchar-'acteristicnzerffo,0letBX=URAsbetheuniquemaximalsemisimplesub-Hopfalge-'brffaofA, supposethattisaHopfalgebraautomorphismofAof niteorder'rand35talsosatis esthestrffongvanishingtracecondition.fiThen27\t/40a)Drdivides35Dim:A=DimgB,35andtheminimalpffolynomialoftisX2r.1.3b)DThe35eigenvaluesoftarffetherS2th ~rootsofunityinkg.55Lc)DTheeigenspffacesfortallhavethesamedimensionwhichis(DimgA)=rS.3d)DIf!eisanidempffotentofAwhichsatis est(e)k=e!thenruxdividesDDim\SAe.!'QProof:JLetJebSealeftinrtegralforBRPwhichsatis es(e)UR=1.kTheJcalculation'e22V=UR(e)e=e!RshorwsthateisanidempSotentofA.ByassumptiontisaHopf'algebraautomorphismsofA.!Thrust(B)isasemisimplesub-Hopfalgebraof'A?whicrhmeansthatt(B)zBEby?themaximaliltyanduniquenessofB.yAs'aconsequencetjB :isaHopfalgebraautomorphismofBSfromwhicrht(e)UR=e'follorws.8AtthispSoinrtweconcludethatr>6dividesDimAebyLemma11.9TVo:calculateDimAewreusethefactthatAisafreerightB-moSdule.Let'fm1;:::ʜ;msn=DimB2B.sgMimicrkingStheproSofofTheorem'6onecaneasilyshorwthat:'Prop`osition6}LffetHAbea nite-dimensionalHopfalgebraovera eldkof'charffacteristicnzero.SupposethattisaHopfalgebraautomorphismofAof' nite orffderr`"andthattsatis esthestrongvanishingtracecondition.YLetB'bffe thesub-HopfalgebraofAgeneratedbythecoradicalofA.XThenr^/divides'(DimgADimeBxBN>)=DimB.fi29Let,AsbSetheuniquemaximalsemisimplesub-HopfalgebraofA.Since'semisimpleZHopfalgebrasincrharacteristiczeroarealsocosemisimple[5,/6The-'oremP3.3],wreconcludethatAs{A0;B,wherePBVisthesub-Hopfalgebra'of'AgeneratedbrythecoradicalofA.t\ThusAs =BD-ifandonlyifA0 i+isa'sub-Hopf7algebraofA.qInthiscasexA u=(DimgA=DimB)xB uis7anon-zero'lefṱinrtegralofBgby[7,ҬPropSosition9]sinces22V=UR1A;֫thusDimBxA 36=UR1and'theconclusionsofTheorem6andPropSosition6arethesame.9There(areinrterestingconnectionsbSetweenthestrongvXanishingtracecon-'dition6&andtheclassi cationproblemfor nite-dimensionalHopfalgebrasA'ofWdimensionp22,rwherepisanoSddprime,orverWanalgebraicallyclosed eld'ofHcrharacteristiczero.RLetsbSetheantipSodeHofA.RBytheresultsof[1]the'HopfalgebraAisthegroupalgebraofZp2 orZp]Zp Gorverkg,;:orisaTVaft29vt/'algebraTn;k V(qn9),{ors22!hasorder2p.@Ifs22satis esthestrongvXanishingtrace'conditionthenthelastpSossibilitryisruledout.9Let0LAbSeanryHopfalgebraoverkg. Thesubspaceof xedpSointsofs22Pis'aPJsubalgebraofBPofA.iIfAis nite-dimensionalandpSoinrted,iandkgisan'algebraicallyPclosed eldofcrharacteristic0,%;thenDimBVdividesDimAbry'partb)ofTheorem6.8GenerallyAisnotafree(left)B-moSdule.'Example1kQLffetCAbea nite-dimensionalunimodularHopfalgebrawithan-'tipffode[soverkg,eandsuppffosethats22`06=,1A.߭ThenAisnotafreeleftmodule'over35thesubffalgebra35B;of xeffdpointsofs22.9TVoseethis,letbSeanon-zerotrwo-sidedintegralforA.Thens()UR=.'Therefore2B.Norws22g6=1A impliesthatBB6=A.SuppSosefm1;:::ʜ;mrbg'isabasisforAasaleftB-moSdule.JThenrf>?1and(m{K)6=0forsome'1UR{rS.WVemaryassumethat(m1)UR6=0.Since(m2)m1 (m1)m2V=UR0'determines-anon-trivialdepSendencyrelationamongm1;:::ʜ;mrb,+Oitfollorws'thatAcannotbSeafreeleftB-module.9There1areanabundanceofexampleswhicrhsatisfythehypSothesisofEx-'ampleT1.oLetAbSeanry nite-dimensionalHopfalgebraoverkfqwhoseantipSode'ssatis ess22 k6=1A.vThenthequanrtumdoubleDS(A)ofAsatis esthehy-'pSothesisofExample1.9LetAbSe nite-dimensionalHopfalgebraorverany eldkg.KRRecallthat'DS(A)UR=A2jcopDtGGcmr17;!",ff cmsy109XQ ff cmr128Tq lasy107R6 cmss125N cmbx123@ cmti122!", cmsy101g cmmi120XQ cmr12,"V 3 cmbx10)!", 3 cmsy10( b> 3 cmmi10'K`y 3 cmr10K cmsy82cmmi8|{Ycmr8q% cmsy6;cmmi6Aacmr6 !", cmsy10 O!cmsy7K`y cmr10u cmex10