; TeX output 2001.11.06:1338?q!j  N cmbx12FINITEٚQUANTUMGROUPSOVERABELIANGROUPSOFPRIMEÁ4EXPONENTKK`y cmr10NICOL@xASUUANDRUSKIEWITSCHANDHANS-J@xURGENSCHNEIDER4(`XQ cmr121.- cmcsc10IntroductionQSincevthediscorveryvofquanrtumgroups(Drinfeld,iJimbSo)andtheir nitedimensionalvXariationsQ(Lusztig,-NManin),theseobjectswrerestudiedfromdi erentpSointsofviewandhadmanyapplications.QTheNpresenrtpapSerispartofaserieswhereweintendtoshowthatimpSortantclassesofHopfalgebrasQarequanrtumgroupsandthereforebSelongtoLietheoryV.QWVe0willassumethattheground- eld) msbm10|isalgebraicallyclosed eldofcrharacteristic0.FxOneofourQmainbresultsistheexplicitconstructionofageneralfamilyofpSoinrtedHopfalgebrasfromDynkinQdiagrams(Theorem5.17).KAlltheFVrobSenius-LusztigkrernelsandtheirparabolicsubalgebrasbelongQtothisfamilyV,8butinadditionwregetmanynewexamples.ZWVeshowthatany nitedimensionalQpSoinrtedMHopfalgebrawithgroupofprimeexponenrt(greaterthan17)isindeedinthisfamily;seeourQmainTheorembSelorw.qAnimportanrtstepintheprooffollorwsfromanothermainresult(TheoremQ7.6),wherewreshowthatalargefamilyof nitedimensionalpSointedHopfalgebrasisgeneratedbyQgroup-likreandskew-primitiveelements,givingadditionalsuppSorttoaconjecturein[AS25].QQIfwg cmmi12AisaHopfalgebra,thenwredenotebyG(A)thegroupofgroup-likeelementsofA.zRecallthatQAf2is,@ cmti12pffointedf2if|G(A)isthelargestcosemisimplesubScoalgebraofA,orequivXalenrtlyifanyirreducibleQA-comoSduleisone-dimensional.QLetbSea niteabeliangroupandwn#u cmex10bqthegroupofitscrharacters.eWVedenotetheunitelementinwbQbry".QTVostateourmainresult,bwrehavetointroSducesomenotation.Alinking6datum !", cmsy10D=of niteCartanQtypffe35forisacollectionD?consistingofTh5aCartanmatrixof nitetrypSe(a2cmmi8ijJ)|{Ycmr81!K cmsy8i;jv#>ofsize6>6forsomeUR1,[K "w],5elemenrtsg1;:::ʜ;g2UR,5crharacters1;:::ʜ;2wbURmsatisfyingэidڹ(gi)c6=UR1;forall;,1iS;Q(1.1)36qjf (gidڹ)i(gj)c=URidڹ(gi) a8:;cmmi6ij ;forall;,1i;j%S;Q(1.2)\5andmafamily(ijJ)1i2andanaturalnrumbSers.,WVeconsider niteabeliangroupsoftheformQƹ(s)UR:=(Z=(p))2sn<.QLet9D/bSealinkingdatumof niteCartantrypeforƹ(s)withCartanmatrix(aijJ)1i;jv"andlinkingQelemenrts(ijJ)1ithegeneratorsyhe;1Ahs.Let+>Yh;1hs+>bSeaZ=(p)-basisof,;candwriteg1z=AYnptqAacmr61/č1 XXY2ptsRAs 0,Qwhere}rt1;;ts뮹arenaturalnrumbSers.yThen}rintherelationsaborve}rreplaceg뫹brytheformalexpressionQynn9tq1/č1 **y2n9tsRAs:QIn(1.5),adG+aiOistheadjoinrtactionofaidڹ,thatisforallxUR2u(DUV),ۍa}(ad\aidڹ)xUR=ai -L(1)AxSb(ai ۟L(2)W)=aixgixgn91 i ʵai:6[QInthiswraythelefthandsideof(1.5) Ҕismeanrtasawell-de nedexpressioninthegenerators.QIn7s(1.7)^,J2+ Sis7sthesetofpSositivrerootsoftherootsystemassociatedtotheCartanmatrix(aijJ);Qthe"roSotvrectors"a ?arede nedinSubsection4.1belorw.QOurmaintheoremisQTheorem1.8.y(a).+LffetvDobealinkingdatumof niteCartantypeforƹ(s)withCartanmatrixʍQ(aijJ),and=assumethatpUR>3=if(aij)hasacffonnected=componentoftypeG2.NThenu(DUV)hasauniqueQHopf35algebrffastructuredeterminedbyYQ(1.9)'yhC=URyh yhe;ai,=ai 1+gi aid;for35all=1URhs;1iS:QThe35Hopfalgebrffau(DUV)ispointed,G(u(DUV))UR'ƹ(s)35anddim{u(D)UR=p2s+j-:+nj.Q(b).NLffetpUR>17.LffetAbeapointed nite-dimensionalHopfalgebrasuchthatG(A)UR'ƹ(s).NThenʍQtherffe35existalinkingdatumDof niteCartantypeforƹ(s)suchthatAUR'u(DUV).LQRffemarks̵1.10.^(i).{In}xSection5wrede nethenotionofa"linkingdatum"forageneral niteabSelianQgroup0/. uInthesituationofthemaintheoremitisalwrays0/pSossibletoreducetothecaseoflinkingQdataewithallenrtriesij }equalto0or1. ThusitfollowsfromTheorem1.8thatthereareonly nitelyQmanry5isomorphismclassesof nitedimensionalHopfalgebraswith xedcoradical|ƹ(s).FVormoreQgeneral niteabSeliangroups,thisisnolongertrue[AS15],[BDG~],[G 8(].?q!5FINITE!QUANTUMGR9OUPSOVERABELIANGROUPSOFPRIMEEXPONENTF 3j Q(ii).ThedimensionsoftheHopfalgebrasinTheorem1.8arevreryspSecialnumbSers.Thisphenom-ʍQenonisshorwningeneralforarbitrary nitegroupsinTheorem7.9.ˍQ(iii).1Let(aijJ)1i;jv#(bSea niteCartanmatrix.Theproblemofactually ndingallthecollections6Qgi>2=ƹ(s),Ki29N[qǍƹ(s)o,K1iS,sucrh8)that(1.1) ^and(1.2)holdhasbSeendiscussedin[AS25].!bItcanQbSestatedastheproblemof ndingallthesolutionsofasystemofalgebraicequationsorverZ=(p)QanditisinprinciplesolvXable.8Notethatinparticular-ˍɼUR2sō33p133[zpP ΍p2ֶ;Qsee[AS25,Prop.88.3].Q(iv).3[Thequestionof ndingallthepSossiblelinkingelemenrtsattachedtoa xedcollectiongidڹ,hi,Q1iS,L(aijJ)1i;jvSݹ,is9Aalsoofcomrbinatorialnature,LseeSection5,andalso[Di <].$OncethesetrwoQproblemsNCaresolvrede ectivelyV,mtheisomorphismclassesoftheHopfalgebrasu(DUV)canbSedeterminedQusing[AS25,Prop.86.3],[AS3,Lemma1.2].%LQ(v). ;As1qaconsequenceofTheorem1.8oneobtainsthecompleteclassi cationofall nitedimen-QsionalpSoinrtedHopfalgebraswithgroupofgroup-likesƹ(1)lk=Z=(p),p6=5;7.$ItisthelistgivenQin[AS25,XTheorem1.3]plustheFVrobSenius-Lusztigkrernelsasdescribedin[AS15].'Indeed,XreplacinginQthe_proSofofTheorem1.8[AS25,}ACor.1.2]bry[AS2,}ATh.1.3]wregettheclassi cationforallprimesQp;6=5r)or7, inviewofTheorem6.8and[AS35,Lemma4.2].dTheonlycasesnotcorveredr)arep;=5,QtrypSeB2r andpUR=7,ktypSeG2.&ThisresultwasindepSendentlyobtainedbyMusson[Msf]usingdi erentQmethoSdsstartingfromourpreviousarticle[AS25].Q(vi).Uptonorw,thedeterminationofall nitedimensionalpSointedHopfalgebrasAwithG(A)UR'Q,gforFa xedgroup,wrasknownonlyforUR=Z=(2)F[N].FOtherclassi cationresultsofpSointedHopfQalgebras-areknorwnforsome xeddimensiond:}di=p22 M1is-easyandfollowsfrom[N],[NZ];odi=p23Qwras.donein[AS15],andbydi erentmethoSdsin[CDu],[SvO*];(d=p242in.[AS35](anddoesnotseemtoQbSeepossibleviatheothermethods);d'=16in[CDRP],d=32in[GS~vn1<];resultsonthecasewhenQhasexpSonenrt2canbefoundin[AD"].Q(vii).ڵThe&classi cationofallcfforadically/graded&pSoinrtedHopfalgebrasofdimensionp25*wasobtainedQin)[GS~vn2<].Itisnotdiculttodeducetheclassi cationofallpSoinrtedHopfalgebrasofdimensionp25QusingTheorem1.8andresultsin[AS35].Q(viii). TheHopfalgebrasu(DUV)canbSede nedforanryCartandatumof nitetypSeDofanQarbitraryP niteabSeliangroup.Prart(a)ofTheorem1.8isaspecialcaseofthegeneralTheoremQ5.17.0*FVorЅsuitablecrhoicesofDUV,տtheFrobSenius-Lusztigkrernelsandtheirparabolicsubalgebrasareof%K?q!Q4YV"NICOL'ҟASTANDR9USKIEWITSCHANDHANS-J'ҟURGENSCHNEIDERj Qtheformu(DUV).0dSeeexample5.12.OtherwiseTheorem5.17prorvidesmanynewexamplesof niteʍQdimensionalHopfalgebrasarisingfromexoticlinkingdata.wQ(ix).߇TheޜresultsofthispapSerhearvilydependonourpaper[AS25]andonpreviouswrorkonquantumQgroups[L1 7o],[L2],[Ro1e],[Ro2],[dCP],[MuKU].ŕQConventions. l0bSethegradedcoalgebraQassoSciatedtothecoradical ltrationofA.ThengrAisagradedHopfalgebra[Mo9]andboththeQinclusion.|:A0 <,!grAandthegradedprojectionɹ:grA!A0 areHopfalgebramaps. rLetwuQRt:=w*grYA2com=w*fx2grA:(id ʠ n9)(x)=x 1g;itisagradedbraidedHopfalgebrainߍ.:(s) 뀍.:(s)YD_QwithQthegradinginheritedfromgrA:4Rn=URn0RJ(n),R(n):=RN\5grA(n).%NoticeQthatgrAcanbSeQreconstructedfromRasabiproSduct:V9grɡ(AUR'RJ#|ƹ(s):QThebraidedHopfalgebraRiscalledthediagrffamofA.8OnehasxRJ(0)UR=|1;Q(2.1)36xRJ(1)UR=P(R);Q(2.2)4?q!5FINITE!QUANTUMGR9OUPSOVERABELIANGROUPSOFPRIMEEXPONENTF 5j QandwreknowfromCorollary7.7bSelowthatcuxRisgeneratedasanalgebrabryR(1)5A:Q(2.3)QLetcV#:=?RJ(1);AitisaYVetter-DrinfeldsubmoSduleofR.TSinceRsatis es(2.1)o,R(2.2)!and(2.3)!cwreQknorwthatRsG'YB(Vp)isaNicholsalgebra,seeSubsection3.2.NNowthereexistsabasisx1;:::ʜ;xX1QofVsandg1;:::ʜ;g m2Rƹ(s),`1;:::ʜ; m29m[qǍRƹ(s)!sucrhthatxi2RV2p8:iRAg8:i ,`1iS.SinceAis niteQdimensional,idڹ(gi)6=1Oforalli[AS15,Lemma3.1]andthereisa niteCartanmatrix(aijJ)1i;jvQsucrhthat(1.2)\holds[AS25,Cor.81.2].ꍑQTVoIgivreanexplicitdescriptionofB(Vp),`weintroSducerffootvectorsIinB(Vp)generalizingtherootQvrectorsۨde nedin[L1 7o].ދWVenotethatLusztig'sroSotvectorscanbSedescribeduptoanon-zeroscalarasQaniteratedbraidedcommrutatorofsimpleroSotvectors.@WVethende netheroSotvectorsinthegeneralQcasebryexactlythesameiteratedbraidedcommutatorwithrespSecttoourmoregeneralbraiding.QAs1Yoneofourmainresults,CwreobtainapresentationbygeneratorsandrelationsandaPBW1GbasisQfor B(Vp)fromthecorrespSondingTheoremforFVrobenius-Lusztigkrernels,usingDrinfeld'stwistingQessenrtially݇inthesamewayasin[AS25].4SeeTheorem4.5.WVecanthendeducepart(a)ofTheoremQ1.8.cFVor)connectedDynkindiagramsitisaconsequenceofTheorem4.5;4inthenon-connectedcaseQwreapplytheideaoftwistingthealgebrastructurebya2-coScyclewhichisgivenbyaHopfalgebraQmap[DTu].8SeeTheorem5.17.QIt/follorwsfromTheorem4.5thatgrAcanbSepresentedasanalgebrabygeneratorsy1;:::ʜ;ysQ(homogeneousofdegree0)andx1;:::ʜ;x9(homogeneousofdegree1),andrelations8yOn9p hG!=UR1;ymyhC=yheym;forall;,1m;hs;Q(2.4)36yyhexjG!=URjf (yhe)xjyh;forall;,1URhs;1j%S;Q(2.5)OpLԤ(ad\xidڹ) 1a8:ijwxjG!=UR0;forall;,1i6=j%S;Q(2.6)Nx pڍ G!=UR0;forall;, h2 +x;Q(2.7)kQandwheretheHopfalgebrastructureisdeterminedbryQ(2.8)(9JyhC=URyh yhe;xi,=xi 1+gi xid;forall;,1URhs;1iS:QBy_[AS15,Lemma5.4],wrecanchoSoseaiу2lPg8:i,r;1Tm(A)28:i suchthattheclassofai]9ingr_A(1)coincidesQwithÙxidڹ.+WVealsokreepthenotationyj)forthegeneratorsofG(A).Itisclearthatrelations(1.3)>andQ(1.4).hold.yNorwrelations(1.5)!>and(1.6),resp.(1.7)B,holdbSecauseofTheorem6.8,resp.LemmaQ6.9.QTheTheoremnorwfollowsfromTheorem6.10.}S& msam10 Q2.2.TheGgeneralcase.TherevaresevreralobstructionstoextendTheorem1.8togeneral niteQabSelian^Mgroups.First,{6itisopenwhetherthediagramofa nitedimensionalpoinrtedHopfalgebraQisgeneratedindegreeone,,ji.e.=whenitisaNicrholsalgebra;lsecond,thereare nitedimensionalQNicrholsalgebraswhicharenotofCartantypSe[N].QFVorliftingsofgr$"AwhenR=isaNicrholsalgebraofCartantypSe,thequantumSerrerelationsofQconnected[vrerticesingeneralstillholdasweshowinTheorem6.8bSelow;howeverthepSowersoftheQroSotkvrectorsarenotnecessarily0.WVeshouldhavea2N RA /=URu J2|G(A);(thekdeterminationofu a~whenQ misYanon-simpleroSotwrasdonein[AS35]fortypSeA2,u:in[BDR]fortypSeB2,u:andin[AS45]fortypSeQAn foranryn(uptosomeexceptionalcasesconcerningtheordersoftheroSotsofunity).I?q!Q6YV"NICOL'ҟASTANDR9USKIEWITSCHANDHANS-J'ҟURGENSCHNEIDERj 3.̘BraidedHopfalgebrasQ3.1.Bipro`ducts.LetRbSeabraidedHopfalgebrain2HbH YDܹ;thismeansthatRisanalgebraandaQcoalgebra0in2HbH tYD"BiandthatthecomrultiplicationR H:URRn!R} d@Rzis0analgebramapwheninR d@RQthevmrultiplicationtwisteffdbythebraiding qcvisconsidered;^]inadditionRadmitsananrtipSode. JTVoQarvoidconfusionswreusethefollowingvXariantofSweedler'snotationforthecomultiplicationofRJ:QR(rS)UR=r2(1) 9 /r2(2) . Let"A=RJ#Hxbe"thebiproductorbosonizationofRl[Mj[W],=[Ra]. RecallthattheQmrultiplicationandcomultiplicationofAaregivenbyDՍ!(rS#h)(s#fG)UR=r(h(1) \|:s)#h(2)f; (r#h)UR=r (1) #(r (2))(1)h(1)$ (r (2))(0) \|#h(2):QThetmapsr+:A!Hʹandt:HH!A,n9(rS#h)=(r)h,(h)=1#h,aretHopfalgebraQhomomorphisms;twre]haveR-=fa2A:(id ʠ n9)(a)=aU 1g. yConrverselyV,letA,HK1bSeQHopfP algebrasprorvidedwithHopfalgebrahomomorphisms#ݹ:A!H=bandP :H!A. i ThenQRn=URfa2A:(id ʠ n9)(a)=av 1ghUisabraidedHopfalgebrain2HbH YD^. oTheaction:ofHUonRistheQrestrictionݤoftheadjoinrtaction(compSosedwith)andthecoactionis(K idZ);RisasubalgebraQofAandthecomrultiplicationisR(rS)UR=r(1) \|n9Sb(r(2))- r(3) \|.TheseconstructionsareinversetoeachQother.8WVeshallmostlyomitinwhatfollorws.QLet#UR:A!RbSethemapgivrenby#(a)UR=a(1) \|n9Sb(a(2)).8ThenQ(3.1)U#(ab)UR=a(1) \|#(b)n9Sb(a(2));QforEalla;b&2AEand#(h)&="(h)Eforallh&2HV;therefore,forEalla&2A,h2HV,wreEhave#(ah)&=Q#(a)"(h)andQ(3.2)'#(ha)UR=h:#(a)=#(h(1) \|an9Sb(h(2))): QNoticealsothat#inducesacoalgebraisomorphismA=XAHV2+ '5RJ.^Infact,ϳtheisomorphismA!QRJ#HcanbSeexpressedexplicitlyas'aUR7!#(a(1) \|)#n9(a(2));aUR2A:QIf%AisaHopfalgebra,MKthewrell-known%adjointrepresentationadkofAonitselfisgivenbyadwx(yn9)UR=Qx(1) \|yn9Sb(x(2)).IfVR isabraidedHopfalgebrain2HbH YD thenthereisalsoabraidedadjoinrtrepresentationQadc@ofRonitselfgivrenbyylad:cihx(yn9)UR=( Sb)(id ʠ c)( iduJ)(x y);Qwhereisthemrultiplicationandc2End(R, RJ)isthebraiding.ϲNotethatifx2P(R)thentheQbraidedadjoinrtrepresentationofxisjustQ(3.3)$adtcx(yn9)UR=(id ʠc)(x y)UR=:[x;y]c.y:QThejelemenrt[x;yn9]cde nedbythesecondequalityforanyxandyn9, wregardlessofwhetherxisprimitive,QwillbSecalledabraidedcommrutator.QWhenAUR=RJ#HV,thenforallb;d2RJ,Q(3.4)ciad(b#1)Ԡ(d#1)UR=(ad\cb(d))#1:^?q!5FINITE!QUANTUMGR9OUPSOVERABELIANGROUPSOFPRIMEEXPONENTF 7j Q3.2.Nicholsalgebras.Let"HxbSeaHopfalgebraandletR޹={n2*ppmsbm8NRJ(n)beagrffaded"braidedHopfQalgebrain2HbH YD2.WVesarythatRعisaNicholsalgebrffaif2.1,2.2and2.3hold,cf.[N],[AS25],[ArG].QA2NicrholskalgebraR൹isuniquelydeterminedbytheYVetter-DrinfeldmoSduleP(RJ);5givenaYVetter-QDrinfeldmoSduleVp, thereexistsaunique(uptoisomorphism)NicrholsalgebraRڹwithP(RJ)T{'V.QItwillbSedenotedB(Vp). #Infact,thekrernelofthecanonicalmap$r:0Tƹ(V)!B(V)canbSeQdescribSediinsevreraldi erentways.FVorinstance,KerlP$=-n0̹Ker%gVS/'un99whereS*nofWI ƹintermsofsimplere ections.TThenweobtainaQreduced*rdecompSositionofthelongestelemenrt!0 5=usiq1 :::ZsiX.P8ofW'fromtheexpressionof!0 vasQproSductofthe!0;I kŹ'sinsome xedorderofthecomponenrts,saytheorderarisingfromtheorderofQthevrertices.8Therefore j\:=URsiq1 :::Zsi8:jY"q% cmsy61u( i8:jO)isanumerationof2+x.QWVe xa nitedimensionalYetter-DrinfeldmoSduleV#orverwithabasisx1;:::ʜ;x:withxi,2URV2p8:iRAg8:i ,Q1URiS.QMajorsexamplesofmoSdulesofCartantrypearetheFVrobenius-Lusztigkrernels.LetN|>>1beanQoSddnaturalnrumberandletq2yj|beaprimitivreN@-throotof1,)notdivisiblebry3incase(aijJ)hasJQaf_compSonenrtoftypSeG2.LetG'=Z=(N@)2v=he1ihei;:letf_j2f};b'G㛹bSetheuniquecrharacterD Qsucrhthathn9(jӹ);e(i)i}=q2d8:i,ra8:ijo.zLetVbSeaYVetter-DrinfeldmoduleorverGwithabasisX1;:::ʜ;XQsucrhthath1pXi,2URV e8:iڍ8:i_;forall;,1iS:QWVe!KdenotebrycthebraidingofV.Lusztigde nedroSotvectorsX J2URB(V),I h22+ =ù[L2 7o].One!KcanseeQfromh[L3 7o]that,uptoanon-zeroscalar,eacrhroSotvectorcanbSewrittenasaniteratedbraidedcom-QmrutatorinsomesequenceX`q1#;:::ʜ;X`a xnhq1* q1 xnhq2* q2 :::hX.P X.P ;for35all=0URhj\NIy1;33if j2I;1j%PS:^QPrffoof.:(a)$Letus rstassumethatthebraidingissymmetric,s*thatisidڹ(gjf )k=j(gidڹ)$foralli;jӹ.ʍQByӹ[AS25,OLemma4.2]wrecanassumemoreoverthattheCartanmatrix(aijJ)isconnected.1;FVromourQassumptionsͳontheordersoftheidڹ(gjf )wrethenconcludethatthebraidinghastheformj(gidڹ)UR=qn92d8:i,ra8:ijQforalli;jKwhereq isaroSotofunitryoforderN6=URidڹ(gi).See[AS25,Lemma4.3].HencetheTheoremQfollorwsdirectlyfromTheorem4.2andRemark3.6.Q(b)ZInthecaseofanarbitrarybraidingwreknowfromLemma4.1of[AS25]thatthereexistsa niteQabSeliangroupGsatisfying:э55TheܟbraidingcofVycanbSerealizedfromaYVetter-DrinfeldmodulestructureorverܟGthatwre5conrtinuedenotingbryVp,cf.8Remark3.6.55TheresexistsacoScycle!0:fMb4GfbjG!4|2 withscorrespondingFֽ24|Gj |Gssucrhthatthe5braidingg/ofVF v~issymmetric.uLet :)HB(Vp)!B(VFO)g/bSetheisomorphismharvingthesame5meaningasin(3.9).55ThebraidingofVF isgivreninthebasis n9(xidڹ)! (xjf )bryamatrix(b2FRAijJ)suchthatb2FRAii d=URidڹ(gi)5andtheorderof(b2FRAijJ)isagainoSddforalliandjӹ. Π?q!Q12YV"NICOL'ҟASTANDR9USKIEWITSCHANDHANS-J'ҟURGENSCHNEIDERj QIf>$:URTƹ(Vp)!B(V),aC$F d:Tƹ(VFO)!B(VF)>denotethecanonicalmaps,aCthenwrehaveacommutativeʍQdiagram&CTƹ(Vp)2$pط߷UO!rB(Vp)Kl7H 3,?38?y3, W?38 W? Wy X xFhTƹ(VFO)s $X.Fط߷UO!B(VFO):*MQClearlyV, n9(Ker$S)=Ker$FO;uif1(rjf )jv2J*isasetofgeneratorsoftheidealKer$withrj 2Tƹ(Vp)z8:j ܞh8:jQthenɎbryPropSosition3.18( n9(rjf ))jv2J?FisasetofgeneratorsoftheidealKer$FO.ՓBythesymmetricQcase?(a),UwreknowthegeneratorsofKer"F'FO.8LetusdenoteXiJ:=" n9(xidڹ).ThenbryLemma3.15andQ(3.11)X,Awrehave n7(ad\~c(xidڹ)21a8:ijw(xjf ))a(='uijJadgc(Xidڹ)21a8:ijw(Xjf )and n7G9x2NX.IRA  G$B=u X2NX.IRA p,A ;$2+8OI whereQuijJ;u 'arenon-zeroscalars.Thisimpliesthe rstclaimoftheTheorem.ThesecondfollorwsinaQsimilarwrayV. mGQLetުIb!VBMg(Vp)bSethebraidedHopfalgebrain2b YDegeneratedbryx1;:::ʜ;xegwithrelations(4.6),3where*Qthexidڹ'sareprimitivre.(LetK,`(Vp)bSethesubalgebraofު3b!VBQ(V)generatedbryx2NX.IRA  , f2׹+8OIx,IxZ2Xӹ;Litis ׍QaYVetter-DrinfeldsubmoSduleofުb!VB9(Vp).0QTheorem4.8.K,`(Vp)35isabrffaided35Hopfsubffalgebra35in2b )YD #ofު+b!VBƹ(V).QPrffoof.:(a).PXAs{intheproSofofTheorem4.5wre rstassumethatthebraidingissymmetric.Ifib6=jӹ,Qthenjf (gidڹ)i(gj)t=1andhencethecorrespSondingSerrerelation(4.6) !:sarysthatxidxjڼ=txjxi.p,Thrus,Qwre caneasilyreducetotheconnectedcase.Insuchcase,jf (gidڹ)=qn92d8:i,ra8:ijdas bSeforeandtheTheoremQisshorwnin[dCP].Q(b).fInOSthegeneralcase,h}wrechangethegroupasintheproSofofTheorem4.5.fTheisomorphismQ 鲹:{yTƹ(Vp)!T(VFO)-respSectstheSerrerelationsuptonon-zeroscalarsbryLemma3.15. Also,~itQmapssubScoalgebrasstableundertheactionofthegrouptosubcoalgebrasbryLemma3.10(iii).]]WVeQconcludefrom(a)thatK,`(Vp)isasubScoalgebraofުb!VB9(V).Js>B̹5.MFPLinkingda32tumandglueingofconnectedcomponentsQ5.1.Linkingdatum.In,WthisSection,wre xa niteabSeliangroup,a niteCartanmatrixQ(aijJ)1i;jv$洹andg1;:::ʜ;g 2'U,1;:::ʜ; 2w}b'Usucrhthat(1.1)#Land(1.2)hold. 1nWVepreservretheQconrventionsandhrypSothesesfromSection4.QDe nition5.1.WVesarythattwoverticesiandj{arffe35linkable(orthatiislinkabletojӹ)ifVxiUR6j;Q(5.2)Lʍxgidgj\6=UR1andQ(5.3)xidj\=UR1:Q(5.4)QIfiislinkXabletojӹ,thenidڹ(gjf )j(gi)UR=1bryo(5.2);{;itfollowsthenfrom(5.4)\thatQ(5.5)Bwjf (gj)UR=idڹ(gi) 1 \|: ՘?q!5FINITE!QUANTUMGR9OUPSOVERABELIANGROUPSOFPRIMEEXPONENTB-"13j QLemma5.6.Assumejthatiandkg,rffesp. 'jand`,arffelinkable. 'Thenaij =ak6`x,ajvi=a`kx. 'InʍQpffarticular,35avertexicannotbelinkabletotwodi erentverticesjandh.QPrffoof.:If>ai` 46=VR0thenaij 6=ajvi=0>(otherwisej%`)andak6` ʹ=a`k=0>(otherwiseikg).:Ifajvk _6=0Qthen+aij 6=URajvi=0+(otherwiseikg)andak6` ʹ=a`k=0+(otherwisej%`).Assumethatai` 3=0=ajvk .QThen{.idڹ(gi) a8:ijn=URi(gjf )j(gidڹ)=1Ok \|(gj)1O`(gidڹ)UR=jf (gk#)i(g`)=1O` \|(gk#)1Ok(g`)UR=k#(gk) ai?k`%=URidڹ(gi) ai?k` [:QThen}NiԹdividesaij ̬ak6` randanalogouslyV,Nkdividesaij̬ak6`x.Sothataij 6=URak6` rbrytheassumptionsʍQonLtheorderofNipandNk#;}brysymmetryV,eajvi F=a`kx.^AssumethatavertexiislinkXabletojiandh.QThen2UR=aii =ajvh K,soj%=h.aQAlinking35datumof niteCartantypffeforisacollectionS bD=URDUT(;(aijJ)1i;jvS;(gidڹ)1i_;(jf )1jv;(ij)1i;+sletMh {˹denotetheorderofʍQYhe,A1 hs.ZLet/D `=DUT(;(aijJ)1i;jvS;(gidڹ)1i_;(jf )1jv;(ij)1i,QwhereQthefLorderofZi&istheleastcommonmrultipleoford_"giandord_"idڹ,51'iVe* .LetfLjVbSetheuniqueQcrharacterHofsuchthatjf (Zidڹ)<=j(gidڹ),_\1iV9e* ,1jV9e* .PThisHiswrellde nedbSecauseord@giQdividesord~ZiOforalli.55BZ:=u(D1),withD1Z=DUT  ;(aijJ)rer3n ',withgenera-ԍ5torsbrer3n+1 ,.T..,b9(insteadoftheaidڹ's)andy1;:::ʜ;ysn<;c55U:=ǚu(D2),>with-D2=DUT  ;(aijJ)n1i;jv㎍rerS;(Zidڹ)n1i㎍rer_;(jf )n1jv㎍rer;(ij)n1i6sucrhthatjvh M6=UR0.Thenidڹ(gjf ghe)UR=jhe(gidڹ) 1ι=1;Qbryo(5.2) &#and(5.4).8Sothatrelations(1.6)\holdand(a)isproven.Q(b).8ThisisequivXalenrtto:thereexistsanalgebramapT:URB!M2(kg)sucrhthat Q(5.23)tTƹ(yk#)UR=z T1(0ɍ T0P idڹ(yk):4&z!E&;Tƹ(bjf )=z & T0VMe*Pijɍ T0 0*ڟz!6Z;$?q!5FINITE!QUANTUMGR9OUPSOVERABELIANGROUPSOFPRIMEEXPONENTB-"17j Q1URkos,Ve* XU+{g1j%S.ݛThenTzisoftheformTƹ(a)UR=z T(a)+Tmidڹ(a)ɍ0*n idڹ(a)C z!Pandi=isthedesiredderivXation. QSo, wreޤneedtoshowthattherelations(5.8), (1.4), (1.5), (1.6), (5.9)holdޤforthematricesin(5.23)!.ʍQThiskisevidenrtfor(5.8).FVor(1.4) itamountstoVe*ijFidڹ(yk#)21=VIe*1ijVjf (yk),whichkfollowsfrom(5.4)QwhenVe*rijb6=UR0.FVorr(1.5) andr(5.9)therargumenrtisclear.FinallyV,thelefthandsideofH(1.6)Pforj%<URh,Qis.A0,?(whereastherighrt-handsidealsovXanishessincejvh K idڹ(gjf ) i(ghe)b=jvhjf (gidڹ)21 \|he(gi)21$޹=bjvh z(ad\idڹ)2a8:ij7jHisahomogeneouspSolynomialini,jHofpSositivredegree.FinallyV,relations(5.9)ʍQfollorwfromthenextLemma.a+_ǍQLemmaK85.24.LffetBbea nitedimensionalpointedHopfalgebrageneratedasanalgebrabygroup-Qlikeelementsandafamilybjf ,"j2J7v,of(hj;1)-primitives,forsomehj X2G(B). Lffetx;bUbetheQalgebrffaopresentedbygeneratorsu1;:::ʜ;urer #Uandz1;:::ʜ;zrer #UwithexactlythesamerffelationsasforUQexcffeptfor I(5.9)3U;{itisaHopfalgebraviaN(1.9)Z.@LetN6=URNJ.@AssumethereexistsaHopfalgebramapQUR:xYbU^C!(B2[ )2copsAsuch35that i,:=UR(zidڹ)andi:=UR(uidڹ)satisfy[Q(5.25)k idڹ(bjf )UR=0;i(gn9)=0;j%2J7v;gË2G(B);Qfor35all1URiVOe* ".fiThen35(u2NRA D)=0forall h2+8OJx.QPrffoof.:There17existsaHopfalgebraprojection$ :xblUNw!l|sucrhthat$S(uidڹ)=0and$S(zidڹ)=ziforʍQalli.8LetKbSethesubalgebraofx$bUgeneratedbryu2NRA D, h2UR+8OJx,andz2NRAi',1iVOe* ".8WVeclaimthatQ(5.26)(u)UR=($S(u))QforEalluUR2K,`.ClearlyV, thisEimpliestheLemma.ByTheorem4.8, wreknowthatKisaHopfsubalgebraQofxabU .7WVeharvetoprovethat(u)(b)UR=($S(u))(b)forbamonomialinthegroup-likresofBandtheQbjf 's.8WVedothisbryinductiononthelengthofthemonomial.9?q!Q18YV"NICOL'ҟASTANDR9USKIEWITSCHANDHANS-J'ҟURGENSCHNEIDERj QWVe rstcrheckthecaseoflength1.8Herewreshowmoregenerallythatbg(u)(gn9)UR=($S(u))(g) and*n(u)(bjf )=($S(u))(bj)QforSallgË2URG(B),j%2JɹandSu2G(x|bU)oruoftheformuiq1 :::ZuitJz,withzE6group-likre,1URi1;:::ʜ;itVOe* ",ʍQand?t2.8NotethateacrhelementinKl^isalinearcombinationofsuchu'ssinceN'v撹2.8ThecaseQwhen'uisagroup-likreisclear.Letu=uiq1 :::ZuitJz,71with'zŹgroup-like,1i1;:::ʜ;itOSVIe* ڹ,71and't2.QThen(u)UR=iq1 :::ZitJ(z),andFn(u)(gn9)p=URiq1(gn9):::ʜitJ(g)(z)(g)=0;ԍCr (u)(bjf )p= gtX UR1rQ(see`[AS15,Lemma5.4]). Letai2Pg8:i,r;1Tm(A)28:i,1iS,sucrh`thatdz ۟KaiismappSedontoxi:foralli.9QThenwreknowfrom[AS25,Lemma5.4]thatforallgË2UR,2wbmwith6=":ElPOPgI{;1 qv(A) ~j6=UR0() LthereissomeQݤ1`:gË=g`;=`;Q(6.6)36Q^PgI{;1 qv(A) "~j=UR|(1gn9):Q(6.7)QRecallthatwrehave xed;(aijJ)1i;jvS;(gidڹ)1i_;(jf )1jv7asinthesituationofSection4.oQTheorem6.8.Lffet35Aanda1;:::ʜ;abeasabove.Q(a).kTherffePisalinkingdatumDUT(;(aijJ)1i;jvS;(gidڹ)1i_;(jf )1jv;(ij)1i 6=n3.IfI2qisoftypffeBnP,D]Cn >orF4,rffesp.G2,assume@furtherthatQNI$6=UR5,35rffesp.fiNI6=7.fiThen35thequantumSerrfferelations1(1.5)!SholdforalliUR6=j%2I.QPrffoof.:Itisknorwnthat(ad\aidڹ)21a8:ijwajɨ2cPig+SI{1aij}i'g8:j;1'(A)2+S1aij}i8:j#ǹ,(seeforinstance[AS25,AppSendix].RPrart9GQ(b)oftheTheoremthenfollorwsfromLemma6.1,(6.6)\and(6.7).QTVokprorvepart(a),wletusassumethatiUR6jӹ.ByLemma6.1,w(6.6)and(6.7)again,waidajqEgjf (gi)ajai,=QijJ(1{gidgjf ), forasomeij 62UR|.1WVecancrhoSoseij=UR0whengidgj\=1orelseifidj\6=".1Thatis, ijQisalinkingdatumfor(aijJ),g1;:::ʜ;g9and1;:::ʜ;;and(1.6)\hold.A'QLemma6.9.Lffet35Aanda1;:::ʜ;abeasabove.fiAssumefurtherthat55the35hypffothesisfromTheorem6.8part(b)holdsforallIF2URX.?q!5FINITE!QUANTUMGR9OUPSOVERABELIANGROUPSOFPRIMEEXPONENTB-"21j 55A35is nitedimensional.ʍ55gn9N8:imi/=UR1,351iS.捑QThen35therffelations1(5.9)!SholdinA.4QPrffoof.:Letus xIF?2TXӹ.?LetxbUٹbSethealgebrapresenrtedbygeneratorsHbai,b:iT2I,y1;:::ʜ;ys YandQrelationsq(5.8),(1.4),(1.5)!,Mandq(1.6);HitqisaHopfalgebravia(1.9).!LetN|$=;@NI A andletKbSetheQsubalgebraSofxbUEgeneratedbryY~ba2NRA ݘ, /2~+8OIx,andSg2n9NRAi},i2I.DBySTheorem4.8,wreknowthatKųisaQHopfvJsubalgebraofxzbU ).NotethatKisagradedHopfalgebrawithtrivialcoradical.BythecrhoiceQofMtheaidڹ'sinAandTheorem6.8,wreseethereisawell-de nedHopfalgebramapxbU>!URAsuchthatbQai67!URaidڹ,i2I.&The6imageofKunderthismapisa nitedimensionalpSoinrtedHopfalgebra;\ithasaQtrivialcoradicalbry[Mo9]andthereforeitistrivial.8ThisimpliestheLemma.ot!QTheorem6.10.Lffet35Abeasaboveandassumethat捍55the35hypffothesisfromTheorem6.8part(b)holdsforallIF2URX.55grKAUR'B(Vp)#|,35hencffeAis nitedimensional.55gn9N8:imi/=UR1,351iS.QThenǫtherffeexistsalinkingdatumD=h=DUT(;(aijJ)1i;jvS;(gidڹ)1i_;(jf )1jv;(ij)1i|moSduleorverwe>| kwith>|braiding(b2FRAijJ).|SinceB(Vp)andB(weeV2FT)harve>|thesamedimension,8QB(weeVp2FT)is nite-dimensional.8Then(aijJ)isof niteCartantrypSeby[AS25,Theorem3.1].50}QLemmak7.2.ڃLffetS=-n0S׹(n)bea nite-dimensionalgradedHopfalgebrain2b |YD#suchthatʍQS׹(0),=|1.kAssumethatS(1)isof niteCartantypffewithbasis(xidڹ)1i_,Ġbraiding(bijJ)1i;jv#handQCartan35matrix(aijJ)1i;jv#asin(3.7).fiForall1URlS,35letqlw=URblKl andNl=URorSd(ql!ȹ).QLffetn1i;joS,}Si6=j,andnassumethatNid;NjԊandorffd(bijJ)areodd,}SandNiZisnotdivisibleby3Qand35>UR7.Q(a)?Assumeijand?letIbffetheconnectedcomponentcontainingi;j. IfthetypeofIisQBnP;Cn ۅorF4,ӮassumecthatNi=isnotdivisibleby5. MIfthetypffeisG2,ӮassumethatNi=isnotdi-Qvisible35by5or7.fiThen(ad\cxidڹ)21a8:ijwxj\=UR0.Q(b)35AssumeiURjand35qidqj\=1ororSd(qidqjf )=ord(qidڹ).fiThen35xixjbijJxjf xi,=UR0.QPrffoof.:De nez10:=pxidڹ,z2:=(ad\cxidڹ)21a8:ijwxjf .hInbSothcaseswrehavetoshowz20=p0.hWVeassumethatQz2C isnot0.ULetgi,2UR;i2wb ,1i;j%S,withbij 6=jf (gidڹ)foralli;jӹ.UThenactionandcoactiononQz1;z2jarefgivrenbys2(z1)UR=gi z1,s2(z2)=gzn91a8:ij Migj z2jandfhz1V=URidڹ(h)z1,hz2V=UR(z1a8:ij Miwjf )(h)z2QforCallh2.CTheCelemenrtsz1,Yz2 arelinearlyindepSendentsincetheyarenon-zeroandofdi erentQdegree.TThebraiding(Bk6lZ)1k6;lK2$Τofthe2-dimensionalYVetter-DrinfeldmoSdulewithbasisz1;z2 isQgivrenbyDՍ#B118ł=URidڹ(gi)=qi;X#B12 =UR(z1a8:ij Miwjf )(gidڹ)=qib1 jvi \|;#B218ł=URidڹ(gzn91a8:ij Migjf )=qzn91a8:ij MibjviJ;X#B22 =UR(z1a8:ij Miwjf )(gzn91a8:ij Migj)=qzn91a8:ij Miqj:QThenB12 B21 =qzn92a8:ij Miѹ.2WVeclaimthat(Bk6lZ)isofCartantrypSeandsatis estherelativeprimenessQcondition,thatisthereareinrtegersA12 ;A21 갹suchthat@qzn92a8:ij Mi`]=URqnn9Aq12Qٍi ,and-(sA12 갹isrelativrelyprimeto{pNid;Q(7.3)@qzn92a8:ij Mi`]=UR(qzn91a8:ij Miqjf ) Aq21ڹ,and0A21 갹isrelativrelyprimetoordH(qzn91a8:ij Miqj):Q(7.4)QInbSothcases2aij 5isrelativrelyprimetoNidڹ,becauseofthehrypothesisonNidڹ.8Thisshorws(7.3).QWVenorwprove(7.4)Vincase(a).ThenNI$=URNi,=Njf ,Banditsucesto ndanintegerA21 ޹relativelyQprimetoNiOwithqzn92a8:ij Mi;#=UR(qzn91a8:ij Miqjf )2Aq212.QFirstassumethataij 6=>0. [Sinceajvi6=>0isrelativrelyprimetoNidڹ,>itisenoughtoconsiderQtheajviJ-thpSorwerofc!(7.4)-. Sinceqzn9a8:ij Mi&==qzn9a8:jYi Mj>brytheCartanconditionfor(bij),zwrehavetosolveQ(2+aijJ)ajvi 6UR((1aij)ajvi vԹ+aij)A21moSd+BNidڹ.$2Since(aij)isof niteCartantrypSe,thepossiblevXaluesQof(2E^aijJ)ajvi are-3,-4,-5,-6,-9(-4,-6resp.(X-5,-9onlyoSccurifthetrypeisBnP;Cn aaorF4yresp.(XG2);Qthe*pSossiblevXaluesof((1naijJ)ajvi !R+aij)*are-3,;-5,-7,(-5,resp.-7*onlyoSccurifthetrypeisBnP;CnQorF4ӹresp.+G2).Hence(2YDaijJ)ajvi and((1aijJ)ajvi (+aij)arerelativrelyprimetoNi'byassumption,Qandtheclaimfollorws.8(Notethata+b꨹isnever0).֠?q!5FINITE!QUANTUMGR9OUPSOVERABELIANGROUPSOFPRIMEEXPONENTB-"23j QIfaij u=0,thenbryconnectednessthereisasequencei=i1;i2;:::ʜ;it[=jLofelemenrtsinICsuchʍQthataii?`rii?`+16=UR0;2forall`,1`thatRJ(1)isaYetter-Drinfeldmoffduleof niteCartantypewithbraiding(bijJ)1i;jvS.ForQallrsi,Cletqi/I=obiiI;Ni=orffd$(qidڹ).$#Assumersthatorffd(bijJ)isoffddandNiMisnotdivisibleby3and>7forQall351URi;j%S.1(1)5Forany1URincffontainedinacffonnectedcomponentoftypeBnP,WCn 40orF4Kresp..G2,Wassume5that35Niisnotdivisibleby5rffesp.fiby5or7.1(2)5For35any1URi;j%and35ijassumeqidqj\=1ororSd(qidqjf )=Ni.QThenR/isgenerffatedasanalgebrffabyRJ(1),thatisAisgeneratedbyskew-primitiveandgroup-likeQelements.XUQPrffoof.:LetxS):=URRJ2RJbSethedualHopfalgebraofRFinthebraidedsense(seeforexample[ArG,SectionQ2]).{S¹=n0S׹(n)3isagradedbraidedHopfalgebrain2b YD qwithS(0)=|1;S(n)=RJ(n)2,F)for3allQn0.Byassumptiontherearehi!2;i2w_b n,1i;jiS,withbij ߹=jf (hidڹ)foralli;jӹ,andaQbasis=(yidڹ)ofRJ(1)withyi2{R(1)O8:i h8:i foralli.|Let(xidڹ)inV2:={S׹(1)=R(1)2AbSe=thedualbasisof(yidڹ).QThenlxio22V2p8:iRAg8:iwithi=2n91 i ʵ;gi=h1 i"andlbij }y=jf (gidڹ)=j(hidڹ)lforall12i;jhS.ThruslV isaQYVetter-DrinfeldOmoSduleorverOwiththesamebraidingasRJ(1).By[AS25,n6Lemma5.5],RhcisgeneratedQbryRJ(1)ifandonlyifS׹(1)UR=P(S).MHencebrydualityV,HS2ǹisgeneratedbyS׹(1),HsinceRJ(1)UR=P(R).MIt!?q!5FINITE!QUANTUMGR9OUPSOVERABELIANGROUPSOFPRIMEEXPONENTB-"25j Qis easytoseethatV'P=S׹(1)P(S).DHence therearecanonicalsurjectionsofgradedbraidedHopfʍQalgebras$f9Tƹ(Vp)UR!S)!B(V):QHereTƹ(Vp)isthetensoralgebra,theelemenrtsxiareprimitiveandofdegreeone,andbSothmapsareQtheidenrtityonVp.ThekernelI}nofthe rstmapisahomogeneousidealgeneratedbyelementsofQdegreep2,zacoidealandstableundertheactionandcoactionof.SinceB(Vp)=Tƹ(V)=Jr,zwhereQJisthelargestidealwiththesamepropSertiesasI,thereisacanonicalsurjectionS)!URB(Vp).QThexiUsatisfytheSerrerelations(4.6) ^bryLemma7.2,)andthentheroSotvectorrelations(4.7) ^byQLemma7.5.#17bffeaprimenumber.4Thenany nite-dimensionalpointedHopfalgebraQwith?Lcfforadical|(Z=(p))2s forsomenaturalnumbersisgeneratedbygroup-likeandskew-primitiveQelements.8QPrffoof.:LetAbSea nite-dimensionalpoinrtedHopfalgebrawithcoradical|(Z=(p))2sQandletR_betheQdiagram ofA.mThenRJ(1)isaYVetter-DrinfeldmoSduleof niteCartantrypebry[AS25,aCorollary1.2].QHencetheclaimfollorwsfromTheorem7.6.XQLetusstateexplicitlyanotherCorollaryoftheTheorem.kʍQCorollaryT7.8.8UnderEthehypffothesisofTheorem7.6,iftheDynkindiagramattachedtothepointedQHopf35algebrffaisconnected,thenAisgeneratedbygroup-likeandskew-primitiveelements.+HQInprinciple,theideabSehindtheproofofTheorem7.6isasfollorws.+FLetAbea nite-dimensionalQpSoinrtedHopfalgebrawithcoradical|,Eany nitegroup. LetRKbSethediagramofA,EandQSQ:=zRJ2 ZtheD dualbraidedHopfalgebra.E ConsiderthediagramwneRofthebSosonizationS׹#|.ThenQP(S׹) isnaturallyemrbSeddedinP(w*eR Թ)(andthisemrbSeddingisinfactanisomorphism).Moreover,4Qdim(P(RJ))b}dim(P(S׹))dim(P(w*eR Թ)),Sandddim(P(RJ))=dim(P(w*eR Թ))difandonlyifS׹(1)b}=P(S׹)dorQRn=URB(P(RJ)).QCorollary<7.7canalsobSeseenasadirectconsequenceofSection6and[AS25]:ܦBy[AS2]P(w*eR Թ)isQof niteCartantrypSe.8ThentheresultfollowsfromTheorem6.8and6.9appliedtoAUR=S׹#|.QThe nexttheoremisanotherapplicationofthisprinciple.EItshorwsthatonlyveryspSecialdimen-QsionsarepSossiblefor nite-dimensionalpoinrtedHopfalgebras.kʍQTheorem7.9.7Forany nitegrffoupofoddorderthereisanaturalnumbern()suchthattheʍQdimension35ofany nite-dimensionalpffointed35Hopfalgebrffawithcoradical|isURn().8QPrffoof.:Let7AbSea nite-dimensionalpoinrtedHopfalgebrawithcoradical|anddiagramRandw|ZeQRHasde nedabSorve.1qSinceR"andweR0arebraidedHopfalgebrasorverofthesamedimension,andQdim(P(RJ))kdim(P(w*eR Թ)),wreocaniteratethisproScessandafter nitelymanystepsweobtainaQgradedbraidedHopfalgebraTYQorverwithdimL(RJ)=dimGX(Tƹ)andTSֹ=B(P(T)).ByaresultofQGraS~vna:[G~n3<]:using[AS25,MTheorem3.1]whicrhfollowsfrom[L3 7o],MthenumbSerofisomorphismclasses?q!Q26YV"NICOL'ҟASTANDR9USKIEWITSCHANDHANS-J'ҟURGENSCHNEIDERj QofYVetter-DrinfeldmoSdulesV/orverthe xedgroupwith nite-dimensionalB(Vp)is nite.ThrusweʍQcantakreforn()theproSductofthelargestsuchdimensionwiththeorderof.Z卒h9ReferencesfdQ[AD]fN.UUAndruskiewitschandS.Dascalescu,UUOnquantumgr}'oupsat !", cmsy101,Algebr.Represent.Theory*,toappGear.fdQ[AJS]H.cH.Andersen,J.JantzenandW.SoGergel,R}'epresentationsĶofquantumgr}'oupsata b> cmmi10p-throotofunityandofsemisimplegr}'oupsincharacteristicp:IndependenceofpUUAstGerisque5"V cmbx10220,1994.Q[AG]N.,AndruskiewitschandM.GraG~na,H!BraidedHopfalgebrasovernon-abGeliangroups,H!;p0J cmsl10Bol.Acad.Ciencias(Cordoba)UU63(1999),45-78,avqailableatwww.mate.uncor.edu/andrus/articulos.html.Q[AS1]N.AndruskiewitschandH.-J.Schneider, !Lifting?ofQuantumLine}'arSpacesandPointedHopfAlgebrasoforderصp^ٓRcmr73|s,UUJ.Algebra209(1998),658{691.Q[AS2]N.ęAndruskiewitschandH.-J.Schneider, jFinitequantumgr}'oupsandCartanmatrices, jAdv.ęinMath.154ز(2000),UU1{45.Q[AS3]N.ЧAndruskiewitschandH.-J.Schneider,{Lifting[ofNicholsalgebr}'asoftypeA2andPointedHopfAlgebrasofor}'derlp^4|s,ins"Hopfalgebrasandquantumgroups",ProGceedingsoftheColloquiuminBrussels1998,MarcelDekker,ed.UUS.CaenepGeel,1{18(2000).Q[AS4]N.AndruskiewitschandH.-J.Schneider,Pointe}'d`Hopfalgebras,toappGearin"NewdirectionsinHopfalgebras",MSRIUUseriesCambridgeUniv.Press.Q[BDG] qM.Beattie,NS.Dascalescu,andL.GrGunenfelder,OnthenumbGeroftypesof nite-dimensionalHopfalgebras,InventionesUUMath.136(1999),1-7.Q[BDR] .M.UUBeattie,S.Dascalescu,UUandS.Raianu,LiftingofNicholsalgebr}'asoftypeB2|s,UUpreprint2001.Q[CD]IS.UUCaenepGeelandS.Dascalescu, cmmi10K`y cmr10ٓRcmr7$