; TeX output 2002.01.09:1936?q!j ETN cmbx12AٚCHARACTERIZATIONOFQUANTUMGROUPSKK`y cmr10NICOL@xASUUANDRUSKIEWITSCHANDHANS-J@xURGENSCHNEIDER v䍍Q- cmcsc10Abstract.F=W*ep classifypGointedHopfalgebraswith niteGelfand-Kirillovdimension,vwhicharedo-fdQmains,whose1groupsofgroup-likeelementsareabGelian,andwhosein nitesimalbraidingsarepositive.RՍ"- cmcsc10IntroductionQXQ cmr12SincetheappSearanceofquanrtumgroups[KR",Sk,Dro,Ji0],"kthereweremanyattemptstode neQthemqinrtrinsicallyV.;ImpSortantdescriptionsoftheso-called"nilpSotent"partsweregivenbyRingelQ[Ri 7],Lusztigg[L2 7o]andRosso[Ro].Horwever,thegquestionof ndinganabstractcrharacterizationofQthequanrtizedenvelopingalgebrasremainedopSen.QInD3themainTheorem5.2ofthispapSer,ZwreclassifyallHopfalgebrasoveranalgebraicallyclosedQ eldofcrharacteristic0whichare.5 !", cmsy105pSoinrted,thatzisalltheirsimplecomodulesareone-dimensional,andharvezanabeliangroup5ofgroup-likreelements,55domainsof niteGelfand-Kirillorvdimension,and55harvepSositivein nitesimalbraiding(seeSection1).QThe> rsttrwo>conditionsarenatural.5ThepSositivitryconditionshouldberelatedtotheexistenceofQarealinrvolution.QInTheorem4.2,wredescribSetheseHopfalgebrasbygeneratorsandrelations.Theyarenaturalgen-QeralizationsofquanrtizedenvelopingalgebraswithpSositiveparameter.TTVoproveourmainTheorem,QwrecombinetheliftingmethoSdforpoinrtedHopfalgebras[AS15,AS2ӻ,AS4]withacrharacterizationQobtainedbryRossoofthe"nilpSotentpart"ofaquantizedenvelopingalgebraintermsof nitenessofQtheGelfand-Kirillorvdimension[Ro].]QAmongthemaindi erencesbSetrweenthenewHopfalgebrasandmrultiparametricquantizeden-Qvreloping%algebras,tletusmentionthatwehaveoneparameterofdeformationforeachconnectedQcompSonenrtoftheDynkindiagram(thisisexplainedasfollows:?inthe"classicallimit",.onemayQharve3di erentscalarmultiplesoftheSklyaninbracketsinthedi erentconnectedcompSonents)andLQlinkingrelations,see(4.12)!,generalizingtheclassicalrelationsg cmmi12E2cmmi8idFjFjf Ei,=URijJ(KiKܞ!K cmsy8|{Ycmr81 i 9).QNotethatwrearenotassumingthattheHopfalgebrashavean,@ cmti12a-prioriassignedDynkindiagramQasin[W u,KW];itcomesfromourhrypSothesis,andhereiswherewerelyonRosso'sresult[Ro].Qxff< UUV*ersionUUofJanuary9,2002.fdThisUUpapGerisavqailableatgeneralizinganideaofTVakreuchi.&In-jSection4,wreconstructanewfamilyQof2pSoinrtedHopfalgebraswithgenericbraidingandestablishthemainbasicpropertiesofthem.TheQapproacrh~issimilarto[AS45]butinsteadofdimensionarguments,weusethetechnicalresultsontheQcoradical ltrationobtainedinSection3;theseresultsshouldbSeusefulalsoforotherclassesofHopfQalgebras.InSection5,%awreproveourmainTheorem.AkeypSointisLemma5.1,%awhichimpliesthatQaIwideclassofHopfalgebraswith niteGelfand-Kirillorvdimensionisgeneratedbygroup-likeandQskrew-primitiveelements.ōQAffcknowledgements.WVeCthankN.ReshetikhinforrevivingourinrterestinthisquestionandJ.AlevQfor(inrterestingconversations._ThispapSerwasbSegunduringvisitstotheMSRI,intheframeworkQofthefull-yrearProgramonNoncommutativeAlgebra(August1999-May2000).oWVethanktheQorganizers+forthekindinrvitationandtheMSRIfortheexcellentworkingconditions.iPartoftheQwrorkofthe rstauthorwasdoneduringavisittotheUniversityofReims(OctobSer2001-JanuaryQ2002);heisvrerygratefultoJ.AlevforhiskindhospitalityV.(Ii1.lPreliminariesQNotation.Let) msbm10|bSeanalgebraicallyclosed eldofcrharacteristic0."Ourreferencesare[M =],D[Sw]forQHopfRalgebras;'[KLy],=forgrorwthofalgebrasandGelfand-Kirillovdimension;'and[AS55]forpSointedQHopf=algebras.IWVeusestandardnotationforHopfalgebras:},`pSb,,denote=respSectivrelythecomulti-Qplication,3theUanrtipSode,theUcounit;qwreuseashortversionofSweedler'snotation:(x)UR=x(1) Z lx(2) \|,Qx꨹inacoalgebraCܞ.ȍQLet&H|bSeaHopfalgebrawithbijectivreantipSode.+YWVe&denotebyG(HV)bSethegroupofgroup-QlikreselementsofHV;(andbyPgI{;h c(HV)thespaceofgn9,hskew-primitiveelementsxofHV,thatiswithQ(x)W=gM x+x h, where^gn9;hW2G(HV);thenP(H)W:=P1;1 \|(HV). Thebraidedcategoryof-JQYVetter-DrinfeldmoSdulesorverHisdenotedbry2HbH YD,cf.8theconventionsof[AS55].QTheadjoinrtrepresentationad!ofaHopfalgebraAonitselfisgivenbyad?x(yn9)7=x(1) \|ySb(x(2)).$QIfiR9isabraidedHopfalgebrain2HbH 3YD%thenthereisabraidedadjoinrtrepresentationadrc^ڹofRQonitselfgivrenbyadmXcx(yn9)J =(r Sb)(id ʠ c)( id=)(x y);չwhereisthemrultiplicationandQc$_2End&(R GRJ)isthebraiding. ,6Ifx$_2P(R)thenthebraidedadjoinrtrepresentationofxisQadcܘx(yn9){=(id ʠc)(x t y){=:[x;y]c.y:Theelemenrt[x;y]c%de nedbrythesecondequalityforanyxQanduyn9,regardlessofwhetherxisprimitivre,willbSecalledabraidedcommrutator.bHWhenAlй=RJ#HV,Qthenforallb;dUR2RJ,adG+(b#1)([߹(d#1)=(ad\cb(d))#1.AۍQIfzQisanabSeliangroup,AYVetter-DrinfeldmoSduleorveris|-moduleVwhicrhQisZalsoa|-comoSdule,vandsucrhthateachhomogeneouscompSonentVg,vg32,isZa|-submoSdule.QThrus,va'vectorspaceV'providedwithadirectsumdecompSositionV n=pgI{2G;2PK$a6cmex8b#~V2pRAg GisaYVetter-9QDrinfeldmoSduleorverHB=UR|.a?q!yA!CHARA9CTERIZA:TIONOFQUANTUMGROUPS3j QBraidedvectorspaces.Aebbraidedevrectorspace(V;c)isa nite-dimensionalvectorspaceprovidedQwithanisomorphismcй:V~ DVq@!V DVhewhicrhisasolutionofthebraidequation,Ithatis(c Qid>)(id ʠ c)(c idm)*=(id c)(c idm)(id c): ExamplesofbraidedvrectorspacesareYVetter-DrinfeldQmoSdules:,ifV22URHbURH nYD[,thencUR:V xVV!V xVVp,c(v wR)UR=v(1):w xVv(0) \|,isasolutionofthebraidQequation.QDe nitionD1.1.㭹Let(V;c)bSea nite-dimensionalbraidedvrectorspace. rrWVeshallsaythattheʍQbraiding|cUR:Vfl V!V Vis|diagonalifthereexistsabasisx1;:::ʜ;x0:ofVandnon-zeroscalarsqijQsucrhthatc(xi xjf )UR=qijJxj xidڹ,1i;j%>f߹.8Thematrix(qijJ)iscalledthematrixofthebraiding.+QFVurthermore,wre#shallsaythatadiagonalbraidingwithmatrix(qijJ)isindeffcomposable#ifforallQiUR6=jӹ,thereexistsasequenceiUR=i1,i2,.T..,it=URjkofelemenrtsoff1;:::ʜ;Sgsuchthatqi;cmmi6s;iqsAacmr6+1qiqs+1 H;isF6=UR1,Q1URst1.8Otherwise,wresaythatthematrixisdecompSosable.QWVe attacrhagraphtoadiagonalbraidinginthefollowingwayV.TheverticesofthegrapharetheQelemenrtsSoff1;:::ʜ;Sg,randthereisanedgebetrweenSiandjĹiftheyaredi erenrtandqijJqjvi 66=UR1.Thus,Q\indecompSosable"meansthatthecorrespondinggraphisconnected.%ThecomponenrtsofthematrixQareytheprincipalsubmatricescorrespSondingtotheconnectedcomponenrtsofthegraph.RIfiandjQarevrerticesinthesameconnectedcompSonent,thenwewriteiURjӹ.&,WVeshalldenotebyXt_thesetofQconnectedTcompSonenrtsofthematrix(qijJ).pIfIf2uXӹ,thenVI̡denotesthesubspaceofVĹspannedbyQxidڹ,iUR2I.QWVeshallsarythatabraidingcisgenericifitisdiagonalwithmatrix(qijJ)whereqii g4isnotaroSotQof1,foranryi.QLetp|u=C.8WVeshallsarythatabraidingcispffositiveifitisgenericwithmatrix(qijJ)whereqii h$isQapSositivrerealnumbSer,foralli.QWVeshallsarythatadiagonalbraidingcwithmatrix(qijJ)isofCartanItypffeifqii e6=~1foralli, andQthereJareinrtegersaij I.withaii t0=*|2,C31iS,andJ0aij u`algebras.LetV22URHbURH nYD[.%AbraidedgradedHopfalgebraRn=URn0RJ(n)in2HbH ,YD"ٹiscalledQaNichols35algebrffaofVif|UR'RJ(0)andV'URR(1)in2HbH YD,and^1(1)5Pƹ(RJ)UR=R(1),1(2)5RisgeneratedasanalgebrabryRJ(1).QTheNicrholsalgebraofV"eexistsandisuniqueuptoisomorphisms;ӜitwillbSedenotedby.%n eufm10B(Vp).QItdepSends,asanalgebraandcoalgebra,onlyontheunderlyingbraidedvrectorspace(V;c).0TheQunderlyingBnalgebraiscalledaquantumo1symmetricalgebrffaBnin[Ro]. @2WVeshallidenrtifyV޹withtheQsubspace,~ofhomogeneouselemenrtsofdegreeoneinB(Vp).bSee[AS55]formoredetailsandsomeQhistoricalreferences.ʼQGivrenVabraidedvectorspaceofanyofthetypSesinDe nition1.1,wewillsaythatitsNicholsQalgebraisofthesametrypSe.(QLemmab;1.2.S[AS25,LLemma_4.2]. LffetVQobea nite-dimensionalYetter-DrinfeldmoduleoveranʍQabliangrffoup. v3LetX=&fI1;:::ʜ;INDgbeanumerationofthesetofconnectedcomponents. v3ThenQB(Vp)l'B(VIq1Z") 뀉z UW US::: 뀉z UW(uFB(VIX.N ")ޛasbrffaidedޛHopfalgebrffaswiththebraidedtensorproductalgebraQstructurffe35 35뀉z UW..& msam10ȯQLifting/zmetho`dforpointedHopfalgebras.Recallb2thataHopfalgebraAispSoinrtedifanyQirreducible A-comoSduleisone-dimensional.3Thatis,TuifthecoradicalA0 equalsthegroupalgebraQ|G(A).QLetʄAbSeapoinrtedHopfalgebraletA0 _=[|G(A)AA1:::g}bSeʄthecoradical ltrationandQlet)&grUAsn=n0̹grA(n))&bSetheassociatedgradedcoalgebra,xwhicrhisagradedHopfalgebra[M =].QThevSgradedprojectionG:Cgr=AC!grA(0)'|G(A)vSisaHopfalgebramapandaretractionoftheQinclusion.dLet$R͹=kfa2A:(id ʠ n9)(a)=aU 1gbSethealgebraofcoinrvXariantsofn9;jR=ɹisa"!QbraidedDbraidedvrectorspace(V;c),SwhereV޹:=nRJ(1)=Pƹ(R)>andcn:VW V!V V,is>thebraidingQinߍ.G(A) 뀍.G(A)"YD*.aqIt.willbSecalledthein nitesimal?brffaidingofA.ThedimensionofVƹ=lVPƹ(RJ),calledthe_QrffankgйofA,orofRJ.XThesubalgebraR20OSofRgeneratedbryR(1),whicrhistheNicholsalgebraofVp:QRJ20<'URB(Vp).8See[AS55]formoredetails.%{}2.NicholsalgebrasofCar32tantypeQNicholsalgebrasofdiagonaltyp`e.In>thissection, (V;c)denotesa nite-dimensionalbraidedQvrectorHspace;(weassumethatthebraidingcisdiagonalwithmatrix(qijJ),0withrespSecttoabasisQx1;:::ʜx.QLetbSethefreeabeliangroupofrank:pwithbasisg1;:::ʜ;g.7WVede necrharacters1;:::ʜ;sofQbryW-idڹ(gjf )UR=qjviJ;1i;j%S:WQWVeconsiderVasaYetter-DrinfeldmoSduleorver|bryde ningxi,2URV2p8:iRAg8:i ,foralli.9?q!yA!CHARA9CTERIZA:TIONOFQUANTUMGROUPS5j QWVebSeginwithsomerelationsthatholdinanryNicholsalgebraofdiagonaltypSe.ύQLemma2.1.Lffet35RLbeabraidedHopfalgebrain2*ppmsbm8|nX|sYD! ,suchthatV,!URPƹ(RJ).ʍQ(a).fiIf35qii |isarffoot35of1oforffderN6>UR1forsomei2f1;:::ʜ;Sg,thenx2NRAi n2Pƹ(RJ).Q(b).}LffetLoiUR6=j%2f1;:::ʜ;SgsuchthatqijJqjvi 6=qn91r ii,zwherfferisanintegersuchthat0URrfع1 QWVenorwrecallavXariationofawell-knownresultofReshetikhinontwisting[Re ދ].Let(webV v; bc)bSeQanother}braidedvrectorspaceofthesamedimensionasVp,suchthatthebraidingԋbc isdiagonalwithQmatrix(wGbqij )withrespSecttoabasis?bx1W;:::}bʜx+.8WVede necrharactersb1;:::ʜ;bofby *b-iu(gjf )UR=̙bqjvi;1i;j%S:EQWVeconsiderwPUbVKUasaYetter-DrinfeldmoSduleorverbryde ning?bxiR&2wbURV2\pbp8:iRAg8:i,foralli.QProp`osition2.2.Assume35thatforalli;j,qii =̙bURqiiandh֍Q(2.1)qijJqjvi 6=̙bURqijM*bqjviVt:QThen35therffeexistsanN-gradedisomorphismof|-comodules Ë:URB(Vp)!B(webV v)35suchthatQ(2.2) n9(xidڹ)UR=3bxi g~;1iS:QLffetGp:{7!|2 d$bffeGtheuniquebilinearformsuchthatn9(gid;gjf ){7=~bqijq1 ij ʵ,LifGij,andGiseffqualtoʍQ135otherwise;nisagrffoup2-cocycleandwehave,forallgn9;hUR2,b[ n9(xy)=URn9(g;h) (x) (y);x2B(Vp)g;yË2B(V)he;Q(2.3)LʍRb n9([x;y]c.y)=URn9(g;h)[ (x); (y)]c.y;xUR2V pڍgc;yË2B(Vp)O he:Q(2.4)WQPrffoof.:See[AS55,Prop.83.9andRemark3.10]. }QRemarkQ 2.3.ಹInthesituationofthepropSosition,wresaythatB(Vp)andB(webV v)aretwist-effquivalent$D;QnotethatB(Vp)istrwist-equivXalenttoaB(webV v)withRWbqij=̙bURqjviforalliandjӹ,.sincealltheqijJqjvi'sharveQsquareroSotsin|.ύQLemma 2.4.Assumethatthebrffaidingwithmatrix(qijJ)isgenericandofCartantypewithgeneralizedQCartan35matrix(aijJ).fiThen(aij)issymmetrizable.M?q!Q6YV"NICOL'ҟASTANDR9USKIEWITSCHANDHANS-J'ҟURGENSCHNEIDERj QPrffoof.:ByZ[K "w,vEx.;2.1],itisenoughtoshorwthataiq1*iq2aiq2*iq3::: iaiqt"q% cmsy61 0ittTait\piq1=)aiq2*iq1aiq3*iq2:::ait\piqt1tTaiq1*it+forʍQalli1;i2;:::ʜ;itʹ.8ButFӍ;*qn9a8:i1*i2.2a8:i2*i3:::Ca8:it1 0ita8:it\pi1 Íiq1*iq1](=URqn9a8:i2*i1.2a8:i2*i3:::Ca8:it1 0ita8:it\pi1 Íiq2*iq2==qn9a8:i2*i1.2a8:i3*i2:::Ca8:it\pit1a8:i1*it Íiq1*iq1YҴ;Qbrysubstitutingq؍n9a8:i1*i2 :oiq1*iq16m=3q؍n9a8:i2*i1 :oiq2*iq2:,Mthenq؍n9a8:i2*i1 :oiq2*iq2=3q؍n9a8:i3*i2 :oiq3*iq3andsoon.9TheclaimfollorwsbSecauseqiq1*iq1ȹisnotaQroSotofone.j$QThex,follorwingresultisduetoRosso,whosketchedanargumentin[Ro,Th.m2.1].WVex,includeaQproSofforcompleteness. ΍QLemmakd2.5.C[Ro].`Lffet"|UR=C.Assume"thatthebrffaidingwithmatrix(qijJ)ispositiveandofCartanQtypffei withgeneralizedCartanmatrix(aijJ).Then(aij)issymmetrizable,vwithsymmetrizingdiagonalQmatrixG(didڹ);andtherffeisacollectionofpositivenumbers(qIM)I2X-suchthat(qijJ)istwist-equivalenttoQ(wGbqij ),35wherffe`njb#qij_=URqzn9d8:i,ra8:ij ܍IȤfor35all;Ui;j%2I:ӠQThat35is,thebrffaidingassociatedto(wGbqij )isofDJ-type.QPrffoof.:WVe\canassumethatthebraidingisindecompSosable;%writeIFչ=URf1;:::ʜ;g. ByRemark2.3,y>wreQcan9qassumethatqij &Q=mqjviJ,M$foralli;j@2I.%:b 6{11^=qzn9d8:j 8:j ܍I蕹.QThrusqjvj f=URqzn9d8:j ܍I 8;andforalli;j%2I,qijJqjvi 6=qzn9d8:i,ra8:ij ܍I=qzn9d8:ja8:jYi ܍ISع,andtheclaimfollorws._\QRemark2.6.Thediagonalbraidingwithmatrix"hBz pqqn921ɍ›qn921Ɛq_z! C_;"iQwhereqXisnotaroSotofone,isgenericofCartantrypebutnotofDJ-trype."QWVenorwstateaveryelegantdescriptionofNicholsalgebrasusedbyLusztiginafundamentalwayQ[L2 7o]. ΍QProp`osition2.7.zLffet!(V;c)beasaboveandassumethatqij Yʹ=qjvi lforalli;j.1LetB1;:::ʜ;B .#?bQLet\A]S:=Q[vn9;v21 ʵ];let[n]i-:=ōv2nRAi1vn i[z,Й 썑viv1 i2R,[rS]idڹ!]S=[1]i[2]id:::/v[rS]i.FLet\B(Wƹ)A ; betheA-subalgebraof࣍QB(Wƹ)generatedbryally yߍn9(rTiq1 Tiq2:::ZTiX.P1 (0yߍn9(cX.P&) 5hiX.P` UR0suchthatadrɟcB(xidڹ)2rb(xjf )=0. m獑QCorollaryo2.12.U2Lffet̐(V;c)beabraidedvectorspaceofdiagonaltypewithindecomposablematrix.QAssume35thatB(Vp)has niteGelfand-Kirillovdimension.Q(a).fiIf35therffeexistsisuchthatqii =UR1,then=1.Q(b).fiIf35thebrffaidingisgenericthenitisofCartantype.-QPrffoof.:ThisfollorwsfromTheorem2.11andLemma2.1(e).vQTheoremb<2.13.A}[Ro,#Theorem21]NLffet(V;c)bea nite-dimensionalbraidedvectorspacewithpositiveʍQbrffaiding.fiThen35thefollowingareequivalent:Q(a).fiB(Vp)35has niteGelfand-Kirillovdimension.Q(b).fi(V;c)35istwist-effquivalenttoabraidingofDJ-typewith niteCartanmatrix.-QPrffoof.:WVecanassumethatthematrix(qijJ)isindecompSosable.(b)=)(a)follorwsfromTheoremQ2.10 P(ii).(a)=)(b).(V;c) PisofCartantrypSebyCorollary2.12(b)withCartanmatrix(aijJ).WVeQknorwS*that(aijJ)issymmetrizablebyLemma2.4andthat(V;c)istwist-equivXalenttoabraidingofQDJ-trypSebyLemma2.5.8ByTheorem2.10(i),theCartanmatrix(aijJ)is nite.aOLQRemark,2.14.See_[AS25]fortheanalogousproblemofcrharacterizing nite-dimensionalbraidedQvrectorspaceswithdiagonalbraidingsuchthatB(Vp)has nitedimension.!э3.;Coradicall32ygradedcoalgebrasQInthisSectionwreproveageneralcriteriontodeterminethecoradical ltrationofcertainHopfQalgebras.&WVegeneralizeamethoSdofTakreuchi[T|l],whocomputedthecoradical ltrationofUq(gn9)inQthiswray;seealso[MuKU].8WVe rstextendthede nitionofcoradicallygradedcoalgebras[CM@].QLetTUR1bSeanaturalnrumber.8IfiUR=(i1;:::ʜ;iT)2N2T,thenwresetjijUR=i1j+UN+iT.WӍQDe nitionٷ3.1. nAn N2T-grffadedcoalgebra is]acoalgebraCprorvidedwithanN2T-gradingCZ=ʍQ>32@cmbx8i2NTCܞ(i)sucrhthatC(i)Vj~C(j) C(ij).:1AnN2T-gradedcoalgebraCǶiscfforadically3graded)ifQthen-thtermofthecoradical ltrationisލCn=UR>i2NT;jijn*Cܞ(i); 8n2N:QWVedenotebryi7:URC1!Cܞ(i)theprojectionassoSciatedtothegrading.ʍQAnN2T-gradedcoalgebraCTisstrictlyHcfforadicallygradedWIifCܞ(0)Wu=C0,d:thecoradicalofCܞ,andQi;j A:URCܞ(i+jXع)!C(i) C(j),i;j A=UR(i j~),isinjectivre,foralli;jUR2N2TQ.QLemman3.2._(a).aLffet%ACbeastrictlycoradicallyN2T-gradedcoalgebraandletDxbeastrictlycoradi-QcffallyAN2S-gradedcoalgebra.\ThenCB} eDgisstrictlycoradicallyN2T.:+Sq-gradedwithrespecttothetensorQprffoduct35grading.Q(b).fiIf35Cisstrictlycfforadically35N2T-graded,thenitiscoradicallygraded. ՠ?q!yA!CHARA9CTERIZA:TIONOFQUANTUMGROUPS11j Q(c).fiIf35Ciscfforadically35N-graded,thenitisstrictlycoradicallygraded.Y򍍍QPrffoof.:(a)follorwsfromthede nition.4GWVeprove(b)byinductiononn,ߟthecasenUR=0bSeingpartofʍQtheQ hrypSothesis.AssumenUR>0.IfQ c2jijnCܞ(i),othen(c)2Cܞ(0)q C(n)+jjjn.BytherecursivrehypSothesis,"(c)UR2Cܞ(0) C+jjj;MV:URN;U qM6!N qM@,A_M (n m)UR=m(1)eqn m(0)sis aniso-ʍQmorphism)withinrverse)18OM \|(n p m)UR=Sb21 B޹(m(1)) pn m(0).WVe)applyidINq1* \ NX.T=Gd 18OH$;Nq1* \ NX.T6a  pid HQto( )21 \|jӹ(z)andgetàFu1j (v1)(1)G*u2 (v2)(1)G*P (vT.:1})(1)uTGrLGȓ h (v1)(0)$ (v2)(0)$ UN (vT)(0) kg;Qtothisexpression,wreapplyidJNq1 18ONq2* \ NX.T H$;Nq1J iduJNq2* \ NX.T HL˹andget u1j u2 (v2)(1)G*P (vT.:1})(1)uTGuC h v1j (v2)(0)$ UN (vT)(0) kg;QiteratingthisproScedure,wreobtainz.8Thisshowsthatj{isbijective.G덑QTheoremn3.4.Wekeffepthenotationsabove. WeassumethatNl T9=2qi2N֥Nl!ȹ(i)isacoradicallyʍQgrffadedVHopfalgebrffain2HbHYD _,qforalllC,1lT,andVthatHiscffosemisimple. LetVU@(i)=Q(N1(i1) UN NT(iT) HV),0forŘallidf=(i1;:::ʜ;iT)2N2T.ThenŘUJ=>i2NTU@(i)isacfforadicallyQN2T-grffaded35coalgebra.QPrffoof.:ThetensorproSductcoalgebraN14 0^ 0NTsG HnisastrictlycoradicallyN2T-gradedcoalgebraQbryĥLemma3.2(a)and(c)sinceeachNlmiscoradicallygradedandHiscosemisimple.SinceeachQNl!ȹ(i)%isaYVetter-DrinfeldsubmoSduleofNl,lDthemapjinLemma3.1ishomogeneous.XHence,itQfollorwsfromLemma3.3thatUcisstrictlycoradicallyN2T-gradedcoalgebra.fThenUiscoradicallyQN2T-gradedbryLemma3.2(b).\ȍvz4.Afamil32yofpointedHopfalgebrasQInthisSection,wre x{j55afreeabSeliangroupof niteranks,֍55a niteCartanmatrix(aijJ)UR2Z2UR0,forallIF2Xӹ."$QLetDUV20:bSeagenericdatumof niteCartantrypeorverafreeabeliangroup20yof niterank,DformedQbry;(a20RAijJ)UR2Z26sucrhthatS55'(gidڹ)UR=g2n90뀍I{(i)&Q,forall1URiS;55i,=UR20뀍I{(i)&Q',forall1i;j%S;*Z55ij 6=fXUR8 ԍUR<UR:󶍍 id jf 20뀍I{(i)(jv);عifn9(i)UR<(jӹ)(퍍 id jf j(gi)20뀍I{(jv)(i);عifn9(i)UR>(jӹ)eq,forall1URiofWI ƹintermsofsimplere ections.TThenweobtainaQreduced*rdecompSositionofthelongestelemenrt!0 5=usiq1 :::ZsiX.P8ofW'fromtheexpressionof!0 vasQproSductofthe!0;I kŹ'sinsome xedorderofthecomponenrts,saytheorderarisingfromtheorderofQthevrertices.8Therefore j\:=URsiq1 :::Zsi8:jY1u( i8:jO)isanumerationof2+x.QWVe xa nite-dimensionalYetter-DrinfeldmoSduleVgorverwithabasisx1;:::ʜ;x3withxi,2URV2p8:iRAg8:i ,Q1URiS.8Notethat Q(4.8)tʤV p8:iڍg8:iTURV p8:jڍg8:j9in _|ڍ_|(/YD:;ꦹforall(/1i;j%S;i6=j:Ϡ?q!Q14YV"NICOL'ҟASTANDR9USKIEWITSCHANDHANS-J'ҟURGENSCHNEIDERj QFVorsuppSosethatV2p8:iRAg8:iPT԰<=Vzp8:j,g8:j Nk,andiUR6=jӹ.8Thengi,=gjf ,i,=jPandoCidڹ(gi) 2V=URi(gjf )j(gidڹ)=i(gi) a8:ij :]QThrusaij 6=UR2,sinceidڹ(gi)isnotaroSotofone;butthisisacontradiction.QLusztig=de nedroSotvrectorsX , 2 _2+ [L2 7o],in=thecaseofbraidingsofDJ-typSe;LtheseareQthecjelemenrtsb1;:::ʜ;bP u6intheproSofofTheorem2.10;theycanbeexpressedasiteratedbraidedQcommrutators./As|in[AS45],thisde nitioncanbSeextendedtogenericbraidingsof niteCartantypSe,Q rstinthetensoralgebraTƹ(Vp),andtheninsuitablequotienrts.QWVe xaZ-basisYhe;1URhs꨹of.3xQTheoremi4.2.gLffet^D==DUV((aijJ);(qIM);(gidڹ);(i);(ijJ))^bffeagenericdatumof niteCartantypeforʍQtheYfrffeeabeliangroupof niterank. LetU@(DUV)bethealgebrapresentedbygeneratorsa1;:::ʜ;a,Qyn91諍1 ʵ;:::ʜ;y2n91RAsand35rffelations'<$y n91ڍm ʵyn91Oh,=URyn91Oh ʵy n91ڍm;y n91ڍmy n91ڍm =1;1m;hs;Q(4.9)Lʍ8yheaj,=URjf (Yhe)ajyh;1URhs;1j%S;Q(4.10)c.Q(ad\aidڹ) 1a8:ijwaj,=UR0;1i6=j%S;ij;Q(4.11)Po6aidajjf (gi)ajai,=URijJ(1gidgjf );1i,:bSeafreeabeliangroupofrankVb* &.fLetj߹beQtheuniquecrharacterofsuchthatjf (Zidڹ)UR=j(gidڹ),1i;j%VOb* ".QLetD1˹bSethegenericdatumorvergivrenby(aijJ)rbr3n.0LetQB:=URU@(D1),withgeneratorsbrbr3n+1 ,.T..,b9(insteadoftheaidڹ's)andy1;:::ʜ;ysn<.QLet=D2AbSethegenericdatumorver=givrenby(aijJ)n1i;jv㎍rbrSݹ,(Zidڹ)n1i㎍rbr_,(jf )n1jv㎍rbr,with=emptylinkingQdatum.8LetUc:=URU@(D2)withgeneratorsu1;:::ʜ;urbr9(insteadoftheaidڹ's)andz1;:::ʜ;zrbr.QThentheanalogueof[AS45,#sLemma5.19]holdsreplacingthedualofBubrytheHopfdual.,NoticeQthattheargumenrtinloffc.2cit.workssincealgebramapsandskew-derivXationsareintheHopfdual.QThe(proSofisactuallyeasierbecausetherearelessrelationstocrheck.FinallyV,xAwe(formtheHopfQalgebra(U pB]m) -asinloffc. ۦcit.,andconsiderthequotienrtx ReAof(U pB]m) -brythecentralHopfQsubalgebrawU|[(zioI  ogn91 i ʵ)D:1iVb* ].Thesameargumenrtasinloffc.Scit.shorwsthatxAeAQ'DU@(DUV)asQHopf̛algebras..Ontheotherhand,ҝthemonomialsbncq1/č1Y bncq2/č2 Y:::#bcX.P%P Ueyn9,cj\2URN,1j%Pƹ,yË2mF̛formQa*basisof(U7 &B]m).SincesplitsastheproSductofandthegroupgeneratedbry(zi; gn91 i ʵ),Q1 niVkb* ׿,owreUconcludethattheimagesofthemonomialsbncq1/č1Y bncq2/č2 Y:::#bcX.P%P Ueyn9,cjpx2 nN,1jAPƹ,yx2Qformabasisofx eA .QStep;IV.8TheclaimabSouttheGK-dimensionfollorwsfromthepreviousstep..TheclaimabouttheQcoradical5 ltrationalsofollorwsfromthepreviousstep,togetherwithTheorem3.4.Indeed,TheoremQ3.4appliessinceNicrholsalgebrasarecoradicallygradedbyde nition.QStepu/V.HTheexistenceof ,follorwsfromthede nitionofNicholsalgebras,sincebythestate-QmenrtabSoutthecoradical ltrationwehaveamonomorphismofYVetter-DrinfeldmoSdulesV H!QU@(DUV)1=U(D)0.ԏSince8therestrictionof 7qtothe rsttermofthecoradical ltration(B(Vp)#|)1Qis injectivre,$ Disinjective[M =,$Th.2 5.3.1].It issurjectivebythePBW-basisclaim.2 Therefore DisQbijectivre.QStepVI.eWVe nallyprorveethatU@(DUV)isadomain. `Herewrefollow[DCK,ĝCorollary1.8]. `WVeQinrtroSduceanN2P.:+1H- ltrationonU@(DUV)bythedegreede nedbyɨdeg%a(bncq1/č1Y bncq2/č2 Y:::#bcX.P%P Ueyn9)UR=(c1;c2;:::ʜ;cP;MX1jvPcj F!hrt jf ); ;cj\2URN;1j%PS;yË2;#iQhere khrt 5isitheheighrtoftheroSot asin[DCK]."WVeclaimthatthisisanalgebra ltration.IfQ =*e1 1{Ϲ+w+e , wresetg u=gnn9eq1/č1 N:::gtn9ei? ,  =neq1/č1 :::ftei?.bRecallthatbk=a i?k , 1kGPƹ.bTVo)?q!Q16YV"NICOL'ҟASTANDR9USKIEWITSCHANDHANS-J'ҟURGENSCHNEIDERj Qprorvetheclaimonehastovrerifythatforallko>URlЍQ(4.15)צbk#blp i?l-(g i?k )bl!bkx=XUURc2NPIcbncq1/č1Y bncq2/č2 Y:::#bcX.P%P Ue;"֍Qwherec0Z2UR|,andc=UR0unlessdega(bncq1/č1Y bncq2/č2 Y:::#bcX.P%P Ue)YQTVo!pdescribSetheisomorphismsbetrween!pHopfalgebrasU@(DUV)andU(DUV20 eG),Iwre rstformulateaLemmaQwhicrhisneededinthisgeneralformintheproSofofthemaintheorem5.2.̍QLemma4.3.vLffetbea nitelygeneratedabeliangroup,=AapointedHopfalgebrawithcoradicalQA0V=UR|,35V22|nX| YD$bwith|-bffasisxi,2V2p8:iRAg8:i ;gi2;i2wb ;1iS:QAssume35thatgrAPUR԰n:=B(Vp)#|asgrffaded35Hopfalgebrffas,andQ(4.16)ai,6=UR";33for35all)1iS:QThen35the rsttermofthecfforadical35 ltrationofAisA1V=URA0jПM'؍gI{2;1i.PgI{g8:i,r;ge(A) 8:i:"JQIfe(gid;i)>6=(gjf ;j)eforalli>6=j,0thenetheveffctorspacesPgI{g8:i,r;ge(A)28:iareone-dimensionalforallQgË2UR;1iS.#QPrffoof.:ByWassumption,A1=XA0P S԰ k=;Vp#|,andtheelemenrtsxidڹ#gL2PgI{g8:i,r;ge(B(Vp)#|)28:i,1iS,QgË2UR,x2PgI{g8:i,r;ge(A)28:i :forma|-basisofVp#|.8Hence򩍒A1=XA0PV԰.>=M'؍1i-Q^(A1=A0) 8:i:"bQLeth1;;hs ԹbSegeneratorsof.lFix16i.lThenforanrya62A1 fwithaf2(A1=XA0)28:i,ՔandQ1l1Cs,ZtheD.adjoinrtactionofhleonaisgivenbyhl da=idڹ(hl!ȹ)a+vl,ZforD.somevl|2A0.EsSinceQisabSelian,thevrectorspacespannedbya;v1;;vs}ǹis-stablecontaininga.ThisshowsthatA1ϏisQloScally1f niteundertheadjoinrtactionof. WVeclaimthatA1jiscompletelyreducibleas-module.QIndeed,`letDUbSeanrylocally nite-module,`let12wb ,andDletU@2()E=1fu2U:9s>0DsucrhQthatp(gcz*(gn9))2sn<(u)\=08gQ2g.8ThenU=2PKbU@2()(seeforinstance[D9,gTh.81.3.19]).Norw,QA0VUR(A1)2""(A1)2(") H,butbryo(4.16)"w,theyareallthreeequal.8Theclaimfollows.AF?q!yA!CHARA9CTERIZA:TIONOFQUANTUMGROUPS17j QHence}A1 =OA0ϟLd'1i)u(A1)28:i.+BythetheoremofTVaftandWilson[M =,Theorem5.4.1],A1 =ʍQA0j+(L UXgI{;h2&f]PgI{;h c(A)).8HenceA1V=URA0juYMflgI{;h2;"6=2PKb8b-PgI{;h c(A)  ;$hQandthelemmafollorws.WQNotethat(4.16)%Xholdsforgenericbraidings,sinceinthiscasenoidڹ(gi)isaroSotof1.QIf1A;Bj7areHopfalgebras,SwredenotethesetofallHopfalgebraisomorphismsfromAtoBbryQIsom(A;B).獑QTheorem5,4.4.PLffet%D{andDUV20begenericdataof niteCartantypefor.:ThentheHopfalgebrasQU@(DUV)35andU(DUV20 eG)arffeisomorphicifandonlyifDisisomorphictoDUV20|.QMorffehprecisely,4leta1;;a aresp.a20RA1;;a20y abehthesimplerffoothvectorsinU@(DUV)resp.U@(DUV20 eG)ofQThefforem!4.2,%Tandletg1;;glrffesp.`g2n90RA1;;g2n90ylbffe!thegroup-likeelementsinDw1resp.`DUV20.`ThentheQmapIsom(U@(DUV);U(D 0 eG))UR!IsomAy(D;D 0eE);ӍQgivenbyUR7!(';;( idڹ)), wherffe'(gn9)UR=(g);'(gidڹ)=g20뀍I{(i)&Q;(aidڹ)= ia20뀍I{(i)&Q, forallgË2UR;1iS,isQbijeffctive.iQPrffoof.:Let Vresp.Vp20 xQbSetheYVetter-Drinfeldmoduleofthein nitesimalbraidingofAD:=U@(DUV)Qresp. 0xofA20:=&U@(DUV).Let&ʹ:A!A20`bSeanisomorphismofHopfalgebras. 0xTheninducesQisomorphismsA0V!URA20RA0] andA1!URA20RA1.Hencede nesanisomorphismofgroups':!,andforQallgn9;hUR2;2wb ,alinearisomorphismMPgI{;h c(A) P r԰ 5Z=P'(gI{);'(h)$dG(A 09) '-:1B:QByTheorem4.2, theassumptionsofLemma4.3aresatis edforA;VI{07(i) wE;1@(A 09)x-:07(i)>e;ꦹwith "'(gidڹ)UR=g n90ڍI{(i)&Q;i' 1ι= 0ڍI{(i)&Q;ꦹforall(/1iS:칍QMoreorver,ysince]Uforalli,Pg8:i,r;1Tm(A)28:i PandPg>I{07(i) wE;1@(A)x-:07(i)areone-dimensionalwithbasisai/anda20RAidڹ,thereQarenon-zeroscalars i,2UR|with(aidڹ)= ia20뀍I{(i)&Q,forall1iS.QThenu theelemenrts(aidڹ);1URiS,satisfyu theSerrerelations(4.11)!|,andtheysatisfy(4.12)%ifandQonlyyifthetriple(';;( idڹ))yisanisomorphismofgenericdata.&ThrusthemapIsom$(U@(DUV);U(D20 eG))UR!QIsom(DUV;D20eE)|@inthetheoremiswrell-de nedandinjective.SurjectivityofthismapfollowsfromtheQdescriptionoftheHopfalgebrasU@(DUV)andU(DUV20 eG)inTheorem4.2.)QThemainreasonwhrytheproSofoftheprecedingtheoremworksistheknowledgeofthecoradicalQ ltration.8ThesameideasallorwtodetermineallHopfsubalgebrasofU@(DUV).U?q!Q18YV"NICOL'ҟASTANDR9USKIEWITSCHANDHANS-J'ҟURGENSCHNEIDERj Rƹ5.aJPointedHopfalgebraswithgenericbraidingsQWVe aregoingtoshorwthattheclassofHopfalgebrasdescribSedintheprevioussectionhasanQinrtrinsicdescription.QTheKmfollorwingkeyLemmaimpliesthatpSointedHopfalgebrasbSelongingtoanaturalclassareQgeneratedbrygroup-likeandskewprimitiveelements.SQLemmal5.1.(a). rLffet8S˹=.n2NS׹(n)beagradedbraidedHopfalgebrasuchthatS׹(0).=C:1,ʍQV&:=MS׹(1)Xis nite-dimensionalandgenerffatesSl/asanalgebra.AssumethatSl/has niteGelfand-QKirillov35dimensionandthatVϥhaspffositivebraiding.fiThenS isaNicholsalgebra.Q(b).Lffet_RybeasSin(a),jexceptthatweassumePƹ(RJ)=R(1)_insteffadofgenerationindegree1.QThen35RLisaNicholsalgebrffa.QPrffoof.:(a).2:B(Vp)ַhas niteGelfand-KirillorvdimensionsinceitisaquotientofS׹.2:Assume rstthatQthematrixisindecompSosable.'WVecanthenapplyTheorem2.13; let(aijJ),(d1;:::ʜ;d)andqϹsucrhQthatqijJqjvi 6=URqn92d8:i,ra8:ijforalli6=j{andqii =qn92d8:ii &Ź.QLetiyW6=jӹ.x]WVeclaimthatz29[=adڟcS(xidڹ)21a8:ijw(xjf )=0inS׹.x]Indeed,letz19[=xidڹ,suppSosethatz29[6=0Qandconsiderthetrwo-dimensionalsubspaceWnofSgeneratedbrytheprimitiveelementsz1andz2.QWVeclaimthatB(Wƹ)has niteGelfand-Kirillorvdimension.For,KletT]bSethesubalgebraofSQgeneratedEbryWƹ;|thenthegradedHopfalgebragr(T#C)has niteGelfand-Kirillorvdimension,fandQconrtainsB(Wƹ).QAlso,thebraidingofWnisgivrenbythematrix:'jDSz <1\+qii9 qzn91a8:ij Miiqij?荍MUqzn91a8:ij Miiqjvi9dqzn9(1a8:ijM)-:2 Mii!qzn91a8:ij Mijqzn91a8:ij MjviqjvjԾz!=URz !_qn92d8:iMqn92d8:i,r(1a8:ijM)%cqijɍ Tqn92d8:i,r(1a8:ijM)%cqjviNqn92d8:i,rd8:ia8:ijM+d8:j^z!kQBy8Theorem2.11andLemma2.1,[thereexistsk0sucrhthat1=qn92d8:i,rk6+2d8:i(1a8:ijM)<qijJqjvi,[henceQ0UR=didkŹ+2di(1aijJ)+diaij 6=URdi(kŹ+2aijJ),aconrtradiction.8Thisshowsthatz2V=UR0.QTherefore,wreϧhaveanepimorphismofbraidedgradedHopfalgebrasB(Vp)!S׹,bryStepISIlofQTheorem4.2,whicrhistheidentityindegree1.8HenceB(Vp)UR'S׹.ڎQAssume+norwthatthematrixisdecompSosable.aLeti,jbelongtodi erenrtcomponenrts;3inparticularQqijJqjvi m=1. WVebclaimthatxidxj=qijJxjf xi. Ifbnot,letz1U:=xiu0andnot1forallI),dRf=B(Vp)whereVH22|nX|YD&9hasabasisQxi,2URV2p8:iRAg8:i ;1iS.QSince"grAP%԰ =B(Vp)#|"asgradedHopfalgebras,1andidڹ(gi)%6=1"forall1%iS,it"follorwsfromQLemma4.3thatthe rsttermofthecoradical ltrationofAis+܍Q(5.1)A1V=URA0jПM'؍gI{2;1i.PgI{g8:i,r;ge(A) 8:i:!HVQWVe #canthencrhoSoseai2>Pg8:i,r;1Tm(A)28:i suchthattheclassofainingrqRA(1)coincideswithxidڹ.PLetQy1;;ysXbSefreegeneratorsofG(A).8Itisclearthatrelations(4.9)\and(4.10)%Xhold.QLetiUR6=jӹ.8WVeclaim:Q(i).8Thereexistsno`,1UR`S,sucrhthatgzn91a8:ij Migj\=URg`,z1a8:ij Miwj=UR`.Q(ii).8IfiURjӹ,thenz1a8:ij Miwj\6=".QWVeprorve(i).8Assumethatgzn91a8:ij Migj\=URg`,z1a8:ij Miwj=UR`㎹forsome`.8Then?Raqtn9d8:i,rai?i`6IX=URh`;gidihi;g`iUR=qzn92d8:i,r(1a8:ijM) ܍I)Nghjf ;gidihi;gjf iUR=qzn9d8:i,r(2a8:ijM) ܍I%c;Qwre7concludethat2=aij +Kai`.The7onlypSossibilityisaij 2ܹ=0andl+=i.Thengj N=1whicrhisQimpSossible.QWVenprorve(ii).ĊAssumethatz1a8:ij Miwjٹ=5",i6=jӹ,ijӹ.ĊEvXaluatingnatgidڹ,wreget15=qzn91a8:ij Miiqij =QqiiI(qijJqjvi)21 \|qij U= qiiqn91 jvi ʵ,osoUIthatqii T= qjviJ.xEvXaluatingatgjf ,wreget1 =qzn91a8:ij MjviqjvjL;henceUIqjvj V=qzn9a8:ijM1 Mii.QHence0UR=didڹ(1aijJ)+djf ;thisisaconrtradiction.oQIf~ciUR6=jӹ, then(ad\aidڹ)21a8:ijwaj\2Pig+SI{1aij}i'g8:j;1'(HV)2+S1aij}i8:j' *bryLemma2.1(b).Ifijӹ, takinginrtoaccount9GQ(5.1)x,(i)and(ii),wreseethatthequantumSerrerelations(4.11)%XholdinA.QFinallyV,assumethatizt6jӹ;if06=(ad\aidڹ)aj~2Pg8:i,rg8:j;1J˹(A)28:i,r8:j%,thenidj~="bry(5.1)!~and(i).=SoQthatonaidajjf (gi)ajai,=URijJ(1gidgj)onforsomeij 62UR|,whereij=UR0whenidj\6=".ButwrecanalsoQcrhoSoseij ?=[0whengidgj e=1.wByD(4.7)k,&wrecanrescaleageneratoraiwithij ?6=0toharveij ?=1.QHence,(ijJ)isalinkingdatumfor(aij),g1;:::ʜ;g9and1;:::ʜ;;and(4.12)%Xholds.}7?q!Q20YV"NICOL'ҟASTANDR9USKIEWITSCHANDHANS-J'ҟURGENSCHNEIDERj QWVeharvefoundapSositivedatumD+forandconstructedahomomorphismofHopfalgebrasʍQ'N:U@(DUV)!A. )Norw:gr':gr}U(DUV)!gr}A:isanisomorphismbryTheorem4.2;indeedgr'isQsurjectivreoandtherestrictionofgr'tothe rsttermofthecoradical ltrationisinjective;‚thusgr'Qisinjectivre[M =,Th.85.3.1].Hence'isisanisomorphism.66ፒh9ReferencesfdQ[AS1]N.AndruskiewitschandH.-J.Schneider, !7': cmti10Lifting?ofQuantumLine}'arSpacesandPointedHopfAlgebrasoforderfd b> cmmi10p^ٓRcmr73|s,UUJ.Algebra8"V cmbx10209(1998),658{691.Q[AS2]ff&,UUFinitequantumgr}'oupsandCartanmatrices,UUAdv.inMath.154(2000),1{45.Q[AS3]ff&,%LiftingofNicholsalgebr}'asoftypeA2andPointedHopfAlgebrasoforderp^4|s,%in"HopfalgebrasandquantumUUgroups",ProGceedingsoftheColloquiuminBrussels1998,ed.S.Caeneppel(2000),1{16.Q[AS4]ff&,UUFinitequantumgr}'oupsoverabeliangroupsofprimeexponent,UUAnn.Sci.Ec.Norm.SupGer.,toappear.Q[AS5]ff&,mPointe}'d{HopfAlgebras,minP\RecentdevelopmentsinHopfalgebraTheory",mMSRIP[seriesCambridgeUniv.UUPress;toappGear.Q[CM]CfW.MChinandI.Musson,KThec}'oradical ltrationforquantizeduniversalenvelopingalgebras,KJ.MLondonMath.SoGc.UU53(1996),pp.50{67.Corrigenda,J.LondonMath.Soc.(2)61(2000),319{320.Q[DCK] gC.DeConciniandV.G.Kac,ER}'epresentationsofquantumgr}'oupsatrootsof1,Ein\OpGeratorAlgebras,UnitaryRepresentations,UUEnvelopingAlgebras,andInvqariantTheory",ed.A.Connesetal(2000);Birkhauser,471{506.Q[D]J.UUDixmier,Envelopingalgebr}'as,AmericanMathematicalSoGciety*,Providence,RI,(1996).Q[Dr]nV.UUDrinfeld,Quantumgr}'oups,ProGceedingsoftheICMBerkeley1986,A.M.S.Q[Ji]M.pJimbGo,Aq[-di er}'enceanalogueofU(:%n eufm10g)andtheY;angBaxterequation,Lett.pMath.Phys.10(1985),pp.63{69.Q[K]V.UUKac,In nite-dimensionalLiealgebr}'as,UUCambridgeUniv.Press,Thirdedition,1995.Q[KW]D.KazhdanandH.W*enzl,R}'econstructing Lmonoidalc}'ategories,I.M.GelfandSeminar111{136,Adv.SovietMath.UU16(1993),Amer.Math.SoGc.,Providence,RI.Q[KL]G.[KrauseandT.Lenagan,xGr}'owthjofAlgebrasandGelfand-KirillowDimension,xRevised[edition,GraduateStudiesUUinMathematicsvol.22,Amer.Math.SoGc.,1999.Q[KR]fP*.ZP.KulishN.Yu.Reshetikhin,AQuantumline}'arproblemforthesine-Gordonequationandhigherrepresenta-tions,UUZap.Nauchn.Sem.Leningrad.Otdel.Mat.Inst.Steklov.(LOMI)101(1981),101{110,207.Q[L2]G.UULusztig,Intr}'oductiontoquantumgr}'oups,UUBirkhauser,1993.Q[MiS]_A.MilinskiandH-J.Schneider,iPointe}'dcIndecomposableHopfAlgebrasoverCoxeterGroups,iContemp.Math.267UU(2000),pp.215{236.Q[M]S.UUMontgomery*,Hopfalgebr}'asandtheiractionsonrings,UUCBMSLectureNotes82,Amer.Math.SoGc.,1993.Q[Mu]E.UUMGuller,SometopicsonF;r}'obenius-Lusztigkernels,I,UUJ.Algebra206(1998),624{658.Q[Re]N.щReshetikhin,Multip}'arameterquantumgr}'oupsandtwistedquasitriangularHopfalgebras,Lett.щMath.Phys.20,(1990),UUpp.331{335.Q[Ri]eC.UURingel,Hallalgebr}'asandquantumgroups,UUInventionesMath.101(1990),583{591.Q[Ro] cmmi10K`y cmr10ٓRcmr7b