; TeX output 2001.02.01:18377 YN{N cmbx12SOMEٚPROPERTIESOFFACTORIZABLEHOPFNALGEBRASgK`y cmr10H.-J.UUSCHNEIDER `Z- cmcsc10Abstract.AEAdirectproGofwithoutmodularcategorytheoryis ZgivenIofarecenttheoremofEtingofandGelaki[EG]onthedimen-Zsionsofirreduciblerepresentations.?F*actorizableHopfalgebrasareZcharacterized intermsoftheirDrinfelddouble,andtheircharacterZringsUUandthegroup-likeelementsoftheirdualsaredescribGed.jqƍ61XQ cmr121.- cmcsc10IntroductionBहEtingofandGelaki[EG6]recenrtlyobtainedafundamentalresultin6the RrepresenrtationtheoryoftheDrinfelddoubleofasemisimpleHopf6algebra_.g cmmi12HLorveranalgebraicallyclosed eldofcharacteristic0.qUsing6the*theoryofmoSdularribboncategories,theyshorwed*thatthedimen-6sionofanrysimpleDS(HV)-moduledividesthedimensionofHV.BInthisnote,adirectproSofoftheirresultisgivren.BlMoregenerally6let8(A;RJ)bSeaquasitriangularandsemisimpleHopfalgebra.!;Assume6thatŎ(A;RJ)isfactorizableinthesenseof[RS,](seesection2).ɑThen6bry0~Theorem3.2,AthesquareofthedimensionofanysimpleA-moSdule6dividesH"thedimensionofA.QNThetheoremofEtingofandGelakiisan6immediateIcorollarybrytakingA'=DS(HV)IwiththeusualRJ-matrixof6the6double.ThemainingredienrtsoftheshortproSofofTheorem3.2are6the8classequationofKacandZhru(see[LWs]foranelementaryproSof8), and6Drinfeld'scenrtralelementconstructionin[D9].Drinfeld'sconstruction6isreviewredinsection2.xmHereitisessentialtonote(seeTheorem2.1)6thatօDrinfeld'sproSofof[D9, 3.3]shorwsmorethanwhatisactuallystated6in[D9,3.3].BMoreorvervthispapSerconrtainstwonewresultsabSoutfactorizableHopf6algebrasN7orveranarbitrary eld.cLet(A;RJ)bSeafactorizableHopfal-6gebra.*It;isnotdiculttoseethatDrinfeld'sconstructionde nesan6algebraisomorphismbSetrweenacrharacterringCܞ(A)andthecenter6Zܞ(A)OofA.QByTheorem2.3,nthisisomorphisminducesagroupisomor-6phismUbSetrweenthegroup-likeelementsofthelineardualA2!K cmsy8 aYandthe6ट :ff< B 1991': cmti10MathematicsSubje}'ctClassi cation.Primary:q16W30;UUSecondary:16G10. u Keyoiwor}'dsandphrases.F*actorizable-Hopfalgebras,5irreduciblerepresentations, DrinfeldUUdouble. 营,o cmr91*7 &e62+H.-J.!SCHNEIDERY6हcenrtralgroup-likeelementsofA.(Thislastresultanswersaquestionof6SoniaSNatale.*nWhenAisthedoubleofa nite-dimensionalHopfalge-6braHV,thegroupisomorphismin2.3wrasalreadyobtainedbyRadford6[R93e,PropSositions9and10].BInvsection4,anewcrharacterizationoffactorizableHopfalgebras6is{givren.Let(A;RJ)bSeaquasitriangularand nite-dimensionalHopf6algebra. IfX(A;RJ)isfactorizable,thenitisstatedin[RS,](without6anexplicitproSof8)thatthedoubleD(A)isisomorphictoa2-cocycle6trwist%ofthetensorproSductHopfalgebraA* !", cmsy10 A. Tsang%andZhurecently6rediscorveredthisresultinthecasewhenAisthedoubleofanotherHopf6algebraHV.-TheirproSofusesthespecialstructureofthedoubleD(HV).6InTheorem4.3,Bastrongerresultisobtainedinamoreconceptualwray:6(A;RJ).'isfactorizableifandonlyifthedoubleDS(A)isisomorphic(in6a#spSeci cwray)#toa2-cocycletrwistofthetensorproductAw A.The6proSofof4.3follorwsfromthetheoremonHopfmodules.BThisV¬eoriginatedontheoSccasionofsomelecturesIV garveV&atthe6UnivrersitygofC ordoba,AArgenrtina,ingthefallof98.Iwouldliketothank6Nicol asAndruskiewitscrh,theFOMECandFVaMAFfortheinvitation.͍{\2."TCentralandgroup-likeelementsBहInA]thispapSer,W algebrasandcoalgebrasarede nedorverA]theground-6 eldjkg;)comrultiplicationandcounitofacoalgebraandtheantipSodejof6aHopfalgebrawillbSedenotedbry,#-andS׹.IfAisanalgebra,the6dualTvrectorspaceA2 isan(A;A)-bimoSdulewith(afG)(b) *=f(ba)Tand6(fGa)(b)UR=f(ab)foralla;bUR2A.BLetAbSeaHopfalgebraandR˹aninrvertibleelementinAȂ A.6ElemenrtsǔtinAX AǔwillbSewrittensymbSolicallyastUR=t2|{Ycmr81! Xt22.FVollowing6Drinfeld,thepair(A;RJ)iscalled-@ cmti12quasitriangular5Sif=(1)Q,x2RJ21 x1RJ22.=URRJ21Nx1j RJ22x2,forallxUR2A.=(2)Q,(RJ21N) RJ22.=URRJ21 rS21: RJ22rS22:=(3)Q,RJ21 (RJ22N)UR=RJ21rS21: rS22 RJ22N:6हHereǿR=ͤrMandthesymrbSolicnotationsR=ͤRJ21} A/RJ22 =rS21T rS22 Q(to6indicate:trwodi erentsummationindices)and(x)UR=x1A C=x2areused.6De ne+@R21 M=ERJ22 ֣RJ21=W(RJ)whereb:R ֣R܏!R ֣RDis+@theusual6trwistmap,andH΍QbUR:=R21 Rn=rS 2RJ 1 rS 1RJ 2N:6हNoteUthatb(x) =(x)bU޹forallx 2AU޹sinceRo(andRJ1諍21ˤbSothsatisfy6(1).BThezxquasitriangularHopfalgebra(A;RJ)iscalledfactorizable`([RS,]if6themap/3a A2cmmi8R H=UR:A V!A;(fG):=b 1f(b 2)forallfQ2URA2I|; 7 &ebSOME!PR9OPER:TIESOFFACTORIZABLEHOPFALGEBRAS'=q3Y6हis"anisomorphismofvrectorspaces;eA2isthekg-lineardualofA.NImpSor-6tanrt}GexamplesoffactorizableHopfalgebrasaretheDrinfelddoublesof6arbitrary"g nite-dimensionalHopfalgebras[RS,,Jt2.10](see2.41)bSelorw).BInA[D9],WDrinfeldinrtroSducedabasicconstructionofcentralelements6ofaquasitriangularHopfalgebra.8Asin[D9]let؍U Cܞ(A)UR:=ffQ2A Vj forall03x;yË2A:fG(xyn9)=f(yn9Sן 2r۹(x))g:e6हLet,Zܞ(A)denotethecenrterofA.uThenextcrucialresultisdueto6Drinfeld}[D9,1.2and3.3]. $Horwever}theformrulationbSelowin(b)is6more]Qgeneral;[D9,y3.3]justsarysthatthemapde nesbyrestriction6anůalgebrahomomorphismonCܞ(A).,FVorcompletenesstheshortproSof6willbSerepeated(inadi erenrtnotation).H6Theorem2.1(Drinfeld[D9]).BLffet(A;RJ)beaquasitriangularHopfal-6gebrffa35andUR=R H:A2V!A.fiThen35forallgË2URCܞ(A)andfQ2A2,B(a)35(gn9)UR2Zܞ(A),andB(b)35(fGgn9)UR=(f)(gn9).6Prffoof.Z(a)FVorallgË2URCܞ(A)andx2A,؍Aox(gn9)b˗=URx1b 1gn9(Sן 1 S(x3)x2b 2)Opb˗=URb 1x1gn9(b 2x2S׹(x3))Щ3sincegË2URCܞ(A)c ;ꦹand(x)bUR=b(x)36b˗=UR(gn9)x:ԍBह(b)FVorallfQ2URA2andgË2Cܞ(A),8 (fGgn9)[=URRJ 2NrS 1(fGgn9)(RJ 1rS 2)[=URRJ 2NrS 1fG(R J1ڍ1r S2ڍ1)gn9(R J1ڍ2r S2ڍ2)[=URRJ 2NrS 2s 1t 1fG(RJ 1t 2)gn9(rS 1s 2)벎bry(2)and(3)withR=r=s=t[=URRJ 2NrS 2s 1gn9(rS 1s 2)t 1fG(RJ 1Nt 2)[=URRJ 2N(gn9)t 1fG(RJ 1t 2)36[=UR(fG)(gn9)/since(gn9)iscenrtralby(a):ebcffxff ̟ff ̎ ̄cffH6Lemma2.2.LffetAbea nite-dimensionalunimodularHopfalgebra.6Then35dim(Cܞ(A))UR=dim(Z(A)).6Prffoof.ZLet 72A2bSeanon-zeroleftinrtegral.ThenA!A2,a7!a;6हis $bijectivre.SSinceAisunimoSdular,foralla;b52A,(ab)=(bSן22r۹(a))6[OS,3.2,2)a)].$Henceforalla;x;yË2URA,(a)(yn9Sן22r۹(x))=(aySן22r۹(x))=6(xayn9):pThrusa9z2Cܞ(A)pifandonlyifforallx9z2A,5(xa)=(ax),6thatisaUR2Zܞ(A).Hfcffxff ̟ff ̎ ̄cff 7 &e64+H.-J.!SCHNEIDERYBहIfC[isacoalgebra,8thenG(Cܞ)R=fc2C.j(c)=c c;(c)R=1g6हwilldenotethesetofallgroup-likreelementsofCܞ.O6Theorem2.3.Lffeti(A;RJ)beafactorizableHopfalgebra.ThenR R:6A2V!URA35inducffesbyrestrictionB(a)35analgebrffaisomorphismCܞ(A)UR!Z(A),35andB(b)35agrffoupisomorphismG(A2)UR!G(A)\Zܞ(A):6Prffoof.Z(a)By[R94e,PropSosition3],AisunimoSdular(foranotherproof6seeNH4.4bSelorw).cByTheorem2.1,g0R A۹de nesaninjectivealgebramap6Cܞ(A)UR!Z(A)whicrhisbijectivebyLemma2.2.B(b)?Letf.92:G(A2)=Alg(A,k)1B2.8FThen?f2Cܞ(A)?sincefGSן22 Y=:f;fS6हbSeing theinrverse off inthegroupG(A2).&Hence(fG)UR2Zܞ(A) bry(a),6and=v$(fG)UR=RJ 2NrS 1f(RJ 1rS 2)UR=RJ 2Nf(RJ 1)rS 1f(rS 2)UR=uvn9;o6हwhereuܹ:=RJ22NfG(RJ21);v:=rS21f(rS22).oItfollorwseasilyfrom(3)and(2)6thatLuandv1aregroup-likreelementsofA.+Thusde nesaninjective6grouphomomorphismfromG(A2)toG(A)\Zܞ(A).BTVopprorvesurjectivityofthismaptakegË2URG(A)\Zܞ(A):ByLemma2.26gn921S%ispinCܞ(A)sincegn921iscenrtral.9Thenforallpa2A2;pp(gn921 ʵ)=6p(gn9)g21]inA2;sinceforallxUR2A;܍Go(p(gn9 1 ʵ))(x)UR=p(x1)(gn9 1x2)UR=((pgn9))(g 1x)UR=p(gn9)(g 1)(x);q6हsincex)isaleftinrtegral.bHenceitfollowsfromTheorem2.1(b)that6forallpUR2A2;|(p)(gn9 1 ʵ)UR=(p(gn9 1))=p(gn9)(g 1):6हTherefore+theleftA-moSdulegeneratedbry(gn921 ʵ)isone-dimensional6(sinceo=6:A2!Aisbijectivre).ƞHencethereisanalgebrahomomor-6phismUR:A!kQŹwithp(gn9)=((p))forallp2A2.BItsremainstoshorwthat()UR=g'orsequivXalently((p))UR=p(())6forallpUR2A2.8De neu:=RJ21N(RJ22);vË:=URrS22(rS21):FVorallpUR2A2;܍z*((p))UR=(RJ 2N)(rS 1)p(RJ 1rS 2)UR=p(uvn9);ꦹandTp(())UR=p(RJ 2NrS 1(RJ 1)(rS 2))UR=p(vn9u):o6हThrus>theclaimfollowsfromtheequalityuvS=vn9u:Sincede nesan6algebrahomomorphismCܞ(A)UR!Z(A),()=S׹((S));hencemtxuvË=UR()=S׹(rS 1)(S(rS 2))S(RJ 2N)(S(RJ 1))UR=vn9u;q6हsinceRn=URS׹(RJ21N) S(RJ22N)bry[K "w,VISII.2.4].cffxff ̟ff ̎ ̄cffO6Remark2.4.ڹ1)LetHibSea nite-dimensionalHopfalgebraandAUR=6DS(HV)=H2cop _HMtheDrinfelddoubleofH.WElemenrtsp_ h;p26HV2Z;h2Hof]DS(HV)willbesimplydenotedbryph.ThenHV2copMandH,M7 &ebSOME!PR9OPER:TIESOFFACTORIZABLEHOPFALGEBRAS'=q5Y6हare1dsubHopfalgebrasofDS(HV)andcomrultiplicationandmultiplication6inDS(HV)aregivrenbygx(ph)UR=p2h1j p1h2;ꦹandzhpUR=p(Sן 1 S(h3)h1)h2;e6हforallhUR2HF:;p2HV2 [K "w,IX.4.1].BLet(eidڹ)and(fi)bSedualbasesinHAandHV2Z.ThenD(HV)isqua-6sitriangular҇withRJ-matrixR=#u cmex10P>iiAei[ fi 7a[K "w,L~IX.4.2]. |Thrusb=6टPCQi;jO2fjf ei *ejfi\and&(Fƹ)j=PCi;jfjf eidF(ejfidڹ)&forallF \2jDS(HV)2.^[Since>6theelemenrtsfjf eidڹ,1~i;j+_n;andS׹(fi)S(ejf ),1~i;j+_n;arebases6ofeDS(HV),IS[cffxff ̟ff ̎ ̄cff6Remark3.4.ڹTheNoriginalproSofofCorollary3.3(1)usesthetheoryof6moSdularcategories.YInparticular,Ointhe rstlineoftheproofof[EG6,6LemmaS1.2],>theauthorsneedatraceformrulainmoSdularcategories.6FVoraproSoftheyreferto[Ki f]whoreferstoTuraev'sbSook[T|l].~@The6lucrkyJreadermay ndaproSofin[T|l,)II,3.2.2.(ii)]makinguseofvXarious6trwistingsinthegraphicalcalculus.BHorwever,#inthecaseofthemoSdularcategoryofall nite-dimensional6represenrtations{ofasemisimplefactorizableHopfalgebra(A;RJ)(in6[EG6]kAistheDrinfelddoubleofasemisimpleHopfalgebraHV),this6formrulaisinfactaspSecialcaseoftheclassequation3.1intheversion6ofLorenz[LWs].BLet Vid;0URim;withV0V:=kg,bSeacompletesetofrepresenrtatives6of;theisomorphismclassesofthesimpleleftA-moSdules.,FVoralli,Plet6ibSec8thecrharacterofVidڹ,~Oandi"q% cmsy6 =URS׹(i)bSethecrharacterof(Vi)2. By6de nition,thematrix(sijJ)in[EG6]isgivrenbynsij 6=URidڹ((jv^));0i;j%m;6हwhereUR=R.+9ThetraceformrulaneededintheproSofof[EG6,Lemma61.2]thenis:!9FVorallj,X ㇍4iR sjviJsijv=URdim(A):BहTVo/prorvethisformula,̮letEi* inZܞ(A)bSethecentralprimitiveidem-6pSotenrtDbofVid;0 im.F FVorDballi,Zde neeiR:= 21 \|(Ei),ZandletieR:=O(R21 )21˹=Sן21 S(RJ22N)A RJ21Z=ORJ22 S׹(RJ21)is6anotherRJ-matrixforA[K "w, VISII.2],thatis(A;w*eR ҹ)isagainquasitrian-6gular,thereisalwraysasecondprojection$a6cmex8eR H:URDS(A)!A.BFinallyV,if)%UR:A!B6/isahomomorphismofbialgebras,thenA2co%޹:=6fxUR2Ajx1j %(x2)=x 1g.56Lemma4.1.Lffet2(A;RJ)bea nite-dimensionalquasitriangularHopf6algebrffa.De ne@Ë:=URR;6e mչ:=eR;:=R,qand@letiUR:DS(A)2coʕ!D(A)6bffe35theinclusionmap.fiThenB(1)35 UR:A2V!DS(A)2co uC, (fG):=f2S׹(n9(f1));isbijeffctive.B(2)35S׹UR=?e mi :A22  pV (!|DS(A)2co2(ipʕ!D(A)2epUR! A:6Prffoof.Z(1)Themap iswrell-de nedsinceforallfQ2URA2,QeC (fG)1j n9( (f)2)UR=f4S׹(n9(f1)) n9(f3S׹((f2))UR= (fG) 1;6हsinceI,isaHopfalgebrahomomorphism. Theinrverseof isgivren6bry,id= ׼",ݹsinceforallf2A2,=j(id ` ") (fG)=f;,ݹandifx=PpiՈfidai*2`L7 &e68+H.-J.!SCHNEIDERY6DS(A)2co uC,Qfi,2URA2빹and+ai2A+foralli,QthenPbi;:fi2ai1 $n9(fi1ai2)=x$ 1;6हhenceh;s>xUR=X ㇍ ifi2ai1Sn9(ai2)Sn9(fi1))UR= (X ㇍8?iUVfid"(ai)):'Bह(2)FVoranryfQ2URA2;jڍGeGjN&( (fG))tU=?eUR m׹(f2f1(RJ 1N)S׹(RJ 2))OptU=URf2(Sן 1 S(rS 2))rS 1f1(RJ 1N)S׹(RJ 2)tU=URS׹(RJ 2NS 1 S(rS 1)fG(RJ 1S 1 S(rS 2))tU=URS׹(RJ 2NrS 1fG(RJ 1rS 2))since,t&Sן 1 S(rS 1) Sן 1 S(rS 2)UR=rS 1: rS 236tU=URS׹((fG)):Nbcffxff ̟ff ̎ ̄cff6हLeta&HN|bSeabialgebrawithcomrultiplicationandaugmentation". An6inrvertibleelementF2]H} 'Hriscalleda2-cffocycleforFHroragauge6trffansformationXif_fF12 ( id uL)(Fƹ)==URF23 (id uL )(Fƹ)inH H HF:;ꦹandOp("(FƟ 1aʹ)FƟ 2==UR1=FƟ 1a"(FƟ 2):6हIf6{FAisa2-coScycleforHV,IpthenthetrwistedbialgebraH2F 3 isde nedas6follorws.8HV2F Q=URHasanalgebrawiththenewcomultiplication}FO(x)UR:=Fƹ(x)F 1forall3.(xUR2HF:;6हandGHtheoldaugmenrtation". NIfH4isaHopfalgebra,qthenHV2F Cisa6HopfcalgebrawithanrtipSodecSFO(x):=vn9S׹(x)v21nforallx2HV,ђwhere6vË:=URFƟ21aS׹(FƟ22);vn921 =S׹(G21)G22;G:=FƟ21(see[K "w,XV.3andXV.6]).BTheoremM2.9in[RS,]describSestheDrinfelddoubleofafactorizable6Hopf"algebraAasa2-coScycletrwistoftheusual(componenrtwise)"tensor6proSductbialgebraA A꨹withoutgivinganexplicitproof.BInBthecasewhenA!=DS(HV)BisitselftheDrinfelddoubleofanother6Hopf^algebraHV,zTsangandZhru[TZ,Theorem2]recenrtlyrediscovered6thisresult."MTheirproSofdependsonthestructureofAasthedoubleof6HV.BIntthenexttheoremastrongerresultisshorwnbyusingtheconcept6ofHopfmoSdulesinthenextlemma. FVactorizableHopfalgebrascanin6factbSecrharacterizedbythepropSertythattheirdoubleisisomorphic6(inaspSeci cwray)toa2-cocycletrwistoftheusualtensorproduct6bialgebra.;6Lemma4.2.Lffet&A;BandDzbebialgebrasand'UR:D!A; Ë:D!B6bialgebrffamaps.bDe neù:CD!A BNbys2(x)C:='(x1) n9(x2)for6all35xUR2DS. q7 &ebSOME!PR9OPER:TIESOFFACTORIZABLEHOPFALGEBRAS'=q9YB(1)35If(DS;RJ)isaquasitriangularHopfalgebrffa,thenY1F:=UR1 n9(S׹(RJ 1N)) '(RJ 2) 1<6isa2-cffocycleforA4 BUwithcffomponentwisebialgebrastructure,and6Ȅ:URD!(A B)2F Bis35abialgebrffamap.B(2)"AssumethatBisaHopfalgebrffaandthatthereisabialgebra6homomorphismg 6m:QB*W!Dwith n9 6m=id(Bw.:Then:D!A B6isUbijeffctiveifandonlyiftherestrictionof'de nesanisomorphism6bffetween35DS2co andA.1x6Prffoof.Z(1)canbSecrheckeddirectlyV.B(2)&ItiseasytoseethatȄ:URD!A Bֹis&amapofrighrt(B;B)-Hopf6moSdules.vHerejD=isarighrtB-modulebryrestrictionvia n:URBX!D,and6aprighrtB-comoSduleby(id & [ n9);)A BdvisparightB-comoSdulebyid [,6andېtherighrtB-moSdulestructureisgivenby(a b)cUR:=a'( (c1)) bc26हfor]allaUR2A]andb;cUR2B.Hence]brythetheoremonHopfmoSdules[M =,61.9],isbijectivreifandonlyifDS2coB=DS2co  !(A B)2coBP԰=A;x7!6'(x);꨹isbijectivre.cffxff ̟ff ̎ ̄cff6Theorem4.3.Lffet (A;RJ)bea nite-dimensionalquasitriangularHopf6algebrffa.2De newKFR :=h1 RJ22e RJ21 135andR :hDS(A)!A A;xh7!6R(x)K:=e dV(x1) n9(x2)35withC :=KR &and"*e!/W:=eR. 34ThenFR aisa62-cffocycle35forA A35withcffomponentwise35bialgebrastructure,<R H:URDS(A)!(A A) FX.R6is35aHopfalgebrffahomomorphism,andthefollowingareequivalent:B(1)35R &isbijeffctive.B(2)35(A;RJ)isfactorizable.6Prffoof.ZLet(eidڹ)and(fi)bSedualbasesofAandA2.8ApplyLemma4.26withfD:=URDS(A)andRJ-matrixPivneii fidڹ,B':=?e m׹, Ë:=n9,andf n:A!6DS(A)theinclusionmap.8The2-cocycleofLemma4.2(1)thenis֕IX ㇍Pi]1 n9(S׹(eidڹ)) e -(fi) 1ۦ)=URX ㇍ i1 S׹(eidڹ) fi(RJ 2N)S׹(RJ 1) 1 0Xۦ)=UR1 S׹(X ㇍8?iUVeidfi(RJ 2N)) S(RJ 1N) 1􍍍ۦ)=UR1 S׹(RJ 2N) S(RJ 1N) 1ۦ)=URFR:-BहByLemma4.2(2)andLemma4.1,*conditions(1)and(2)arebSoth6equivXalenrttothebijectivityofthemapDS(A)2coʕ!URA;x7!?e m׹(x):#Bcffxff ̟ff ̎ ̄cff6Remark4.4.ڹRadfordU[R94e, PropSosition3]shorwedUthatafactoriz-6ablebHopfalgebraisunimoSdular.}Anotherproofofthisresultfollorws 7 &e610+H.-J.!SCHNEIDERY6हfromtheisomorphismDS(A)PUR԰n:=(A A)2FX.RinTheorem4.3andtheuni-6moSdularitry ofthedouble[R93e,Theorem4].Letbealeftinrtegralin6A.ݵThen&| isaleftinrtegralin(A A)2FX.R .ݵSinceDS(A)isunimodular,6 isarighrtintegral.8HenceisarightintegralinA.ʍReferences6ल[D]KnV./G.Drinfeld,٥OnIalmostc}'ocommutativeIHopfalgebr}'as,Leningrad/Math.J. Kn."V cmbx101UU(1990),321{342.6[EG]PP*.;EtingofandS.Gelaki,uSomehpr}'opertiesof nite-dimensionalsemisimpleKnHopfalgebr}'as,UUMathematicalResearchLetters5(1998),191{197.6[K]KnC.UUKassel,QuantumGr}'oups,Springer,GTM155,NewY*ork1995.6[Ki]KA.fA.KirillovJr.,"0OnYaninnerpr}'oductYinmo}'dularcategories,"0J.fAmer.Math.KnSoGc.UU9(1996),1135{1169.6[LR]O QR. 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