; TeX output 2001.05.10:20027 *6Aacmr6Con9temp1oraryMathematicsfeY:N cmbx12Pointedindecomp`osableHopfalgebrasoverCoxetergroupsz'9 K`y 3 cmr10AlexanderfMilinskiandHans-JM8urgenSc!hneider,@!"V cmbx101.҆In9troQductionHK`y cmr10EverypGointedHopfalgebracanbGedecomposedasacrossedproductofits 6linkindecompGosablecomponentcontaining1(aHopfsubalgebra)andagroup6algebra[M2]. \jThismeansthatthestructuretheoryofpGointedHopfalgebras6can%bGereducedtothelink-indecomposablecase.tHowever,Ythe%structureofsuch6link-indecompGosable;HopfalgebrasA,inparticularinthecasewhen b> cmmi10Ais nite-6dimensionalandthegroupofitsgroup-likeelementsG(A)isnon-abGelian,Aisnot6known.US.W*estreichhasshownthatifAisapGointed,link-indecomposable,and6quasitriangularHopfalgebraovera eldofcharacteristiczero,UPthenthegroup6वG(A)%ofitsgroup-likeelementsisabGelian[W ㊲].7Moreoveritseemsthatall nite-6dimensional,%link-indecompGosableHopfalgebraswhichhaveappGearedintheliter-6atureUUsofardohaveUUanabGeliangroupofgroup-likes.HInIthispapGerwepresentageneralconstructionwhichassoGciatestoanyCoxeter6groupeW;apGointedlink-indecomposableHopfalgebrawithgroup-likeelementsW6ल(overanarbitrarybase eld% msbm10|).ɘIfW+isoneofthesymmetricordihedralgroups6वDٓRcmr73|s(=S3),D4,S4orvrS5,thevrHopfalgebrais nite-dimensional(seeSection6).6W*edonotknowwhetherourconstructiongivesa nite-dimensionalHopfalgebra6wheneverUUtheCoxetergroupis nite.HInֽSection4weshowthatlink-indecompGosabilityisanin nitesimalpropGerty6in\thefollowingsense.!LetAbGeapointedHopfalgebrawithcoradical|G, ^G#=6वG(A)5thegroup-likeelementsofA.gAsin[AS1N]weconsidertheassoGciatedgraded6Hopfalgebragr ½(A),wherethegradingisde nedbythecoradical ltration.The6coinvqariantelementsR²ofgr ߧ(A)withrespGecttothepro8jectionontothedegreezero6part Lgr (A)(0)=|G LformaHopfalgebrainthebraidedcategoryofY*etter-Drinfeld6moGdules~over|G(see[M1]forthenotionsofbraidedcategoriesandY*etter-Drinfeld6moGdules).:PTheprimitiveelementsV=Pc(RDz)ofR·areaY*etter-DrinfeldsubmoGdule.6ThenMAislink-indecompGosableifandonlyiftheelementsg !", cmsy102d۵Gwithnon-zero6homogeneouscompGonentV 0ercmmi7gKeygwordsandphrases.@PÎointed0Hopfalgebras,Glink-indecomp2^څGbڅG _Y}Dthebraiding6वc:VP lW-!W V)isde nedbyc(vE wD)=(gEwD) v[ٲ,|v%θ2Vg;wظ2Wc.CThe6वG-gradationZandG-actiononVu^ ղ(v1_H pQ vnq~)o:=(gյv1|s) pQ (gvnq~),gո2oG,6वv1|s;:::;vn2xV8;Fifvieĸ2VgO \cmmi5i;1in,thenv1 VW Vvn2Vg,wheregmQ=g1':::|jgnq~.6ThereaisauniquecoalgebrastructureonTc(V8),̒suchthatv"2P(T(V8))forallv"2V6लandTc(V8)isabialgebrain^GbG ǸY}D",thatis(` )(1 c 1)( )=,wherecis6theybraidingoftheY*etter-DrinfeldmoGduleTc(V8)asde nedabove.S)F*oryexamplewe6have(v[wD)=1j^ vwA+v7 w+(g7wD) v+v[wA 1forv"2Vg;w 2V8.O\W*enotethat6the*jcanonical ltrationofTc(V8)isacoalgebra ltration.cyThereforethecoradicalof6theJYcoalgebraTc(V8)equals|,LandT(V8)andallitscoalgebraquotientsarepGointed6irreducible̕coalgebras[M1,5.3.4,5.3.5].D2In̕particularthisimpliesthattheidentity6ofPTc(V8)isconvolutionPinvertibleby[M1,5.2.10].UHenceTc(V8)andallitsbialgebra6quotientsin^GbG Y}D̽areHopfalgebrasin^GbG Y}D.&NotethatTc(V8)isaGN-graded6vectorUUspaceaseachTc(V8)(n)hasabasisconsistingofG-homogenouselements. @HW*eFconsiderthesetx䍑J?eS [ofallidealsandcoidealsIofTc(V8)whicharegenerated6asidealsbyN-homogeneouselementsofdegree2.0LetSQbGethesubsetofallI2x䍑ʧeS 6लwhichareY*etter-DrinfeldsubmoGdulesofTc(V8),thatisstableundertheactionofG6लandgeneratedbyGiN-homogeneouselements..Bothsetsareclosedunderaddition Ǎ6andcontainthezeroideal.!HencethesumsI(V8)v=PIJ2SMIandx䍑;eȵI(V)=PamJ2;"#cmex7eS hJ 6लareUUthelargestelementsinSQandx䍑XeS #.HF*orfallIj2S,&6denoteUUby,_vi;j 2:RDz(i8+j)T΍1+2-!RL R?xDi* j߸>/!"RDz(i) R(j);i;jY0;>S6लtheUU(i;j)-thcompGonentofthecomultiplication.qMoregenerally*,W썑ai1 ;::: ;ik:RDz(i1S+8g+8ik됲)!R(i1|s)8 g R(ik됲);i1|s;:::;ik0;6लdenotesUUthe(i1|s;:::;ik됲)-thcompGonentofthe(kw81)-folditerationof.bOHLemma2.3.L}'etkR߲=L ㌟n0<&RDz(n)beagradedcoalgebrawithgroup-likeelementfd61andRDz(0)=|1.Thenthefollowingar}'eequivalent:H(1)P(R)=R(1).H(2)F;oralln2;1;::: ;1:RDz(n)!R(1)^ nEfisinje}'ctive.H(3)F;oralli;jY0,i;j 2:RDz(i8+j)!RDz(i) R(j)isinje}'ctive.H(4)F;oralln2,n1;1}:RDz(n)!R(n81) R(1)isinje}'ctive.HProof.pZLet.x1ȸ2RDz(n);n2;and(x)=Pލn%i=0yi?whereyi=i;ni(x)26वRDz(i) R(ni)|Jforall0in.F*or|Jallm1,let|Jm bGethecompositionZ6m _:RT΍Drm12߸䍍>!"}Rǟ^ m k@L mj^1n{⍍>}!+JRDz(1)^ m ؜:ڲHence1;::: ;1oistherestrictionofm utoR(m).6ThenUUforall1in81,Pnq~(x)=(i, 8ni˲)((x))=(i 8ni˲)(yiTL):6लThis0formulashowsthat(1))(2)byinductiononn(asin[AS2N,Lemma5.5]),fd6andUUalso(2))(3).<נ7 UWPOINTEDINDECOMPOSABLEHOPFALGEBRASOVERCOXETERGROUPS@5YHल(3)ָ)(4)istrivial,ֶand(4))(1)isclearsinceby(4),RDz(n)}\Pc(R)s=0forfd6allUUn2:7qffdffYffff*HलAllthepreviousresultsholdmoregenerallyforY*etter-DrinfeldmoGdulesover 6HopfZfalgebraswithbijectiveantipGodeZfinsteadofgroupalgebras.W*enowrecallan6impGortantUtoolintroducedbyNichols[N,U3.3]todealwithB(V8)overgroupalgebras6without knowingtherelationsexplicitly*.<(LetV2^žGbžG Y}DbGeof nitedimension.6W*e؍choGoseabasisxid2Vgi j@withgi2G;1iG;؍ofG-homogeneouselements.H/Let E6वI2S TandIXR߲=Tc(V8)=I.mThenR]isagradedHopfalgebrain^GbG 2Y}DٲwithRDz(0)=|16andoRDz(1)=V8.K{F*orall1i)letid:R߸!R6bGethealgebraautomorphismgiven6byUUtheactionofgiTL.VHProposition2.4.'L}'etR;xiTL;i;1iasab}'ove.H1)F;orall1<iG,ther}'eexistsauniquelydetermined(id;iTL)-derivation6वDid:R߸!RwithDiTL(xj6)=i;j (Kr}'onecker`)forallj.H2)I=I(V8),thatisR߲=B(V8),ifandonlyifTލ >% >i=15ker(rز(DiTL)=|1:HProof.pZ1)ZDF*orall1PiG,[letZDDiTL(1)=0,andZDde neforallxP2RDz(n);n1;6लelementsUUDiTL(x)2RDz(n81)byQvn1;1 e(x)= X tմi=1㉵DiTL(x)8 xi:v6लThenUUforalln;m1;x2RDz(n);y"2RDz(m);fGO(xy[ٲ)=(x)(y)d<=(x8 1+DiTL(x) xi,+:::/)(y 1+Di(y[ٲ) xi,+:::/)fdd<=xy 81+xDiTL(y[ٲ) xi,+Di(x)(gi,y[ٲ) xi+:::6लbythede nitionofthebraidedmultiplicationinR RDz.WHenceDiTL(xy[ٲ)=xDi(y[ٲ)+6वDiTL(x)i(y[ٲ):\SincetheelementsxjJareprimitive,^qDiTL(xj6)==ij gfor\alli;j,andDiis6uniquelyUUdeterminedbythisequationsinceRiisgeneratedbyx1|s;:::;x7.H2)g9ByLemma2.1,IXs=I(V8)ifandonlyifPc(RDz)=R(1).rHenceg9theclaim6followsh)fromLemma2.3,lsinceTލ % i=1wker(8(DiTL)z=|1h)ifandonlyifn1;1risinjective6forUUalln2.(g6ffdffYffff*HF*orUUlaterusewenoteVHLemma2.5.L}'etV9;xiTL;Di;1iG;b}'easinProposition2.4.H1)>F;orall1iG;>letB(V8)^(i) ;b}'ethesubalgebraofB(V8)generatedbyall6वxj6;1}iG;j6=i.Assume?>a;b2B(V8)^(i) ӌwith?>a+bxiQɲ=}0.Thena=0and6वb=0.H2)!L}'et1i1|s;:::;in8G,8n1,and!ik6=il2forallk6=l2`.sRThenxi1;lxi2:::;cxin и6=60inB(V8).HProof.pZT*o`prove1)notethatDiTL(B(V8)^(i) N)V=0.sHenceb=DiTL(ar<+bxi)V=0.62)UUfollowsfrom1)byinductiononn.ʭffdffYffffQI7 zQ6ऱ6z$gALEXANDERMILINSKIANDHANS-JV;URGENSCHNEIDERY_$3.ndNon-categoricalTsplittingofY etter-DrinfeldmoQdulesHलInȽthissectionweuseabraidedversionofthefundamentaltheoremonHopf 6moGdulesUUtoproveUUfreenessofcertainextensionsofY*etter-DrinfeldHopfalgebras.HLet4LbGeaHopfalgebrawithbijectiveantipGode,lյHa4Hopfalgebrain^LbL 1cY}D6ल(see[M1,10.6.10]forthegeneralde nition)andV߲arightH-moGduleandaright6वH-comoGduleEQwithstructuremapsV :VQ ٵH!V~5andEQV:V!VQ ٵH.lqThenEQV6लisSarightH-Hopfmo}'dule,T9ifthebraidedHopfmoGduleaxiomholds,thatisifV Eis6a+rightH-linearmapwheretheH-moGdulestructureofV His(V߸ )(id c 6वid)(idA id ).Here,ebanddenotethemultiplicationandcomultiplicationof6वH.qNoteUUthatwedonotassumethatV9isaY*etter-DrinfeldmoGdule.ÒHLemma3.1.L}'etiLbeaHopfalgebrawithbijectiveantipode,r H9aHopfalgebrafd6in^ôLbôL (Y}D?w,ϧandôVarightH-Hopfmo}'dule.(De neV8^coHq=fvy2VVjVɲ(v[ٲ)=v\ [1g.6ThenV8coHӸ 8H!V9; v h7!v[h;86isbije}'ctive.HProof.pZThisUUisshownasforusualHopfalgebras(see[M1,1.9.4]).';ffdffYffffHTheorem3.2.%L}'et]L^0andLbeHopfalgebraswithbijectiveantipode,zand >:6वL^0u"!LsaHopfalgebr}'amap. ;LetR#:beabialgebrain^LbL Y}D6,.WRǟ^0saHopfalgebrain^6ऴLr0b6ऴL0?Y}DO,ܲand#i0:Rǟ^0!R,:RD^!Rǟ^0#algebr}'a#andcoalgebramapssuchthati0=id.6AssumethatiisanL^09-line}'armap, Yandthati;areL-colinear, YwhereRisaleft6वL^09-mo}'duletandRǟ^0taleftL-comodulebyrestrictionvia z.µ!#&R Rǟ^0.pHereweusedfd6thatiisanalgebramapandisacoalgebramap.!ThenonechecksthatRisa6right=Rǟ^0-HopfmoGduleusingthatiisacoalgebramapandL^09-linear, isanalgebra6mapandL-colinear, andiV=id.)mHencethemultiplicationmapKQ Rǟ^0V!VRis6bijective6by3.1.gMoreoveritfollowsfromthede nitionofKthat(K)RŸ K.6T*oseethatKWisasubalgebraofRDz,onechecksthat(idj i)x䍑 :CRWJ!R~k jRòisan6algebra#map,wherethealgebrastructureonRh TRisgivenbythebraidingofRDz,6usingUUthati;areL-colinearalgebramaps.UffdffYffffWyHTheorem3.2istrueinamoregeneralversionwhichonlyusesthebraidings 6वc:Rj VR%!R VRand!c^0:Rǟ^08 Rǟ^0!Rǟ^08 Rǟ^0d!of!RandRǟ^0.+Itsucestoassume6thati;arealgebraandcoalgebramapswithi=id,whichsatisfythefollowing6identities:q(i8 id)c^09( id)=(id8 )c(id i),UUandc(i8 id)=(id8 i)c:HलIfdXce=Lލ1%n=0! Xn xisagradedvectorspacewith nite-dimensionalcompGo-6nentsXnq~,kZn0,theHilb}'ertseriesofXnistheformalpGowerseriesPX$(t)=6टPލAn߷1%Anߴi=0Pֲdima8(Xnq~)t^n.d 7 UWPOINTEDINDECOMPOSABLEHOPFALGEBRASOVERCOXETERGROUPS@7YHCorollarUTyL3.3.L}'et7;LbeaHopfalgebrawithbijectiveantipode,`V'ݸ2^LbL mY}Dj,fd6andRL^0YLaHopfsub}'algebraRwithbije}'ctiveantipode,mandV8^0vYV6asubspacesuch6thatL^09V8^05V8^0,b`(V8^0)L^0 ZV8^0,bwher}'e':V!LZ V-disthec}'oactionofV8.dtThenV6isCaleftL^09-mo}'dulebyrestrictionandV8^0JisaleftL-comodulevia`.~Let':V!V8^06b}'eanL^09-linearandL-colinearmapwith'jV8^05=id.HThenŵV8^0isaY;etter-Drinfeldmo}'duleoverL^09,ԽB(V8^0)isagradedsubalgebraof6B(V8),andB(V)isafr}'eerightB(V8^0)-module._IfVis nite-dimensional,then6वP:(X&eufm7B(V0mW) dividesP:B(V)K.HProof.pZTheWtensoralgebrasTc(V8)andT(V8^0)areHopfalgebrasin^LbL ˸Y}Dqand^6ऴLr0b6ऴL0?Y}DS-aasinthe rstsectionwheretheelementsofVUkandV8^0 #areprimitive.^The6naturalUmap0:Tc(V8)!T(V8^0)Ugivenby'0:V!V8^0]isUanalgebraandacoalgebra6mapFwithi=id,|whereFi:Tc(V8^0)!T(V8)Fdenotestheinclusionmap.BMoreover,|6लisdL-colinearandL^09-linear,whereTc(V8^0)isanL-comoGduleandT(V8)isanL^09-moGdule6via7theinclusionmap O :EL^0ɸ!L.UmTherefore7inducesanL-colinearalgebraand6coalgebraHmapB(V8)!B(V^0)HwhichwillagainbGedenotedby.mXSinceB(V8)and6B(V8^0)aregradedalgebrasandcoalgebrasandisagradedalgebramap(ofdegree60), K~4=B(V8)^co+isW;agradedsubalgebraofB(V).HenceweconcludefromTheorem63.2VwithR߲=B(V8)andRǟ^0=B(V8^0)thatKL B(V8^0)T͍+3= UNB(V8).VrInparticular,B(V)6isUUfreeoverUUB(V8^0),andifV9is nite-dimensional,P:B(V0mW)#PK |˲=P:B(V)K.0ffdffYffffʍHWhenB(V8^0)is nite-dimensional,Corollary3.3alsofollowsfrom[G1ʥ,Theo- 6rem&3.11]withacompletelydi erentproGof.;ButtheargumentinTheorem3.2is6quiteUUgeneralanddoGesnotdependonNicholsalgebras.HAnexampleofthesituationofCorollary3.3willbGegiveninSection5.AThe6nextremarkshowsthattheexistenceofthesplittingmap'inCorollary3.3isa6ratherUUmildassumption.HRemark3.4..̲LetL4GbGeagroup,NG^0QGa nitesubgroupsuchthatthechar-fd6acteristicaof|doGesnotdividetheorderofG^09.-LetV2^GbG Y}DandV8^05VEasubspace6suchzthatG^09V8^05V8^0,&@and`(V8^0)|G^0d +V8^0,&@wherez':V!|G+ VS^isthecoaction6ofV8.Thenthereisalwaysa|G^09-linearand|G-colinearmap'\:VL@!V^0 #ϲwith6व'. jV8^0 5==id. Hence,weareinthesituationofCorollary3.3withL^0Y=. |G^0and6वL=|G.HProof.pZSince|Giscosemisimple,thereisa|G-colinearmapfڧ:V!V8^0with6वfٸjJV8^0g=id.VDe neڵ':V.!V8^0by'(v[ٲ)= 1}&feoord (G0s)"EǟP,g@L2G0Awgf(g^1 Mv)forallv#2JV8.6Then'isaG^09-linearmapwith'jV8^05=id,#and'is|G-colinear,sinceforallh2G6लandεv"2Vh.,*bytheY*etter-Drinfeldconditiong[ٟ^1 Mv2V㐴g@L1 hg2andg[f(g^1 Mv)2V^80vhNUfor6allUUg"2G^09.4ffdffYffffȍxD4.Link-indecompsableTpQoin9tedHopfalgebrasHलLet!AbGeapointedHopfalgebrawithgroup-likeelementsGi=G(A).*F*or 6वg[;h2G,0Pg@L;h (A)=fx2Aj(x)=g8  x+x 1g'uwilldenotethe(g;h)-primitive6elementsUUofA.qǵPg@L;h (A)iscallednon-trivialif|(g8h)$Pg@L;h(A).xɠ7 zQ6ऱ8z$gALEXANDERMILINSKIANDHANS-JV;URGENSCHNEIDERYHलW*eWUrecallthede nitionofthequiverA OofAWUin[M2].wTheverticesofA are 6the@elementsofG;forg[;hO2G,{Ithere@existsanarrowfromhtogXifPg@L;h (A)is6non-trivial.HNote9thatPg@L;h (A)isnon-trivialifandonlyifP㐴g@Lh1 Qϴ;1в(A)orP㐴h1 Qϴg@L;1(A)isnon-6trivial.[gThen5g[;hH2Garec}'onnected(orgsHh)iftheyareinthesameconnected6compGonentJofA ڲasanundirectedgraph. iTheHopfalgebraAiscalledlink-6inde}'composableUUifanytwoelementsinGareconnected.HW*e`callSupp8H(A)&=fg2GjPg@L;1 z(A)UUisnon-trivial;@g`thesupp}'ortofA. oIf6वV²=ޟLRg@L2G jVgԲisaleft|G-comoGdulewithhomogeneouscomponentsVg, kthenwe t6de neUUSupp=(V8)=fg"2GjVgn6=0g.HLetA0H=|GA1:::KbGethecoradical ltrationofAandgr ܗ(A)=A06वA1|s=qA0S8A2=A1S8:::ㄲtheUUassoGciatedgradedHopfalgebra.qThenOgr?(A)T͍+3= UNRDz#|G;6लwhere)R@=y(gr(A)^co^A0䆲isthediagr}'amofA;<RisagradedHopfalgebrain^GbG Y}D *6लandUUPc(RDz)isin^GbG /Y}DӲ(see[AS1N]).㍍HLemma4.1.L}'etDAbeapointedHopfalgebrawithG8=G(A)DanddiagramR.fd6De neN3=fg"2Gjgs1g,andV=Pc(RDz).ThenH1)쪵Nisanormalsub}'groupofGandthec}'onnectedcomponentsofA p:arethe6c}'osetsofN.H2)Nisgener}'atedbySupp(A).H3)Supp(A)=Supp(V).HProof.pZ1)visnotedin[M2,~3.2].Byde nitionthereexistsanarrowbGetween6वg[;h 2Gnifandonlyifgh^1+)orhg^1W2 Supp(A).SuppGosegs 1.Thenthere6areg1|s;:::;gn I32׵G;n1;g1T(=g[;gn I3=1withgiTLg䍐[ٷ1i+1L2Supp(A)or(Supp8(A))^16लforalli,Qandg=cg1|sg䍐[ٷ12 Mg2g䍐[ٷ13 :::Dgn1g^[ٷ1፴n9isgeneratedbySupp(A).Conversely*,Qif6वu1|s;:::;un1u 2Supp(A)or(Supp8(A))^1 t,n1,andg"=u1':::|jun1,de neg1C=g[ٲ,6and@gi+1Ѳbyuid=giTLg䍐[ٷ1i+1 tO;1in:1.jThen@g[isconnectedtogn8=1.Thisshows62).nF*or3),notethatPc(RDz)#|GT͍S+3S=nDA1|s=qA0&by[AS1N,Lemma2.4].nF*orallg[;hS2G6लchoGoseadecompositionofvectorspacesPg@L;h (A)sk=|(g٧}εh)Pg@L;h(A)^09.Thenby6theaTheoremofT*aftandWilson[M1,d5.4.1],LQg@L;h2G'~Pg@L;h (A)^0T͍+3=LA1|s=qA0.Thusafor6allUUg"2GweobtainanisomorphismPc(RDz)g#1T͍+3= UNPg@L;1 z(A)=|(18g[ٲ).=TffdffYffff7OHAs)suggestedbythepreviousLemma,2wesayaY*etter-DrinfeldmoGduleVbҲover 6वGUUislink-inde}'composableUUifGisgeneratedbySupp(V8).HProposition4.2.'L}'etGbea nitegroup.Thenthefollowingareequivalent:H(1)Ther}'eexistsa nite-dimensionalpointedandlink-indecomposableHopfal-6gebr}'aAsuchthatGT͍+3= UNG(A).H(2)BTher}'eexistsalink-indecomposableY;etter-DrinfeldmoduleV#&overGsuch6thatB(V8)is nite-dimensional.HProof.pZAssumeo(1)andletR6bGethediagramofA.8ThenVn'=5CPc(RDz)islink-6indecompGosableUUbyLemma4.1.qBy2.22),|[V8]T͍+3= UNB(V)UUis nite-dimensional. 7 UWPOINTEDINDECOMPOSABLEHOPFALGEBRASOVERCOXETERGROUPS@9YHलAssumeOv(2).oDe neR߲=B(V8),PandletA=RDz#|GbGetheRadfordbiproduct.fd6ItDiswell-knownDandnotdiculttoseethatPg@L;1 z(A)V\=Pc(RDz)g#1ؕ+|(g4n1)D(if6वxS2Pg@L;1 z(A),PwriteQ("R ? pݵid)(x)= z(g̶1); \2|;QandcheckQthatx z(g̶1)S26वPc(A)^co|Gg().i>Hence;Supp(A)=Supp(P(RDz))=Supp(V8)generatesG,@andAislink-6indecompGosableUUby4.1.D1ffdffYffffC@@V5.ORLink-indecompQosableTpoin9tedHopfalgebrasoverCoxetergroupsHलLetn9WȲbGeagroupandT*Wasubsetsuchthatforallg"2W;t2TV;g[tg^1 e2Tc. 6LetUU:Wo8T*!|nf0gUUbGeafunctionsuchthatforallg[;h2WandUUt2Tc,(1;t)l=1;6ल(5.1)u(g[h;t)l=(g[;hth1 t)(h;t):6ल(5.2)D6W*eIcanthende neaYetter-DrinfeldmoGduleV=V8(W;TV;)IoverW~with|-basis6वxtV;t2TV;UUandactionandcoactionofWgivenbyÀg[xtԥ)=(g[;t)x㐴g@Ltg1%3;6ल(5.3)׵`(xtV)ԥ)=t8 xt6ल(5.4)( 6forUUallg"2W;t2Tc.HConversely*,|ifAthefunctionde nesaYetter-DrinfeldmoGduleonthevector6spaceUUV9byx(5.3),(5.4),thensatis es(5.1),(5.2).HNoteUUthatthebraidingcofV8(W;TV;)UUisdeterminedbyKc(xs 8xtV)=(s;t)xsts1y xtګforUUall$ vs;t2TV;6लhenceUUbythevqaluesofonTo8Tc.HOurmainexamplecomesfromthetheoryofCoxetergroups([B.6,ChapitreIV]).6LetlS5bGeasubsetofthegroupWofelementsoforder2.6$F*oralls;s^0Q2Sletm(s;s^09)6bGe5?theorderofss^09.g(W;S)iscalledaCoxetervbsystemandWβaCoxetergr}'oupifW 쬍6लissgeneratedbySwithde ningrelations(ss^09)^m(s;sr0s)r=1foralls;s^0ȸ2Ssuchthat6वm(s;s^09)UUis nite.HLet9(W;S)bGeaCoxetersystem.JF*oranyg"2WCȲthereisasequence(s1|s;:::;sqj)6ofelementsinSYwithg"=s1I 9Isqj.BIfq!isminimalamongallsuchrepresentations,6thenѵqi=dl2`(g[ٲ)iscalledthelengthofg,oand(s1|s;:::;sqj)isar}'educedrepresentation6लofUUg[ٲ.6HDefinition5.1.s²Let(W;S)bGeaCoxetersystem,|andT*=fg[sg^1 ejg"2W;s2fd6वSg.qDe neUU:Wo8T*!|nf0gUUby 2(g[;t)=(1)l `(g@L)&forUUall-g"2W;t2TV:6ल(5.5)6LetQV=kV8(W;TV;)2^WbW Y}D.fTheQassoGciatedHopfalgebraswillbedenotedby6वRDz(W;S)=B(V8)UUandA(W;S):=B(V8)#|Wc.ҍHByPropGosition4.2,A(W;S)isapointedlink-indecomposableHopfalgebra 6withUUgroup-likeelementsWc.6HRemark5.2..̲W*euassumeW;TV;uasinthebGeginningofthissection,andde ne6वV=V8(W;TV;). 7 zQ6ऱ10z$gALEXANDERMILINSKIANDHANS-JV;URGENSCHNEIDERYHल1)Byde nition,?T 9isaunionofconjugacyclassesofWc.hLett1 wl2Tandfd6वT1ftheconjugacyclasscontainingt1|s..LetsiTL;1*iG;锲berepresentativesofthe6right-residueclassesofthecentralizerCW (t1|s);Wg(j):6ल(5.9) HProof.pZLet0t1 =p(12). =6व(t:(i)c=;t1|s)xt2i oforXalli2Z,d wher}'eX(i)isthenaturalnumberwith1(i)lvand6व(i)imoGdߵl2`:HThenxa=xb+ifa;b2Z;abmoGdߵm;andforalli;jY2Z,tiTLxjIJ=[ٲ(ti;tj6)x2ij6wher}'eMgthefunction@isde nedby/(5.5) ifmisodd,[andby/(5.10)",(5.11) 7&ifm=2lfd6iseven,andwher}'e"0|s;"1C2f1;1garegivenby((tt^09)^lȵ;tiTL)="i;0i1.HThefollowingr}'elationsaresatis edinB(V8):ȑ{x2፴iՌ"=0forall!bi:6ल(5.16)HIfmiseven,assume"0C="1|s;andde ne"="0C="1|s:Thenfx1|sx2S+8x2x3+8g+8xm1xm-`="xmx1|s;6ल(5.17)fd x1'|jxm-`="xm CCx1|s;6ल(5.18) (xmx1|s)l-`="(x1|sxm)lȵ:6ल(5.19)HIfm=2lk@+81iso}'dd,thenrx1|sx2S+8x2x3+8g+8xm1xm {+xmx1@=0;6ल(5.20)fd\x1|sx2'|jxm@=xmxm1 x1|s;6ल(5.21)'E(xmx1|s)lȵxm@=(x1|sxm)lȵx1:6ल(5.22)HAssumem=2lGandlar}'eevenand"0C="1=1.fdHL}'etx =Pލ USl% USi=1ɢ(1)^i1 x2i13;andy=Pލ USl% USi=1ɢ(1)^i1 x2iP:HIfx=x+;y"=y Sorx=x;y"=y+,thenӣx2C=0;y[ٟ2Tɲ=0;6ल(5.23)fdlxy[xy+8yxyxTɲ=0;6ल(5.24)6andifchar(|)6=2,thenthesub}'algebrageneratedbyx;yhasdimension8.$HProof.pZLet۵UջV"bGethebraidedsubspacewithbasisx1|s;:::;xm./XThenU6लisaY*etter-DrinfeldmoGduleoverDandUH,=1V8(D;ft1|s;:::;tmg;[ٲ)withrespectto6thetbasisyti p="xiTL;1im;tasshowninExample5.4.rSincetheobviousnatural6mapUUB(U)!B(V8)isanalgebramap,wecanassumethatU3=V8.HF*orall1im,letDi3bGetheskew-derivqationde nedinProposition2.4with6respGect toxid2VtiN.IZAsinthepreviousLemmaitsucestoshowthattheidentities6holdUUafterapplyingDiforalli.7 UWPOINTEDINDECOMPOSABLEHOPFALGEBRASOVERCOXETERGROUPS15YHलThearelations(5.16)"Q^followfrom[ٲ(tiTL;ti)#=(tiTL;ti)#=1aasintheproGofoffd6(5.12)Pn.|HT*oshow((5.17)and(5.20),Zleta=x1|sx2+smx2x3+sm+smxm1xm "xmx1|s;where6व"=1UUwhenmisoGdd.qThen|εD1|s(a)q=t1S8x2"xm _=([ٲ(t1|s;t2)")x0;fd|D2|s(a)q=x1S+8t2x3C=([ٲ(t2|s;t3)+1)x1;T:::o}2Dm1(a)q=xm2 +8tm1xm _=([ٲ(tm1;tm)+1)xm2;yDm(a)q=xm1 8"tm {x1C=("[ٲ(tm;t1S+1)xm1:6लAssume}Kmiseven.F*rom(5.10)"and(5.11)weseethat[ٲ(tiTL;ti+1 tO)[=1}Kforallfd62Pim]1,?and[ٲ(t1|s;t2)P="0|s;[ٲ(tm;t1)P="1|s.GThusDiTL(a)=0forallisince6व"0C="1|s.HIfUUmisoGdd,thenDiTL(a)=0UUforalli,since1="=[ٲ(tk됵;ti)UUforallkP;i.HT*oUUprove(5.18) 8and(5.21),we rstnotethatMٵD1|s(x1(x2'|jxm))=t1S8(x2'xm)=[ٲ(t1|s;t2):::[ٲ(t1;tm)xmxm1 x2|s:6लOnUUtheotherhand,D1|s((xm CCx2)x1)=xm CCx2|s:HलAssumemSҲ=2l2`. By(5.10)#kand(5.11)#H2wecompute[ٲ(t1|s;t2):::[ٲ(t1;tm)S=6व"^ll0|s"䍴l `116=ܵ":HenceDG1Ð(x1'|jxm)=D1|s("xm CCx1):Similarlyweobtainthesame6equalityUUforDm since[ٲ(tm;tm1):::(tm;t1|s)="䍴l `10 N<"^ll1C=":HलIfmisoGdd,then[ٲ(tk됵;tiTL)>`=1forallkP;i,andDiTL(x1'|jxm)>`=DiTL(xm CCx1|s)6forUUi=1andi=m.HFinally*,UUif2im81,UUthenfDiTL((x1'|jxi1 xi)(xi+1tFxm))/~=fdvDiTL((x1'|jxi1 )xi)UUti,8(xi+1tFxm)/~=(x1'|jxi1 )UUti,8(xi+1tFxm)/~=0;6लsinceUUti,8(xi+1tFxm)bGeginswiththeelementxi1 ,andx^2;Zi1Wز=0byx(5.16) .fdHSimilarlywehaveDiTL(xm CCx1|s) =(xm CCxi+1 tO)UUtiv6!(xi1;hxm)=0, Isince6वx^2;Zi+1;g=0.HT*ohshow(5.19)X,m[considerthesequence(s1|s;:::;s2l;) =(t;t^09;t;t^0;:::;t;t^0).oThen6वs1ز=iet=t1|s;s1s2s䍷11%ٲ=iett^09t=t2|s;s1s2s3(s1s2)^1%ٲ=iett^09tt^0t=t3|s;::::Thust1;:::;t2l6लcoincide0ewiththetiTL'scorrespGondingtos1|s;:::;s2l >inLemma5.5.ewHencebyLemma65.5,S((x1|sxm)lȲ)="(t;t09;:::;t;t0)xmxm1 x1|s:6ल(5.25):5ChangingUUtherolesoft;t^0#weobtaininthesamewaygS((xmx1|s)lȲ)="(t09;t;:::;t0;t)x1|sx2'|jxm;6ल(5.26)'67 zQ6ऱ16z$gALEXANDERMILINSKIANDHANS-JV;URGENSCHNEIDERY6लsinceUU(t^09t)^mit^0Q=(tt^0)^iTLt^0=(tt^0)^i1 t=tiforUUalli.qBythede nitioninLemma5.5>Edѵ"(t;t09;t;t0;:::;t;t0)="2l ݱY ti=2f[ٲ(s1'|jsi1 ;siTL)=(t;t09)(tt0;t)(tt0x0tt09t;t0):6लF*romUU(5.10),UU(5.11), [ٲ((tt09)i1 t;t0)=(tiTL;t2l;)="0ȲforUUall%1il2`;6लandڍ]h[ٲ((t09t)iTL;t)=(ti+1 tOt;t)=(ti+1 tO;t)(t;t)=1UUforall 1ilk@81:s6लHenceUUwehave^i"(t;t09;t;t0;:::;t;t0)="l፱0|s;andUUsimilarlyA\x"(t0;t;t0;t;:::;t0;t)="l፱1|s:6लThereforeUUweobtainfrom(5.25),(5.26) 8and(2)theequalityS((x1|sxm)lȲ)=S("(xmx1|s)lȲ);6लandUUtheclaimfollowsfromthebijectivityoftheantipGodeUUS.HT*oh6prove(5.22)t,lweproGceedasinthepreviouscase.kLet(s1|s;:::;s2l `+1.>)bethefd6sequence$(t;t^09;t;t^0;:::;t;t^0;t).aWThen$t1|s;:::;t2l `+1RCcoincidewiththetiTL'scorrespGond-6ingUUtos1|s;:::;s2l `+1inLemma5.5.qByLemma5.5, S((x1|sxm)lȵx1)="(t;t09;:::;t)xmxm1 x1|s;6लwheref@"(t;t^09;:::;t)K=[ٲ(t;t^0)(tt^0;t)[ٲ(tt^0x0tt^0;t)K=(1)^1+l 1˲.Byf@changingt;t^0fd6लweUUobtainڵS((xmx1|s)lȵxm)=(1)1+l 1˵x1x2'|jxm;s6लsinceUU(t^09t)^i1 t^0Q=tmi+1,1i2lk@+81.qThereforetheclaimfollowsfrom(5.21).HThe/lastpartoftheTheoremfollowsfromthefactthattheelementsx;yspanfd6aubraidedsubspaceoftypGeA2|s.ӆW*ritexf=Pލ l% i=1 z^i1 :x2i1anduyY?=Pލ l% i=1 ^i1ܵx2iP,6whereUU z; N42f1gand N4=1.qW*ecomputethebraidingcwithrespGecttox;y[ٲ: c(x8 y[ٲ)=X ti;j㉵ zi1 : jg1B(V)._bAssumeforallt;t^0 ¸2>T9esuchthatthepr}'oductֵtt^0has6evenor}'der2lGthat((tt^09)^lȵ;t)=((tt^0)^lȵ;t^0).6F;orallt2TvletRǟ^(t) mb}'ethesubalgebraofRgeneratedbyallxt09ɵ;t^0Q2TV;t^06=t.6Thenthegener}'atorsxtV;t2TvofRsatisfytherelationsf(5.16)!-{(5.22)"%,andH1)F;orallt ޸2TRandõx2R̊ther}'eareuniquelydeterminedelementsa;b ޸2Rǟ^(t)6such7thatx=a+bxtV.gIf7xisaWc-homo}'geneous7elementinRg,g"2W,then7a2Rg6andb2Rg@Lt].H2)F;orallg=ĸ2Wc,jtheW-homo}'geneouscomponentsR1)andRgIareisomorphic6asve}'ctorspaces.H3)]F;orallg"2Wc,hcho}'oseareducedrepresentationg"=s1'|jsqj,hs1|s;;sq12S,6ofg[,andde nez5ѵxgn=xs1 -Qxsq?:6Thenxg:isanon-zer}'oelementinRg.H4)F;orallg"=t1'|jtpR,wher}'et1|s;;tp39areelementsofS,čc͘xt1 lmxtp B=8 < :P卍 xg;*if(t1|s;;tpR)isar}'educedrepresentationofgG 0;*otherwiseS:6ल(5.27)~HProof.pZ1)T*oprovetheexistenceoftherepresentationwehavetoshowfor6anyns1|s;:::;sq,2~T&andx=xs1-Z;xsq theexistenceofWc-homogeneouselements6वa;bޝ2Rǟ^(t) 6ThisUUfollowsdirectlyfrom(5.19),(5.22).HFinallywehavetoshowthatxt1 lmxtp u=0ifthelengthofg²isandlet^0Wb}'etherestrictionof.{3De neVe=V8(W;TV;)and6वV8^05=V8(Wc^01ȵ;Tc^0;^09).ThenH(1)B(V8)isafr}'eerightB(V^0)-mo}'dule,A_andtheHilbertseriesP:B(V0mW)divides6वP:B(V)K.H(2)AsinThe}'orem+(5.8)0assumeforallt;t^0$52UTFWsuchthatthepr}'oductȵtt^0has6even-Sor}'der2l_that((tt^09)^lȵ;t)=((tt^0)^lȵ;t^0).eL}'etd8ѳeȍ-SB L(V8)-Sresp.d8 ;eȍB<Բ(V8^0)-Sbethetensor6algebr}'aеTc(V8)resp.OTc(V8^0)moduloallthequadraticrelationsֲ(5.16)$and>(5.17)g|,6ल(5.20)Pn.Thend88GeȍB j(V8)isafr}'eerightd88GeȍB(V8^0)-mo}'dule,andPrIdeqBp(V0mW) dividesPrIdeqBp(V)K.HProof.pZDe ne}':V!V8^04forallt2Tলby'(xtV)=xt,Wif}t2Tc^01Ȳ,and}'(xtV)=0,6otherwise.Then^'isa|Wc-colinearmap,`anditisW^01Ȳ-linear,`sinceforallg2e2֌W^01Ȳ,6वg[tg^1 e2Tc^0ifUUandonlyift2Tc^01Ȳ.qHence(1)followsfromCorollary3.3.HT*o?prove(2)let,F:Tc(V8)!T(V8^0)?bGethemapinducedby'.Sincethe6quadratic̤relationsde neinfactprimitiveelementsinthetensoralgebras,d8׫eȍwB p(V8)6andd8eȍXB /(V8^0)XarebraidedHopfalgebras.{qThemapinducesamaponthequotients6sinceDthequadraticrelationsarepreserved.[Thisisclearfor(5.16), zandisalsotrue6forSX(5.17),S(5.20),SsinceSXallthetiTL;1im;SXareinTc^01Ȳ,ifforoneibGothtiandti+16लareinTc^01Ȳ.+TheclaimnowfollowsfromTheorem3.2inthesamewayasCorollary63.3.Q'˄ffdffYffffd7 UWPOINTEDINDECOMPOSABLEHOPFALGEBRASOVERCOXETERGROUPS19Yo6.!aExamplesHलW*eUUconsidersomespGecialcasesofTheorem5.8.HExample6.1.Let}W=5ZSnq~;n2,andT=f(ij)j1ix:(ik+B)8+x:(ik+B) ޵x:(ijg)\?=0;6ल(6.3)bEx:(jgk+B)>x:(ijg)ڲ+8x:(ik+B) ޵x:(jgk+B)+x:(ijg) Jx:(ik+B)\?=0:HExample6.2.Let}W=5ZSnq~;n2,andT=f(ij)j1ix:(ik+B)8x:(ik+B) ޵x:(ijg)Y5=0;6ल(6.6)X x:(jgk+B)>x:(ijg)ڸ8x:(ik+B) ޵x:(jgk+B)x:(ijg) Jx:(ik+B)Y5=0:퍑HलThe&algebrasgeneratedbyallx:(ijg) J;1i:B(Vnq~)bGetheHopfalgebraofExample6.1.,uIfnK>=3>:or64,thenV(6.1)2,(6.2)*andV(6.3)areVde ningrelationsofBq(Vnq~),anddimOB(V3|s)l=612;dimUVB(V4|s)=24^2.[SInbGothcasestheintegralcanbedescribedintermsofthe6longestUUelementintheCoxetergroupsS3|s;S4.{E7 zQ6ऱ20z$gALEXANDERMILINSKIANDHANS-JV;URGENSCHNEIDERYHProof.pZ1)W*e rstconsiderRX=ܑB(V3|s).e=Lett1Y=(12);t2=(23);t3=(13)fd6andQxi{=l/xtiN;1i3..gThenS=ft1|s;t2g;TϾ=l/ft1;t2;t3g.gByQCorollary5.96वRǟ^(s3 )²=[|[x1|s;x2]shasdimensionord'YS3=6.}"ThusdimY!Rof=2dimRǟ^(s3 )/byTheorem65.8̊1).eOntheotherhanditisnotdiculttocheck̊(usingthediamondlemma)6thatpthealgebrawithrelations(6.1),(6.2)uand(6.3)ointhiscasehasdimension612..Thelongestelementiss1|ss2s1.Thereforex1x2x1x3isanon-zeromonomialof6maximalUUlength,henceanintegral.H2)!NowletRR/=>hB(V4|s).*W*edenotes1 ۲=(12);s2=(23);s3=(34);s4=6(24);s5=(14);s6=(13),KanddxicѲ=xsiq2;1i6.HenceS=fs1|s;s2;s3g;Ts=6सfs1|s;:::;s6g.qTheUUrelationsarex^2;ZiC=0,forall1i6,andux1|sx3S+8x3x1৲=0;fdux2|sx5S+8x5x2৲=0;ux4|sx6S+8x6x4৲=0;yryx1|sx2S+8x2x6+8x6x1৲=0; x2|sx1S+8x1x6+8x6x2C=0;yryx1|sx4S+8x4x5+8x5x1৲=0; x4|sx1S+8x1x5+8x5x4C=0;yryx2|sx3S+8x3x4+8x4x2৲=0; x3|sx2S+8x2x4+8x4x3C=0;yryx3|sx5S+8x5x6+8x6x3৲=0; x5|sx3S+8x3x6+8x6x5C=0:6लF*romtheserelationsonecanseethatB(V4|s)=|[x1;x2;x3]D#|[x4;x5;x6].HByCorol-fd6laryUU5.9,|[x1|s;x2;x3]UUhasdimensionord;S4C=24.HLet]=$(1243)inS4|s.Then1^1q=$s4;s2^1q=$s5;s3^1q=$s6.Hence6the[actionofde nesanalgebraisomorphism[ٸ|v:|[x1|s;x2;x3]|v!|[x4|s;x5;x6].6HencedimqB(V4|s)24^2.MUsingthemethoGdoftheproofofLemma5.6,zitispossible6toDishowthatdimB(V4|s)U=24^2.?OnDitheotherhanditfollowsfromthediamond6lemmaUUthatthealgebrawiththeabGoveUUrealationsalsohasdimension24^2|s.HThelongestelementisw0 =7s1|ss2s1s3s2s1.Since0^1Oܲ=7s4s5s4s6s5s4,6वx1|sx2x1x3x2x1x4x5x4x6x5x4cղisbanon-zeromonomialofmaximallength, hencean6integral.<ِffdffYffffjHAccordingnto[FK?,i(2.8)]alsoE56is nite-dimensional.>%Howevernwedonotknow 6whetherUUE5C=B(V5|s).`HExample6.5.AssumeƇchar(|)6=2.B-W*econsiderthedihedralgroupD4BinEx-6ample5.4._MThenT*=ft1|s;:::;t4g.LetbGede nedbyAr(5.5)ϯ,(andV=V8(D4|s;TV;).6ThenBbRe=R+B(V8)isgeneratedbytheelementsxiw=R+xtiN;1i4Bbandde ning6relationsUUarethequadraticrelationsx^2;ZiC=0;1i4UUand噉x1|sx3S+8x3x1g=0;fd噉x2|sx4S+8x4x2g=0;Yx1|sx2S+8x2x3+8x3x4+8x4x1g=0;Yx1|sx4S+8x4x3+8x3x2+8x2x1g=0;C7 UWPOINTEDINDECOMPOSABLEHOPFALGEBRASOVERCOXETERGROUPS21Y6लandUUtherelationsofdegree4 hxy[xy+8yxyx=0; uv[uv+8vuvu=0;6लwhereUUx=x1S+8x3|s;y"=x28x4|s;u=x18x3|s;v"=x2+8x4|s.fdHThedimensionofR*is64,G$andtheintegralcanbGedescribedintermsofthe6longestUUelementoftheCoxetergroupD4|s.HProof.pZThese_relationsarepartoftherelationsofTheorem5.7.XHenceR&is6anUUepimorphicimageofthealgebrax䍑9WeRV9generatedbyx;y[;u;v.withUUtherelations hxy[xy+8yxyx=0; uv[uv+8vuvu=0;6लand^ҵx^2C=0;y[ٟ^2d=0;u^2C=0;v[ٟ^2d=0;xuK޲+ux=0;xv+K޵v[x=0;y[uK޸uy"=0;y[v+vy"=0,fd6whichfollowfromthequadraticrelationsabGove.^UsingLemma5.6itiseasytosee6thatUUdimx䍑eRr=64.HOni2theotherhand,(dim|[x1|s;x2]==ordjD4 =8,(byi2Corollary5.9. ]Since6वt1|st2t1t2t1 =Mt3,:FandIt1t2t2t2t1=Mt4,:FtheIactionoft1|st2 Ude nesanisomorphism6|[x1|s;x2]!|[x3|s;x4].XAgain onecanshowsimilarlytotheproGofofLemma5.6that6|[x1|s;x2]޸|[x3|s;x4]-hasdimension8^2|s.dHenceR߲=|[x1;x2]޸|[x3;x4],5and-wehavea6monomialUUbasisintheelementsxiTL.HThelongestelementinD44{ist1|st2t1t2.Thereforex1x2x1x2x3x4x3x4isanon-6zeroUUintegral.'ffdffYffff.yHInFanearlierversionofthispapGerweshoweddimR=S64inthelastexample 6in"adi erentway*.I-Inadditiontothequadraticrelations,weprovedthefollowing6relations:qIfUUy1C=x1|sx2S+8x4x1;y2=x1x4S+8x2x1,UUthen \y[ٱ21d=0;y[ٱ22=0;y1|sy2S+8y2y1C=0:6लIttisnotdiculttoseethattheserelationsde neanalgebraofdimension64.A|Then 6GraG~na[G2ʥ,5.2.1]notedthatthebraidinginthebasisx;y[;u;vJofV'reducesthe6computationUUofB(V8)tothecaseofA2ȲinLemma5.6.HRemark6.6..̲TheTexampleswehaveconstructedshowthatthesymmetricfd6groupstS3|s;S4andS5andthedihedralgroupD4oGccurasthegroupsofgroup-like6elements\of nite-dimensionalpGointedandlink-indecompGosableHopfalgebras.sIn6theotherexamplesoflink-indecompGosableHopfalgebrasover niteCoxetergroups6weUUdonotknowwhetherthealgebrasB(V8)are nite-dimensional.)cReferences6ह[AÎG]L.- cmcsc10N.nAndr}uskiewitschandM.Gra>UV~na,C@Braidedf;Hopfalgebrasovernonabeliangroups, HYBol.Acad.Ciencias~(Cqordoba)X/2@cmbx863(1999),45-78.6[AS1]O0`N.Andr}uskiewitschandH.-J.Schneider,ÐLiftingofquantumlinearspacesandpointedHYHopf~algebrasoforder2cmmi8p-:3*,XJ.AlgebraX209(1998),658-691.6[AS2]O0`N.Andr}uskiewitschandH.-J.Schneider,FinitevquantumgroupsandCartanmatrices,HYAdv.~inMath.,XtoappUV~na,XA~freenesstheoremforNicholsalgebras,XJ.Algebra,XtoappUV~na,XOn~Nicholsalgebrasoflowdimension,Contemp.Math.,thisvÎolume.6[M1]K$S.Montgomerwwy,cHopfAlgebrasandTheirActionsonRings,cCBMSLecture$Notes82,HYAMS,X1993.6[M2]K$S.Montgomerwwy,Indecomposable|coalgebras,simplecomodules,andpointedHopfalgebras,HYProc.~AMSX123(1995),2343-2351.6[N]HYW.Nichols,XBialgebras~oftypeone,XComm.AlgebraX6(1978),1521-1552.6[R]HYD.eRadf}ord,3pThe{structureofHopfalgebraswithaprojection,3pJ.Algebra3G92(1985),3p322-347.6[Ro]JZPM.RDosso,XQuantum~algebrasandquantumshues,Invent.math.133(1998),399-416.6[Sb]IK`P.Scha}uenburg,ACharacterizationoftheBorel-likeSubalgebrasofQuantumEnvelopingHYAlgebras,XComm.~AlgebraX24(1996),2811-2823.6[W]HS.Westreich,QuasitriangulariHopfalgebraswhosegroup-likeelementsformanabelianHYgroup,XProc.~AMS124(1996),1023-1026.6[WJo]LS.!eWDor}onowicz,kDi erential,calculusoncompactmatrixpseudogroups(quantumgroups),HYComm.~Math.Phys.X122(1989),125-170.HMawwthematischesInstitut,FUniversit>UVat-M>UVunchen,Theresienstrae39,D-80333M>UVunchen,6GermanyHE-mail1addressD:N<2Cscmtt8milinski@rz.mathematik.uni-muenchen.de,@hanssch@rz.mathematik.uni-muenchen.de%;7  2Cscmtt8/2@cmbx8.- cmcsc10+- cmcsc10*': cmti10(X&eufm7'%n eufm10&qymsbm7% msbm10"#cmex7!"V cmbx10 K`y 3 cmr10N cmbx12#fcmti8q% cmsy62cmmi8Aacmr6|{Ycmr8 !", cmsy10 O!cmsy7 0ncmsy5 b> cmmi10 0ercmmi7O \cmmi5K`y cmr10ٓRcmr7Zcmr5u cmex10