; TeX output 2001.08.02:1038rjXc.}N cmbx12POINTEDٚHOPFALGEBRASBK`y cmr10NICOL@xASUUANDRUSKIEWITSCHANDHANS-J@xURGENSCHNEIDER v䍍Q- cmcsc10Abstract.F=ThisqasurveyonpGointedHopfalgebrasoveralgebraicallyclosed eldsofcharacteristicfdQ0.ZW*e1propGosetoclassifypointedHopfalgebras b> cmmi10Aby rstdeterminingthegradedHopfalgebraQgr A>ŲassoGciatedtothecoradical ltrationofA.jBTheAٓRcmr70|s-convqariants>elementsformabraidedHopfQalgebraRϲinthecategoryofY*etter-DrinfeldmoGdulesoverthecoradicalA0=C msbm10|,thegroupofQgroup-likeelementsofA,andgr-A !", cmsy10'RDz#A0|s.W*ecallthebraidingoftheprimitiveelementsofRQtheiin nitesimalbraidingofA.IfthisbraidingisofCartantypGe[AS2<],.thenitisoftenpossibleQto$-determineRDz,.toshowthatR7isgeneratedasanalgebrabyitsprimitiveelementsand nallytoQcompute>alldeformationsorliftings, thatispGointedHopfalgebrassuchthatgr A'RDz#|.RIn>theQlast3Chapter,9asaconcreteillustrationofthemethoGd,wedescribGeexplicitlyall nite-dimensionalQpGointedHopfalgebraswithabeliangroupG(A)andin nitesimalbraidingoftypeA 0ercmmi7n-(uptosomeQexceptionalcwcases).-Inotherwords,fwecwcomputealltheliftingsoftypGeAnq~;jthisresultisourmainQnewUUcontributioninthispapGer.T43- cmcsc10ContentsQXQ cmr12InrtroSduction'2Q1. BraidedHopfalgebrasX`F4Q2. NicrholsalgebrasrA15Q3. TrypSesofNicholsalgebrasA21Q4. NicrholsalgebrasofCartantypSe"I31Q5. Classi cationofpSoinrtedHopfalgebrasbytheliftingmethoSde34Q6. ProintedHopfalgebrasoftrypSeg cmmi12A2cmmi8nﷹ42QReferences62Qoىff< 𝍑 2000': cmti10MathematicsSubje}'ctClassi cation.Primary:q17B37;UUSecondary:16W30.fc Keywor}'dsandphrases.PointedUUHopfalgebras, nitequantumgroups. uƍܱThisܱworkwaspartiallysuppGortedbyANPCyT,AgenciaCordobaCiencia,CONICET,DAAD,thefdGraduiertenkollegUUoftheMath.qInstitut(UniversitatMGunchen)andSecyt(UNC).t*,o cmr91*rjXQ2ANDR9USKIEWITSCH!ANDSCHNEIDERc.}(IntroductionQAHopf algebraAorver a eld) msbm10|iscalled-@ cmti12pffointed [Sw], [M19],ifallitssimpleleftorrighrtcomoSdulesQareone-dimensional.SThecoradicalA|{Ycmr80ofAisthesumofallitssimplesubScoalgebras.ThrusAisQpSoinrtedifandonlyifA0isagroupalgebra.QWVewillalwraysassumethatthe eld|isalgebraicallyclosedofcrharacteristic0(althoughseveralQresultsofthepapSerholdorverarbitrary elds).QItbUiseasytoseethatAispSoinrtedifitisgeneratedasanalgebrabygroup-likeandskew-Qprimitivrecelements.Inparticular,Qgroupalgebras,univrersalenvelopingalgebrasofLiealgebrasQandItheqn9-deformationsoftheunivrersalenvelopingalgebrasofsemisimpleLiealgebrasareallQpSoinrted.QAnessenrtialtoSolinthestudyofpoinrtedHopfalgebrasisthecfforadical35 ltration #A0V !", cmsy10URA1A;W#u cmex10[ n!K cmsy80An=A_QofA.8Itisdualtothe ltrationofanalgebrabrythepSowersoftheJacobsonradical.8FVorpSointedQHopfsalgebrasitisaHopfalgebra ltration,andtheassoSciatedgradedHopfalgebragrڴAhasaQHopfalgebraprojectiononrtoA08'=x#|;=G(A)thegroupofallgroup-likeelementsofA.v=ByaQtheoremofRadford[Ra],grQAisabiproSduct-gr\APUR԰n:=RJ#|;QwhereRisagradedbraidedHopfalgebrainthecategoryofleftYVetter-DrinfeldmoSdulesorver|Q[AS25].QThisIdecompSositionisananalogofthetheoremofCartier{Kostanrt{Milnor{Mooreonthesemidi-Qrect5proSductdecompositionofacocommrutative5Hopfalgebrainrtoanin nitesimalandagroupQalgebrapart.QThe,vrectorspaceV¹=URPƹ(RJ)oftheprimitiveelementsofRvisaYVetter-DrinfeldsubmoSdule.% WeQcallitsbraiding62ecUR:VG V!V VdQtheBin nitesimalbraidingofA.Thein nitesimalbraidingisthekreytothestructureofpSointedQHopfalgebras.QThe~~subalgebra.%n eufm10B(Vp)ofRȹgeneratedbryVisabraidedHopfsubalgebra.aAsanalgebraandQcoalgebra,B(Vp)vonlydepSendsonthein nitesimalbraidingofV.Inhisthesis[N]publishedinQ1978,>Nicrhols studiedHopfalgebrasoftheformB(Vp)#|underthenameofbialgebrasoftypSeQone. 4WVecallB(Vp)theNichols,algebrffaofV. 4TheseHopfalgebraswrerefoundindepSendentlylaterQbryWVoronowicz[WVo]andotherauthors.QImpSortanrtexamplesofNicholsalgebrascomefromquantumgroups[Dr1j].7IfgisasemisimpleQLie\algebra,NU2@0RAq `(g),qBnotaroSotofunitryV,andthe nite-dimensionalFVrobSenius-Lusztigkrer-nQnels-~u20RAq \|(g),~3qaroSotofunitryoforderN@,arebSothoftheformB(Vp)#|withz=Z2 resp.Q(Z=(N@))2;zk&ݹ1:([L3 7o],[Ro1e],[Sbg,],and[L2 7o],[Ro1e],[MuKU]). 0(Here,therearesometecrhnicalQconditionsonN@).  rjXIPOINTED!HOPFALGEBRASW3c.}QIngeneral,theclassi cationproblemofpSoinrtedHopfalgebrashasthreeparts:nq(1)u$StructureoftheNicrholsalgebrasB(Vp).Vލq(2)u$Theliftingproblem:,DeterminethestructureofallpSoinrtedHopfalgebrasAwithG(A)UR=u$sucrhthatgrQAPUR԰n:=B(Vp)#|.q(3)u$Generationindegreeone:KDecidewhicrhHopfalgebrasAaregeneratedbygroup-likeandu$skrew-primitiveelements,thatisgrQAisgeneratedindegreeone.nQWVecconjecturethatall nite-dimensionalpSoinrtedHopfalgebrasoveranalgebraicallyclosed eldQofcrharacteristic0areindeedgeneratedbygroup-likeandskew-primitiveelements.QInthesequel,wredescribSethestepsofthisprogramindetailandexplainthepositivreresultsQobtained?sofarinthisdirection.8OItisnotourinrtention?togivreacompletesurveyonallaspSectsQofpSoinrtedHopfalgebras.QWVe willmainlyrepSortonrecenrtprogressintheclassi cationofpoinrtedHopfalgebraswithQabffelian꨹groupofgroup-likreelements.QIfthegroupisabSelian,5andVIisa nite-dimensionalYVetter-Drinfeldmodule,5thenthebraidingQisgivrenbyafamilyofscalarsqij 62UR|;1iS,intheformT0c(xi xjf )UR=qijJxj xid;ꦹwhere'Vx1;:::ʜ;x9isabasisofHuV:eQMoreorverqthereareelemenrtsg1;:::ʜ;g[2\,'andcharacters1;:::ʜ;[2wb\@suchthatqij @=\jf (gidڹ):QThegroupactsonxinviathecrharacteridڹ,andxiisagidڹ-homogeneouselemenrtwithrespSecttotheQcoactionof.8WVeinrtroSducedbraidingsofCartantypSe[AS25]where9EqijJqjvi 6=URqzn9a8:;cmmi6ij Mii U;1i;j%S;ꦹandz(aijJ)isageneralizedCartanmatrix.QIfO(aijJ)isaCartanmatrixof nitetrypSe,hthenthealgebrasB(Vp)canbeunderstoodastrwisting$QoftheFVrobSenius-Lusztigkrernelsu20 \|(g),gasemisimpleLiealgebra.QBydeformingthequanrtumSerrerelationsforsimpleroSotswhichlieintwodi erentconnectedQcompSonenrts$oftheDynkindiagram,wede ne nite-dimensionalpSointedHopfalgebrasu(DUV)inQterms&ofa"linkingdatumD$|of niteCartantrypSe\[AS45]./TheygeneralizetheFVrobenius-LusztigQkrernelsu(g)andareliftingsofB(Vp)#|.QInWsomecaseslinkingdataof niteCartantrypSearegeneralenoughtoobtaincompleteclassi -Qcationresults.QFVorಹ17ands1,QwrehavedeterminedthestructureofQall nite-dimensionalHopfalgebrasAwithG(A)UR'.8Theyarealloftheformu(DUV)[AS45].QSimilardataallorwaclassi cationofin nite-dimensionalpSointedHopfalgebrasAwithabSelianQgroupG(A),Gwithoutzerodivisorsand niteGelfand-Kirillorvdimension,inthecasewhentheQin nitesimalbraidingis"pSositivre\[AS55].6Butthegeneralcaseismoreinvolved.6WVealsohavetoQdeformtheroSotvrectorrelationsoftheu(g)209s.QThe*structureofpSoinrtedHopfalgebrasAwithnon-abffeliangroupG(A)iswidelyunknown.QOnebasicopSenproblemistodecidewhicrh nitegroupsappearasgroupsofgroup-likreelementsQof nite-dimensionalpSoinrtedHopfalgebraswhicharelink-indecompSosableinthesenseof[M29].QInourformrulation,thisproblemisthemainpartoffollowingquestion:givena nitegroup,QdetermineallYVetter-DrinfeldmoSdulesV7WorversucrhthatB(Vp)is nitedimensional.JOntheonewrjXQ4ANDR9USKIEWITSCH!ANDSCHNEIDERc.}Qhand,"thereAareanrumbSerAofsevrereconstraintsonVn[GS~vn3<].0SeealsotheexpSositionin[A,"5.3.10].QOnF7theotherhand,]itisvreryhardtoprovethe nitenessofthedimension,]andinfactthiswasQdoneonlyforafewexamples[MiS],f[FKx],[FP]whicrhareagainrelatedtoroSotsystems.TheQexamples.orverthesymmetricgroupsin[FKx]wereintroSducedtodescribethecohomologyringofQthe agvXarietryV.j;Atthisstage,9themaindicultryistodecidewhencertainNicholsalgebrasoverQnon-abSeliangroups,forexamplethesymmetricgroupsSnP,are nite-dimensional.$QTheBlastChaptercanservreasaconcreteillustrationofthetheoryexplainedinthispapSer.QWVedescribSeexplicitlyall nite-dimensionalpoinrtedHopfalgebraswithabeliangroupG(A)andQin nitesimalXbraidingoftrypSeAn W(uptosomeexceptionalcases).ThemainresultsinthisChapterQarevnew,jandcompleteproSofsaregivren.KTheonlycaseswhichwereknownbSeforearetheeasyQcaseA1[AS15],andA2[AS35].QThenewrelationsconcerntheroSotvrectorsei;j X;1Di!HVG Vp2  VsSidY[X.V2 jev {X.VN d !9u|VG I\ c ύUO!VQand,bVp2@gGgUO!dcVp2  IhidOp"V8G"q% cmsy6~\ jbX.V~D>!Vp2  VG Vp2 evПX.Vt jid 괟"V8G~ S !@*I Vp2g)n)UO!v%Vp2ōQare,3respSectivrelyV,the$identityofVmandVp2\t.A$braidedcategoryisrigidifanyobjectVmadmitsaQleftdual[Kas,Ch.8XIV,Def.2.1].Q1.2.BraidedvectorspacesandYetter-Drinfeldmo`dules.ʍQWVebSeginwiththefundamenrtalTBQDe nition1.1.D3_Leti]V͹bSeavrectorspaceandc,:V] Vn!V V͹ai]linearisomorphism.ThenQ(V;c)iscalledabrffaided35vectorspace,ifcisasolutionofthebrffaidequation,thatisXd(c iduJ)(id ʠ c)(c id)UR=(id ʠ c)(c id)(id ʠ c):Q(1.2)QItiswrell-knownthatthebraidequationisequivXalenrttothequantum35Yang-Baxtereffquation:6R12 R13R23 UZ=URR23R13R12:Q(1.3)䍑QHerewreusethestandardnotation:R13 UZ:URV: )V V!URV V VMVisthemapgivrenbyP[jrj )ʹid j rS2j,QifRn=URPjfrj rS2j.8SimilarlyforR12 ,R23.>QThe]equivXalencebSetrween]solutionsof3(1.2)!`andsolutionsof3(1.3)isgivrenbytheequalityc=QRJ.8FVorthisreason,someauthorscall(1.2)\thequanrtumYang-Baxterequation.>rjXQ6ANDR9USKIEWITSCH!ANDSCHNEIDERc.}QAn|easyandforthispapSerimportanrtexampleisgivenbyafamilyofnon-zeroscalarsqij 2Q|;i;j%2URI,whereVisavrectorspacewithbasisxid;i2I:Thenec(xi xjf )UR=qijJxj xid;forall$E;NV:URMy mN6!N mM isgivrenQbry^ocMa>;NĹ(m n)UR=m(1):n m(0) \|;mUR2M;n2N;Q(1.5)kQThesubScategoryof2HbH)TYD&)consistingof nitedimensionalYVetter-Drinfeldmodulesisrigid.QNamelyV,g ifV22[hHb[hH tYD%(yis nite-dimensional,thedualVp2 ܹ=[hHom(V;|)isin2HbH 4LYD$withthefol-Qlorwingactionandcoaction:s2u#u$(hfG)(vn9)UR=f(Sb(vn9))forallhUR2HV,fQ2Vp2\t,vË2Vp.u#u$IffQ2URVp2\t,thens2(fG)=f(1) f(0)G$isdeterminedbrytheequation~|f(1)f(0) \|(vn9)UR=Sb 1 B޹(v1)fG(v0);vË2V:QThentheusualevXaluationandcoSevaluationmapsaremorphismsin2HbH YD.QLetCVp,JW bSetrwoC nite-dimensionalYVetter-DrinfeldmodulesorverCHV.WVeshallconsidertheQisomorphismUR:WƟ2 r Vp2 !(VG Wƹ)2givrenby:"(' n9)(v wR)UR= (v)'(wR);'2WƟ a; Ë2Vp \t;vË2V;w2Wr:Q(1.6)ԍQRemark1.3.8_ҹWVeEyseethataYetter-DrinfeldmoSduleisabraidedvrectorspace. ISConverselyV,-aʍQbraidedvrectorspace(V;c)canbSerealizedasaYVetter-DrinfeldmoduleorversomeHopfalgebraHQifandonlyifcisrigid[Tk1].8Ifthisisthecase,itcanbSerealizedinmanrydi erentways.0 QWVerecallthataHopfbimoSduleorveraHopfalgebraH9issimrultaneouslyabimoduleandQaUbicomoSdulesatisfyingallpossiblecompatibilitryconditions. yThecategory2HbH nM2HbHofUallHopfQbimoSdulesorverHisabraidedcategoryV.Thecategory2HbH 1YD ᑹisequivXalent,AasabraidedcategoryV,toQthecategoryofHopfbimoSdules.Thiswrasessentially rstobservedin[WVo]andthenindepSendentlyQin[AnDe,AppSendix],[Sbg,],[Ro1e].RrjXIPOINTED!HOPFALGEBRASW7c.}QIf$Hzisa nitedimensionalHopfalgebra,qthenthecategory2HbH hYD"zQisequivXalenrttothecategoryQof2moSdulesorver2thedoubleofH [Mj1;S].Thebraidingin2HbH KYD#qcorrespondstothebraidinggivrenQbry the\canonical"RJ-matrixofthedouble.6VInparticular,ifHaisasemisimpleHopfalgebrathen2QHbQHjYD7isasemisimplecategoryV.\Indeed,itisknorwnthatthedoubleofasemisimpleHopfalgebraQisagainsemisimple.fQThe3kcaseofYVetter-DrinfeldmoSdulesorver3kgroupalgebrasisespeciallyimportanrtfortheappli-QcationsMtopSoinrtedHopfalgebras.IfHD=|,,vwhereisagroup,thenanHV-comoSduleVisjustQaf-gradedvrectorspace:0VĹ=(IgI{2΢Vg,whereVgG=fv2VĹjs2(vn9)=gmD  vg.WVefwillwrite2b 5YD-JQfor*thecategoryofYVetter-DrinfeldmoSdulesorver*|,9andsarythatV]22.b. "YD RisaYetter-DrinfeldQmoSduleorver(whenthe eldis xed).QRemark1.4.8_ҹLet&bSeagroup,5VWaleft|-module,5andaleft|-comodulewithgradingVXM=ʍQgI{2΢Vg.8WVede nealinearisomorphismcUR:VG V!V Vbry3h#c(x yn9)UR=gy x;ꦹforall(/x2Vg;gË2;yË2V:Q(1.7)QThenCda)u$V22URbUR $FYDifandonlyifgn9VhCURVgI{hgAacmr61Uforall>1g;hUR2:\Hb)u$IfV22URbUR $FYD ,then(V;c)isabraidedvrectorspace.c)u$ConrverselyV,ifHVCisafaithful-moSdule(thatis,ifforallg>2u;gn9v=vkforHallv>2Vp,impliesu$gË=UR1),andif(V;c)isabraidedvrectorspace,thenV22b $FYD .QPrffoof.:a)isclearfromthede nition.QBy˸applyingbSothsidesofthebraidequationtoelemenrtsoftheformxC y z;xg2Vg;yB2QVhe;z52URV;꨹itiseasytoseethat(V;c)isabraidedvrectorspaceifandonlyifV.c(gn9y gz)UR=gn9hz3 gy;ꦹforall(/g;hUR2;yË2Vhe;z52V:Q(1.8)QLetuswritegn9yË=URPa2!pxaϹ,iwherexaY!2URVaforalla2.Thenc(gn9yrM gz)=Pa2!pagz xa:HenceQ(1.8)8meansthatagn9z5=URghz,ȋforallz2URV\tanda2sucrhthatthehomogeneouscompSonentxaӹisQnotzero.8Thisprorvesb)andc).E Wcffxff ̟ff ̎ ̄cffQRemark1.5.8_ҹIfLisabSelian,e?aYVetter-DrinfeldmoduleorverLH闹=A|isnothingbuta-gradedQ-moSdule.QAssumethatisabSelianandfurthermorethattheactionofisdiagonalizable(thisisalwraysQthex#caseifis nite).RThatis,V┹=F$2PK$a6cmex8bVp2c,wherex#Vp2 =fv]2Vjgn9v=(gn9)vXforall&g2g.QThenV¹=URgI{2;2PKb"kV pڍgc;Q(1.9)drjXQ8ANDR9USKIEWITSCH!ANDSCHNEIDERc.}QwherefV2pRAg =URVp2 E\Vg. ConrverselyV,anyvectorspacewithadecompSosition(1.9)isaYVetter-DrinfeldʍQmoSduleorver.8ThebraidingisgivenbyXFIc(x yn9)UR=(g)y x;ꦹforall(/x2Vg;gË2;yË2Vp c;2wb :8QItvisusefultocrharacterizeabstractlythosebraidedvectorspaceswhichcomefromYVetter-QDrinfeldC~moSdulesorverC~groupsorabeliangroups.CaThe rstpartofthefollorwingde nitionisdueQtoM.TVakreuchi.QDe nition1.6.D3_Let(V;c)bSea nitedimensionalbraidedvrectorspace.u#u$(V;c)[+isofgrffouptype[+ifthereexistsabasisx1;:::ʜ;x ofVandelemenrtsgidڹ(xjf )2Vfor[+allu$i;j{sucrhthatXFy!~c(xi xjf )UR=gidڹ(xj) xid;1URi;j%S;Q(1.10)u$necessarilygi,2URGL(Vp).u#u$(V;c)isof niteVgrffouptype(resp.ofabelianVgrouptype)ifitisofgrouptrypSeandthesubgroupu$ofGL(Vp)generatedbryg1;:::ʜ;g9is nite(resp.8abSelian).u#u$(V;c)isofdiagonal35typffeifVhasabasisx1;:::ʜ;x9sucrhthat{c(xi xjf )UR=qijJxj xid;1i;j%S;Q(1.11)u$forsomeqij 5in|.8Thematrix(qijJ)iscalledthematrixofthebraiding.u#u$If(V;c)isofdiagonaltrypSe,thenwesaythatitisindeffcomposableifforalliandjӹ,thereu$existsasequenceiQ=i1,i2,:::G,it&=Qj~ofelemenrtsoff1;:::ʜ;Sgsuchthatqis;iqs+1qiqs+1 H;is6=Q1,u$1Pst1.4Otherwise,Swre>saythatthematrixisdecompSosable.4WVecanalsoreferthenu$tothecompSonenrtsofthematrix.8QIf,V |22o bo >YD!fis nite-dimensionalwithbraidingc,then(V;c)isofgrouptrypSeby(1.4).)lCon-QvrerselyV,assumedthat(V;c)isa nite-dimensionalbraidedvectorspaceofgrouptypSe. ILetbetheQsubgroupofGL(Vp)generatedbryg1;:::ʜ;g.De neacoactionbys2(xidڹ)UR=gihn xiforalli.ThenVQisaYVetter-DrinfeldmoSduleorverwithbraidingcbryRemark1.4,c).|QAbraidedvrectorspaceofdiagonaltypSeisclearlyofabeliangrouptrype;'Titisof nitegroupQtrypSeiftheqijJ'sarerootsofone.$թQ1.3.BraidedHopfalgebras.QThenotionof\braidedHopfalgebra"isoneofthebasicfeaturesofbraidedcategories.WVeQwill(!dealinthispapSeronlywithbraidedHopfalgebrasincategoriesofYVetter-Drinfeldmodules,Qmainlyorveragroupalgebra. urjXIPOINTED!HOPFALGEBRASW9c.}QLet[HHܹbSeaHopfalgebra. yFirst,thetensorproductin2HbH tYD%iallorwstode nealgebrasandQcoalgebrasin2HbH YDd.NamelyV,analgebrainthecategory2HbH YD$%isanassoSciativrealgebra(RJ;m),Qwhereӕmɹ:Rb I>R!R߹istheproSduct, withunitu:|!RJ, sucrhthatR߹isaYVetter-DrinfeldQmoSduleorverHandbSothmanduaremorphismsin2HbH YD.DQSimilarlyV,aZcoalgebrainthecategory2HbH ؞YD"0isacoassoSciativrecoalgebra(RJ;),whereUR:Rn!QR9 R@is'HthecoproSduct,6qwithcounit":R!|,sucrh'HthatR@isaYVetter-DrinfeldmoSduleoverHQandbSothand"aremorphismsin2HbH YD.QLetOnorwRJ,SbSetwoalgebrasin2HbH hYD ¹. hThenthebraidingc:SP Ra!R SallorwsOtoQprorvidetheYVetter-DrinfeldmoSduleR aSwitha"twisted"versionofalgebrain2HbH 6YDϱ.NamelyV,QtheproSductinR SismR S?:=UR(mR ; mS)(id ʠ c iduJ):)$ۍRߍ>MR S] R S  UO!`R SKl\pidr c id3,5?385?5y3, 1?38 1? 1y(1=B >MR R S] Ss$mX.R mX.S vuV/!R S :(z/QWVe5shalldenotethisalgebrabryRJ J뀉z UW nS׹.Thedi erencewiththeusualtensorproSductalgebraisQthepresenceofthebraidingcinsteadoftheusualtranspSositionW.#FQDe nition1.7.D3_Abrffaided35bialgebrain2HbH LyYD"isacollection(RJ;m;u;;"),where u#u$(RJ;m;u)isanalgebrain2HbH YD.Qu#u$(RJ;;")isacoalgebrain2HbH YD.u#u$UR:Rn!RJ J뀉z UW nRisamorphismofalgebras.u#u$uUR:|!Rand":Rn!|aremorphismsofalgebras.QWVesarythatitisabrffaided35Hopfalgebrffain2HbH LyYD"ifinaddition:u#u$TheidenrtityisconvolutioninvertibleinEnd(RJ);itsinverseistheantipSodeofRJ.QAcgrffadedcbraidedHopfalgebrain2HbH |YD#yiisabraidedHopfalgebraR|in2HbH |YDprorvidedwithaʍQgradingRn=URn0RJ(n)ofYVetter-DrinfeldmoSdules,sucrhthatRisagradedalgebraandagradedQcoalgebra.#FQRemark1.8.8_ҹThereVisanon-categoricalvrersionofbraidedHopfalgebras,Fgsee[Tk1].oAnybraidedQHopfalgebrain2HbH YD"givresrisetoabraidedHopfalgebrainthesenseof[Tk1]byforgettingtheQaction4andcoaction,>andpreservingthemrultiplication,comultiplication4andbraiding.gFVortheQconrversesee[Tk1,<:Th.c 5.7].AnalogouslyV,onecande negradedbraidedHopfalgebrasintheQspiritof[Tk1].V|QLet}R1bSea nite-dimensionalHopfalgebrain2HbH +YD /. ThedualS=RJ2 W5isabraidedHopfQalgebra[rin2HbH tYD#hwithmrultiplication2bRandcomultiplication21 \|2bR,wcf.=(1.6)I;thisisRJ2bop:inQthenotationof[ArG,Section2]. rjXQ10ANDR9USKIEWITSCH!ANDSCHNEIDERc.}QIn thesamewrayV,if RKQ=2n0RJ(n)isagradedbraidedHopfalgebrain2HbH dYD%Iwith nite-QdimensionalhomogeneouscompSonenrts,thenthegradeddualSH=RJ2 n=n0RJ(n)2 fisagradedQbraidedHopfalgebrain2HbH YD.iQ1.4.Examples.Thequantumbinomialformula.QWVe(shallprorvidemanyexamplesofbraidedHopfalgebrasinChapter2.HerewediscussaveryQsimpleclassofbraidedHopfalgebras.QWVeE rstrecallthewrell-knownEquantumbinomialformula. JLetU¹andVNbSeelementsofanQassoSciativrealgebraover|[qn9],qXanindeterminate,suchthatVpU6=URqn9U@V.8Then<ng(U댹+Vp) n=mX UR1innqō'nYy)1wi.Oq7 pq<U@ iV ni;jif(/nUR1:Q(1.12) QHereV&qō/nYy16i6kq?ß pqGy=ō^ (n)q![z? ΍(i)q!(ni)q!EfZ;where3N(n)q!UR=mY 1inp(i)q;and(r(i)qP= :ПX 0jvi1(uqn9 jC::QByspSecialization,(1.12)%holdsforq2i|.]Inparticular,ifU7andV,areelemenrtsofanassociativreQalgebraorver|,andqXisaroSotof1oforderdividingn,suchthatVpU6=URqn9U@Vthen (U댹+Vp) n=URU@ n ܹ+V nD:Q(1.13)QExample1.9.bSeamatrixsucrhthatۍ*qijJqjvi=UR1; 1i;j%S;i6=j:Q(1.14)QLetNiObSetheorderofqiiI,whenthisis nite.ʍQLetRbSethealgebrapresenrtedbygeneratorsx1;:::ʜ;x9withrelationsXxN8:imi =UR0; ifBord-~qii <1:Q(1.15)LʍXxidxj\=URqijJxjf xi; 1iQ1.6.Someprop`ertiesofbraidedHopfalgebras.QInƭthisSection,wre rstcollectseveralusefulfactsabSoutbraidedHopfalgebrasinthecategoryQofYVetter-DrinfeldmoSdulesorveranabeliangroup.lWVebeginwithsomeidenrtitiesonbraidedQcommrutators.QInVthefollorwingtwoLemmata,qRp8denotesabraidedHopfalgebrain2b %YD.}Leta1;a2; 2 R]QbSeelemenrtssuchthatai,2URR2J8:iRAg8:i ܹ,forsomei2wbUR ,gi2UR.ZQLemma1.10.;(a).쉍:8P[[a1;a2]c.y;a3]c!+2(g1)a2[a1;a3]c˹=UR[a1;[a2;a3]c.y]c!+3(g2)[a1;a3]c.ya2:Q(1.22)Q(b).fiIf35[a1;a2]c˹=UR0and[a1;a3]c˹=UR0then[a1;[a2;a3]c.y]c˹=UR0.Q(c).fiIf35[a1;a3]c˹=UR0and[a2;a3]c˹=UR0then[[a1;a2]c.y;a3]c˹=UR0.Q(d).fiAssume35that1(g2)2(g1)2(g2)UR=1.Thenʡu^ڹ[[a1;a2]c.y;a2]c˹=UR2(g1) 1 \|1(g2)[a2;[a2;a1]c.y]cQ(1.23)QPrffoof.:Lefttothereader.gcffxff ̟ff ̎ ̄cffF:QThefollorwingtechnicalLemmawillbSeusedatacrucialpoinrtinSection6.1.QLemma1.11.;Assume35that2(g2)UR6=135and^o1(g2)2(g1)2(g2)=UR1;Q(1.24)Lʍ^o2(g3)3(g2)2(g2)=UR1:Q(1.25)QIfع[a2;a2]c.y;a1]c}=UR0;Q(1.26)uL[[a2;a2]c.y;a3]c}=UR0;Q(1.27)Yl[a1;a3]c}=UR0;Q(1.28)Qthen<[[[a1;a2]c.y;a3]c.y;a2]c˹=UR0:Q(1.29) "rjXIPOINTED!HOPFALGEBRAS13c.}QPrffoof.:WVecompute:X[[[a1;a2]c.y;a3]c.y;a2]c\=URa1a2a3a2j2(g1)a2a1a3a23(g1)3(g2)a3a1a 2ڍ2Lʍ\-e+3(g1)3(g2)2(g1)a3a2a1a2j2(g1)2(g2)2(g3)a2a1a2a3\-e+2(g1) 22(g2)2(g3)a 2ڍ2a1a3j+2(g1)2(g2)2(g3)3(g1)3(g2)a2a3a1a2\-e2(g1) 22(g2)2(g3)3(g1)3(g2)a2a3a2a1:QWVelindexconsecutivrelythetermsintheright-handsidebyromannumbSers:=6(I);:::ʜ;(VpIII).ʍQThen(II)+(VpII)UR=0,bryo(1.25)&and(1.28)!.8Now,_K(I)^$=ō/ 1[z\ ΍3(g2)(1+2(g2))bxa1a 2ڍ2a3j+ō2(g2)3(g2)۟[z?@ ΍1+2(g2)DQa1a3a 2ڍ2Qq^$=ō/ 1[z\ ΍3(g2)(1+2(g2))bxa1a 2ڍ2a3j+ō2(g2)3(g2)3(g1)۟[z^ ΍s1+2(g2)c a3a1a 2ڍ2Lʍ^$=UR(Ia)+(Ib);Qbryo(1.27)&and(1.28)!.8Bythesameequations(1.27)%Xand(1.28),wrealsohave⠍J(VpIII),=URō332(g1)222(g2)2(g3)3(g1)33[zAX ΍'1+2(g2)a 2ڍ2a3a1jō2(g1)222(g2)2(g3)3(g1)3(g2)22۟[z ΍9D1+2(g2)a3a 2ڍ2a1,=URō332(g1)222(g2)2(g3)33[zc ΍1+2(g2)fia 2ڍ2a1a3jō2(g1)222(g2)2(g3)3(g1)3(g2)22۟[z ΍9D1+2(g2)a3a 2ڍ2a1Lʍ,=UR(VpIIIa)+(VIIIb):QWVenextuse(1.26)%XtoshorwthatoP(Ia)+(Vp)+(VI)+(VIIIa)*.=UR0;(Ib)+(III)+(IVp)+(VIIIb)*.=UR0:QIn=thecourseoftheproSofoftheseequalities,wreneed(1.24)%Wand(1.25)!E."This nishestheproofofʍQ(1.29)X.#1cffxff ̟ff ̎ ̄cff"QLetH>bSeaHopfalgebra.%Thentheexistenceofaninrtegralfor nite-dimensionalbraidedHopfQalgebrasimpliesŦrjXQ14ANDR9USKIEWITSCH!ANDSCHNEIDERc.͍QLemma1.12.;LffetqR=ҟL**N U_*n=0#RJ(n)bea nite-dimensionalgradedbraidedHopfalgebrain2HbH YDʍQwithRJ(N@)UR6=0.^Therffeexists2RJ(N@)whichisaleftinteffgralonR4andsuchthatRJ(i)v R(NQi)UR!|;x yË7!UR(xyn9),35isanon-deffgenerate35pairing,forall0URiN@.fiInpffarticular,荍 dimKRJ(i)UR=dimR(Ni):QPrffoof.:This@isessenrtiallyduetoNichols[N,U1.5].:eInthisformulation,UoneneedstheexistenceofQnon-zeroinrtegralsonRJ;thisfollowsfrom[FMSV].8See[AG,Prop.83.2.2]fordetails.@=cffxff ̟ff ̎ ̄cffqQ1.7.TheBkin nitesimalbraidingofHopfalgebraswhosecoradicalisaHopfsubalgebra.QFVortheconrvenienceofthereader,Wwre rstrecallinthisSectionsomebasicde nitionsfromQcoalgebratheoryV.QDe nition1.13.J_LetCFbSeacoalgebra.؍u#u$G(Cܞ)UR:=fx2CFnf0gj(x)=x xg꨹isthesetofallgroup-likreelementsofCܞ.荍u#u$If؆gn9;hUR2G(Cܞ),&thenx2C$is(gn9;h)-skew"primitiveif(x)=x h+g x.2Thespaceofallʍu$(gn9;h)-skrewIprimitiveelementsofC&9isdenotedbyP(Cܞ)gI{;h c.UIfC&9isabialgebraorabraidedu$bialgebra,andgË=URh=1,thenPƹ(Cܞ)UR=P(C)1;1G$isthespaceofprimitiveelemenrts.u#u$The#cfforadicalofCWisC0vx:=tPaDS,1whereDwGrunsthroughallthesimplesubcoalgebrasofCܞ;u$itisthelargestcosemisimplesubScoalgebraofCܞ.8Inparticular,|G(C)URC0.u#u$CFispffointedif|G(Cܞ)UR=C0.u#u$ThetcfforadicalD ltrationofCQistheascending ltrationC0 @C16CjCjv+1"::: :;u$de ned,bryCjv+1|:=yfx2CVj(x)2Cj' Cb+C C0g.This,isacoalgebra ltration:u$Cj\URP0ij*Ci Cjvi \;anditisexhaustivre:8C1=URS UTn0CnP.u#u$A2grffaded%coalgebraisacoalgebraGprorvidedwithagradingGw =n0G(n)suchthatu$G(jӹ)URP0ij*G(i) G(jW{i)forallj%UR0.u#u$Acfforadically2graded4coalgebra[CM@]isagradedcoalgebraG-=n0G(n)4sucrhthatitsu$coradicalz ltrationcoincideswiththestandardascending ltrationarisingfromthegrading:u$(gr g/Cܞ)n =dmnɌgr 0C(m):$Astrictlygrffadedcoalgebra[Sw]isacoradicallygradedcoalgebrau$G꨹sucrhthatG(0)isone-dimensional.u#u$ThegradedcoalgebraassoSciatedtothecoalgebra ltrationofCbisgrVC1=URn0̹grCܞ(n),!whereu$grSCܞ(n)UR:=CnP=Cn1̹,n>0,grQCܞ(0):=C0.8Itisacoradicallygradedcoalgebra.QQWVeshallneedabasictecrhnicalfactonpSointedcoalgebras.QLemma1.14.;[M19,5.3.3].GASwmorphismSofpffointedScoalgebraswhichisinjectiveinthe rsttermQof35thecffoalgebra35 ltration,isinjective.'+ffLʼnff3<;ff3<3<ff0rjXIPOINTED!HOPFALGEBRAS15c.}QLeta norwAbSeaHopfalgebra. WVeshallassumeinwhatfollowsthatthecoradicalA0!isnotonlyQasubScoalgebrabutaHopfsubalgebraofA;thisisthecaseifAispoinrted.QTVo5studythestructureofA,Hwreconsideritscoradical ltration;bSecauseofourassumptionQonA,itisalsoanalgebra ltration[M19].Therefore,theassoSciatedgradedcoalgebragrAisaQgraded Hopfalgebra.FVurthermore,Q|H+E:==A0 'grA(0) isaHopfsubalgebraofgrpA;@andtheQprojection8D:{ gr:A{ !grA(0)8withkrerneln>0̹grA(n),\isaHopfalgebramapandaretractionQof+(theinclusion._WVecanthenapplythegeneralremarksofSection1.5.LetRDrbSethealgebraofQcoinrvXariantskofn9;RƵisabraidedHopfalgebrain2HbH ƯYD" ߹andgrAcanbSereconstructedfromRandQHasabSosonizationgrQAUR'RJ#HV.QThe.!braidedHopfalgebraRGkisgraded,SsinceitinheritsthegradationfromgrPA:ڜRn=URn0RJ(n),QwhereRJ(n)UR=grA(n)\R.8FVurthermore,Risstrictlygraded;thismeans,y!(a).u$RJ(0)UR=|1(hencethecoradicalistrivial,cf.8[Sw,Chapter11]).1(b).u$RJ(1)UR=Pƹ(R)(thespaceofprimitivreelementsofRJ).QItisingeneralnottruethatabraidedHopfalgebraRsatisfying(a)and(b),alsosatis es뱘(c).u$Risgeneratedasanalgebraorver|bryRJ(1).QAXbraidedXgradedHopfalgebrasatisfying(a),tm(b)and(c)iscalledaNicrholsalgebra.IntheQnextNcrhapterwewilldiscussthisnotionindetail.eWNoticethatthesubalgebraRJ206SofRhgeneratedQbryRJ(1),aHopfsubalgebraofR,isindeedaNicrholsalgebra.QDe nition1.15.J_Thebraiding}6IcUR:VG V!V V;LQofV¹:=URRJ(1)=Pƹ(R)iscalledthein nitesimal35brffaidingofA.ʍQThegradedbraidedHopfalgebraRiscalledthediagrffamofA.QThedimensionofV¹=URPƹ(RJ)iscalledtherffankofA. ta2.NicholsalgebrasQLet/ HabSeaHopfalgebra.InthisChapter,@#wrediscussafunctorBfromthecategory2HbH HOYD#toQtheәcategoryofbraidedHopfalgebrasin2HbH YD;HgivrenaYVetter-DrinfeldmoSduleVp, thebraidedQHopfalgebraB(Vp)iscalledtheNichols35algebrffaofV.QThe;structureofaNicrholsalgebraappSeared rstinthepaper"Bialgebrasoftrypeone\[N]ofQNicrhols.andwasrediscoveredlaterbyseveralauthors.,Inourlanguage,zabialgebraoftypSeoneisQjustabSosonizationB(Vp)#HV.CHenceNicrholsalgebrasaretheH-coinrvXariantelementsofbialgebrasQofQtrypSeone,p*alsocalledquantumsymmetricalgebrasin[Ro2e].Severalyearsafter[N],p*WVoronowiczQde nedDNicrholsalgebrasinhisapproachto"quantumdi erentialcalculus"[WVo];gagain,ktheyQappSearednastheinrvXariantnpartofhis"algebraofquanrtumdi erentialforms".wLusztig'salgebrasfQ[L3 7o],4de ned%brythenon-degeneracyofacertaininvXariantbilinearform,4areNicholsalgebras.InQfactmNicrholsalgebrascanalwaysbSede nedbythenon-degeneracyofaninvXariantbilinearformQ[ArG].8ThealgebrasB(Vp)arecalledbitensoralgebrasin[Sbg,].PrjXQ16ANDR9USKIEWITSCH!ANDSCHNEIDERc.}QInasense,Nicrholsalgebrasaresimilartosymmetricalgebras;.indeed,bSothnotionscoincideQin)thetrivialbraidedcategoryofvrectorspaces,8ormoregenerallyinanysymmetriccategory(e.Qg.in]thecategoryofsupServrectorspaces).ButwhenthebraidingisnotasymmetryV,zaNicrholsQalgebra>couldharve>amruch>richerstructure.3EWVehopSethatthiswillbeclari edintheexamples.QOneYtheotherhand,Nicrholsalgebrasarealsosimilartouniversalenvelopingalgebras.However,Qin[spiteofthee ortsofsevreralauthors,jitisnotcleartoushowtoachieveacompact,jfunctorialQde nitionofa"braidedLiealgebra"fromaNicrholsalgebra.QWVebSelievrethatNicholsalgebrasareveryinterestingobjectsofanessentiallynewnature.+Q2.1.De nitionofNicholsalgebras.QWVenorwpresentoneofthemainnotionsofthissurveyV.卍QDe nition2.1.D3_Letf!VbSeaYVetter-Drinfeldmoduleorverf!HV. JAebraidedgradedHopfalgebraʍQRn=URn0RJ(n)in2HbH YD"YiscalledaNichols35algebrffaofVif|'RJ(0)andV'RJ(1)in2HbH YD,andtxPƹ(RJ)UR=R(1);Q(2.1)LʍxRisgeneratedasanalgebrabryRJ(1):Q(2.2)QThedimensionofVwillbSecalledtherffankofRJ.QWVe5needsomepreliminariestoshorwtheexistenceanduniquenessoftheNicholsalgebraofVQin2HbH YD.QLetJVCbSeaYVetter-DrinfeldmoduleorverJHV.nThenthetensoralgebraTƹ(Vp)f=L뾟n0$pT(Vp)(n)QofthevrectorspaceVWadmitsanaturalstructureofaYVetter-DrinfeldmoSdule,Gsince2HbH +YD$׹isaQbraided EcategoryV.Itisthenanalgebrain2HbH %YDN.Thereexistsauniquealgebramap:Tƹ(Vp)!QTƹ(Vp) 뀉z UW UWT(V)sucrhthat(vn9)UR=v 1+1 v,forallvË2URVp.8FVorexample,ifx;y2URVp,then`(xyn9)UR=1 xy+x y+x(1)y x(0)$+yn9x 1:XQWith^Dthisstructure,{,Tƹ(Vp)isagradedbraidedHopfalgebrain2HbH wYD#nwithcounit":T(Vp)!|,Q"(vn9)=0,{ifv2Vp.tTVoshorwtheexistenceoftheantipSode,{onenotesthatthecoradicaloftheQcoalgebra=8Tƹ(Vp)is|,QandusesaresultofTVakreuchi=8[M19,5.2.10].0HenceallthebraidedbialgebraQquotienrtsofTƹ(Vp)in2HbH YD"YarebraidedHopfalgebrasin2HbH YD.QLetusconsidertheclassSofallIFURTƹ(Vp)sucrhthatu#u$I+isahomogeneousidealgeneratedbryhomogeneouselementsofdegreeUR2,u#u$I+isalsoacoideal,i.fie.8(I)URI+ Tƹ(Vp)+T(Vp) I.QNotefthatwredonotrequirethattheidealsIXvareYVetter-DrinfeldsubmoSdulesofTƹ(Vp). Letthenުe!VQS kbSeTthesubsetofSconsistingofallIF2URSwhicrhareYVetter-DrinfeldsubmodulesofTƹ(Vp).-oTheQideals5I(Vp)UR=X%&I2/\%eufm8SI;weI(Vp)=WXzJ[2Ǎfe9SFJ"0-QarethelargestelemenrtsinS,respSectivelyުe!VS #.rjXIPOINTED!HOPFALGEBRAS17c.}QIfIF2URSthenRn:=Tƹ(Vp)=IFչ=n0RJ(n)isagradedalgebraandagradedcoalgebrawithRJ(0)UR=|;V'R(1)Pƹ(R):QIfactuallyIF2ުe!VURS F͹,thenRisagradedbraidedHopfalgebrain2HbH YD.QWVecanshorwnowexistenceanduniquenessofNicholsalgebras.iQProp`osition2.2.N!?Lffet35B(Vp)UR:=Tƹ(V)=weI +(V).fiThen35thefollowinghold:Q1.u$V¹=URPƹ(B(Vp)),35hencffeB(V)isaNicholsalgebrffaofV.Q2.u$I(Vp)UR=w^JeI u}(V).Q3.u$LffetRn=URn0RJ(n)beagradedHopfalgebrain2HbH YD"suchthatRJ(0)UR=|1andR!isgeneratedʍu$asfanalgebrffabyV*:=RJ(1).'ThenthereexistsasurjectivemapofgradedHopfalgebrasu$Rn!URB(Vp),35whichisanisomorphismofYetter-Drinfeldmoffdulesindegree1.Q4.u$Lffetn}Rd=n0RJ(n)beaNicholsalgebraofVp.BThenRd'B(V)asbrffaidedn}Hopfalgebrffasin2u$Hbu$H hYD'-.Q5.u$LffetR(|=2Ldn0#TRJ(n)beagradedbraidedHopfalgebrain2HbH YD#3withRJ(0)2=|1andR(1)2=u$Pƹ(RJ)35=URVp.fiThenB(V)isisomorphictothesubffalgebra35|hViofRLgenerffated35byV.iQPrffoof.:1.WVejharvetoshowtheequalityV =/Pƹ(B(Vp)).LetusconsidertheinverseimageX\iinQTƹ(Vp)]ofallhomogeneousprimitivreelementsofB(Vp)indegreenUR2. Then]XO.isagradedYVetter-QDrinfeld6IsubmoSduleofTƹ(Vp),Z\andforallxUR2X,(x)2x:> 1+1 x+Tƹ(Vp) wC6eIZi(V)+wC6eIZi(V) Tƹ(V):QHence͒theidealgeneratedbryw֊eI (Vp)andXisinުpe!VS ,LandXwe׎I (V)brythemaximalityofw֊eI (Vp).QHence2theimageofXinB(Vp)iszero.'Thisprorves2ourclaimsincetheprimitivreelementsformaQgradedsubmoSdule.Q2.#WVe|rharvetoshowthatthesurjectivemapB(Vp)UR!Tƹ(V)=I(V)|risbijectivre.#ThisfollowsfromQ1.8andLemma1.14.Q3.8ThekrernelI+ofthecanonicalprojectionTƹ(Vp)UR!RbSelongstoުe!VS #;henceIFw^JeURI u}(V).Q4.8follorwsagainfromLemma1.14,asin2.Q5.8follorwsfrom4.cffxff ̟ff ̎ ̄cff8鍑QIfQHU,isabraidedsubspaceofV22URHbURH nYD[,othatisasubspacesucrhthatc(UE qaU@)URU qaU@,owhereQHcQisthebraidingofVp,˶wrecande neB(U@)UR:=Tƹ(U)=I(U)withtheobrviousmeaningofI(U).+ThenQthedescriptionstatedinPropSosition2.8alsoappliestoB(U@).QCorollary2.3.AXTheAassignmentVs7!֕B(Vp)isafunctorfrffom2HbH YD$tothecategoryofbraidedʍQHopf35algebrffasin2HbH LyYD>.QIfUisaYetter-DrinfeldsubmoffduleofVp,ormoregenerallyifUisabraidedsubspaceofVp,Qthen35thecffanonical35mapB(U@)UR!B(Vp)35isinjeffctive. rjXQ18ANDR9USKIEWITSCH!ANDSCHNEIDERc.}QPrffoof.:Ifq):U !Vsisamorphismin2HbH GYD ,thenTƹ():T(U@)!T(Vp)isamorphismofbraidedʍQHopfalgebras.6SinceTƹ()(I(U@))isacoidealandaYVetter-DrinfeldsubmoSduleofT(Vp),MtheidealQgeneratedbryTƹ()(weI +(U@))iscontainedinweI ӹ(Vp).8HencebyPropSosition2.2,Bisafunctor.QThesecondpartoftheclaimfollorwsfromPropSosition2.2,5.rcffxff ̟ff ̎ ̄cffQThedualitrybSetweenconditions(2.1)and(2.2)inthede nitionofNicrholsalgebra,emphasizedQbryParts3and5ofPropSosition2.2,isexplicitlystatedinthefollowing`QLemma2.4.5Lffet9R.=Xn0RJ(n)beagradedbraidedHopfalgebrain2HbHRYD ;1:QHerearesomebasicwrell-knownfactsabSoutthebraidgroup.rjXIPOINTED!HOPFALGEBRAS19c.}QThere5isanaturalprojection:jBn !Sn sending5i\tothetranspSositioniψ:=j(i;i3+1)5forallQi.8TheprojectionXadmitsaset-theoreticalsectionsUR:Sn!Bn determinedbry0Mzs(idڹ)j=URid; 1in1;36u8ls(W!n9)j=URs(W)s(!n9); ifD`(!)UR=`(W)+`(!):8׍QHere`denotesthelenghrtofanelementofSn1withrespSecttothesetofgenerators1;:::ʜ;n1̹.HTheQmap*siscalledtheMatsumotosection.hInotherwrords,:if!0ȹ=iq1 :::ZiX.MisareducedexpressionQof!Ë2URSnP,thens(!n9)=iq1 :::ZiX.M .QLetZqn2X5|,q6=0.=TheZquotienrtofthegroupalgebra|(BnP)bythetwo-sidedidealgeneratedbyQtherelationsI荒`(iqn9)(i+1); 1URin1;Qistheso-calledHeffcke35algebra꨹oftrypSeAnP,denotedbyHq(n).QUsing'thesections,thefollorwingdistinguishedelementsofthegroupalgebra|Bn warede ned:I荑K5Sn:=|XwURI{2*ppmsbm8Sns(n9);Si;j :=XURI{2X8:i;j s();"dHQhereXi;j URSn Bisthesetofall(i;jӹ)-shrues.2TheelementSn Biscalledthequantumsymmetrizer.QGivrenabraidedvectorspace(V;c),\therearerepresentationsofthebraidgroupsn a:Bn!QAut.(Vp2 n8)foranrynUR0,givenbyy+nP(idڹ)UR=id  iduH c id  iduJ;Qwhere cactsinthetensorproSductoftheiandi+1 copiesofVp.Byabuseofnotation,wreshallQdenotebrySnP,Si;j alsothecorrespSondingendomorphisms(Sn),(Si;j X)ofVp2 n6=URTƟ2nJ(Vp).QIfC1=URLn0#/tCܞ(n)isagradedcoalgebrawithcomrultiplication,?wedenotebyi;j :URCܞ(i+jӹ)!VQCܞ(i) C(jӹ),i;j%UR0,the(i;jӹ)-gradedcompSonenrtofthemap.$QProp`osition2.8.N!?Lffet35V22URHbURH nYD[.fiTheni;j =URSi;j X;Q(2.3)ԍB(Vp)UR=M cn0wvTƟ nJ(V)=kgerS(SnP):Q(2.4)%ְQPrffoof.:Seeforinstance[Sbg,].T2Ccffxff ̟ff ̎ ̄cff@bQThisTdescriptionoftherelationofB(Vp)doSesnotmeanthattherelationsareknorwn.IngeneralQit[isvreryhardtocomputethekernelsofthemapsSn inconcreteterms.pFVoranybraidedvectorQspace(V;c),wremayde neB(Vp)byo(2.4);{.- rjXQ20ANDR9USKIEWITSCH!ANDSCHNEIDERc.}Q2.3.Inv@arianceundertwisting.QTwistingwisamethoSdtoconstructnewHopfalgebrasbry"deforming"thecomultiplication;QoriginallyduetoDrinfeld[Dr2j],itwrasadaptedtoHopfalgebrasin[Re ދ].ɍQLetKAbSeaHopfalgebraandF.2hA AKbeaninrvertibleKelement.[LetF :=hFƹF21:A!QA A;itisagainanalgebramap.8Ifs(1 Fƹ)(id ʠ )(F)i=UR(FLn 1)( iduJ)(Fƹ);Q(2.5)36(id ʠ ")(Fƹ)i=UR(" iduJ)(Fƹ)=1;Q(2.6)QthenAF 3(thesamealgebra,butwithcomrultiplicationFO)isagainaHopfalgebra.iWVeshallsayQthatAF isobtainedfromAviatrwistingbyFƹ;FnisacoScycleinasuitablesense.QThereUisadualvrersionofthetwistingopSeration,s\whichamountstoatwistofthemultiplicationQ[DTu].8LetAbSeaHopfalgebraandletË:URAA!|꨹bSeaninrvertible2-cocycle21@,thatisbY[n9(x(1) \|;y(1))(x(2)y(2);z)=URn9(y(1) \|;z(1))(x;y(2)z(2));36\on9(1;1)=UR1;SQforallx;yn9;z2A.ThenA {thesameAbutwiththemrultiplicationbSelorw{isagainaHopfQalgebra,whered|IdxOyË=URn9(x(1) \|;y(1))x(2)y(2) 1 ʵ(x(3);y(3)):퍑QAssumeRnorwthatH@GisaHopfalgebra,mRl;isabraidedHopfalgebrain2HbH l5YD,andAԹ=RJ#HV.QLet :A!Hsand:Hx4!AbSethecanonicalprojectionandinjection.@Let:HiHx4!|bSeQaninrvertible2-coScycle,andde ne \:URAA!|꨹bryc< \:=URn9( );Q sisȘaninrvertibleȘ2-coScycle,hwithinrverseȘ(n921 ʵ) .-ThemapsË:URA 5!H,:HQ!A7{areȘstillQHopf:algebramaps.)fBecausethecomrultiplicationisnotchanged,ŃthespaceofcoinvXariantsof*sisQRJ;thisisasubalgebraofAYthatwredenoteR;themultiplicationinR isgivenbygx:yË=URn9(x(1);y(1))x(0) \|y(0);x;yË2URRn=R:Q(2.7)'QEquation!(2.7)[Nfollorws!easilyusing(1.17)!.ClearlyV,ya=URn9(g;(y))gn7 1 ʵ(g;1)+n9(g;h)gyn7 1 ʵ(g;1)+n9(g;h)gh 1 ʵ(g;(y))Lʍ>ya=URn9(g;h)gy;&yn9:g>ya=URn9((y);g)gn7 1 ʵ(1;g)+n9(h;g)ygn7 1 ʵ(1;g)+n9(h;g)hgn7 1 ʵ((y);g)>ya=URn9(h;g)yg;ꍑQhenceHԓgn9:yË=UR(g;h)gy=UR(g;h)(g)yg=UR(g;h) 1 ʵ(h;g)(g)y:g;QwhicrhisequivXalentto(2.8).8Nowo(2.9) &#followsatonce,and(2.10)%Xfollowsfrom(2.7)\and(2.9):[x;yn9]c =URx:y:c(x yn9)UR=(g;h)xy(g;h) 1 ʵ(h;g)(g)(h;g)yxUR=(g;h)[x;y]c.y:ӿacffxff ̟ff ̎ ̄cffQThe@proSofofthefollorwingLemmaisclear,V sincethecomultiplicationofaHopfalgebraisnotQcrhangedbytwisting.QLemma2.10.;Lffet H`beaHopfalgebraandletR1TbeabraidedHopfalgebrain2HbH 1NYD.LetkL:ʍQHL_lHB!UR|Zbffeaninvertible2-cocycle.[ ThenR*isaNicholsalgebraifandonlyifR YisaNicholsQalgebrffa35inߍHbHYD#KI.ilffLʼnff3<;ff3<3<ff!3.wTypesofNicholsalgebrasQWVenorwdiscussseveralexamplesofNicholsalgebras.WVeareinterestedinexplicitpresentations,Qe. sUR1forsomei2f1;:::ʜ;Sg,thenx2NRAi n2Pƹ(RJ)./(b).u$Lffet軹1URi;j%S;i6=j;suchthatqijJqjvi 6=URq2n9rRAiiI,wherffe0rUR1thenx2NRAi n=0.8sInpffarticular,ifB(Vp)isanintegraldomain,u$then35qhh 4=UR1oritisnotarffoot35of1,forallh./(b).u$If35iUR6=j,then(ad\cxidڹ)2rb(xjf )=(rS)!q8:ii $eQy0k6r4(G91q2n9kRAiiIqijJqjviGz3x2rRAixjf :lrjXQ24ANDR9USKIEWITSCH!ANDSCHNEIDERc.}0(c).u$IfiUR6=jandqijJqjvi 6=q2n9rRAiiI,=rwherffe0rUR0suchthat(ad\cxidڹ)2r8:ij =(xjf )=0.0(e).u$IfB(Vp)isadomainof nitegrffowthandthereexistsksuchthatqk6k =1,thenqii 4H=1,foru$all351URiS,andqijJqjvi 6=1forall1i;j%S;i6=j.(f8).u$If35B(Vp)has nitegrffowthandiUR6=j,35thenu$min lfr>UR0j(ad\cxidڹ)2rb(xjf )=0g=minꘟGjfordNiiIgS fko>0jqijJqjviq2n9kRAii =UR1gG2:XQPrffoof.:Prarts!G(a)and(c)followfromLemma3.6.ܼPart(b)isapparentlywell-knownandappSearsQin[Ro2e]withoutproSof.8Prart(d)is[Ro2,Lemma20].8Therestfollorwsfromtheseparts.*cffxff ̟ff ̎ ̄cffBǍQWVenorwdiscusshowthetwistingopSeration,cf.Section2.3,a ectsNicrholsalgebrasofdiagonalQtrypSe.QDe nition3.8.D3_WVe1"shallsarythattwobraidedvectorspaces(V;c)and(W;d)ofdiagonaltypSe,Qwithmatrices(qijJ)and(wGbqij ),aretwist-effquivalentifdimV¹=URdimWnand,foralli;jӹ,qii =̙bURqii[and鸍qijJqjvi 6=̙bURqijM*bqjviVt:Q(3.1)QProp`osition3.9.N!?Lffetm(V;c)and(Wr;d)bffetwotwist-equivalentbraidedvectorspacesofdiagonalQtypffe,YwithR"matrices(qijJ)and(wGbqij );asaywithrespecttobasisx1;:::ʜx,Yresp.b1x10;:::}bx+.1ThenR"thereQexists35alineffarisomorphism Ë:URB(Vp)!B(Wƹ)35suchthat鸍 n9(xidڹ)UR=3bxi g~;1iS:Q(3.2)QPrffoof.:LetnbSethefreeabeliangroupofrank,ϼwithbasisg1;:::ʜ;g. WVede necrharactersQ1;:::ʜ;,b1;:::;bofbry鸍mjidڹ(gjf )UR=qjviJ;bi$A6(gj)=̙bqjvi;1i;j%S:QWVeconsiderVp,WQasYetter-DrinfeldmoSdulesorverbrydeclaringxi 2V2p8:iRAg8:i ,5bxi2V2\pbp8:iRAg8:i.FHence,QB(Vp);B(Wƹ)arebraidedHopfalgebrasin2b YDRa.QLetË:UR!|2  bSetheuniquebilinearformsucrhthat! ꍍ?n9(gid;gjf )UR=fX8 ԍ<:󶍍7=bqij&@q1 ij ʵ;ij;(퍍1;0i>jӹ;Q(3.3)!QitisagroupcoScycle.WVeclaimthat'UR:W!B(Vp)(1),'(Tbxi ,)=xidڹ,1iS,isanisomorphismQin2b YDRa.8Itclearlypreservresthecoaction;fortheaction,weassumeiURj{andcompute#gjOxi=URn9(gjf ;gidڹ) 1 ʵ(gi;gjf )i(gj)xiLʍ=UR(wGbqij ) 1 \|qijJqjvixi,=̙bqjvixid;~,rjXIPOINTED!HOPFALGEBRAS25c.}QandalsopgiOxj.=URn9(gid;gjf ) 1 ʵ(gj;gidڹ)j(gi)xjLʍ.=̙bURqijqn91 ij ʵqijJxj\=̙bURqijxjf ;Qwherewrehaveused(2.8)"andthehypSothesis(3.1)".iThisprovestheclaim.iByLemma3.6,? 'ʍQextendstoanisomorphism'UR:B(Wƹ)!B(Vp); Ë='21G$isthemapwreareloSokingfor.$$!cffxff ̟ff ̎ ̄cffQRemarks3.10.Ds(i).'Theemap %de nedintheproSofismruchemorethanjustlinear;{bry,(2.7)andQ(2.10)X,wrehaveZ n9(xy){=URn9(g;h) (x) (y);x2B(Vp)g; yË2B(V)he;Q(3.4)J n9([x;y]c.y){=URn9(g;h)[ (x); (y)]c.y;xUR2V pڍgc; yË2B(Vp)O he:Q(3.5)Q(ii). Anbraidedvrectorspace(V;c)ofdiagonaltypSe,Hwithmatrix(qijJ),istrwist-equivXalenttoʍQ(Wr;d),withasymmetricmatrix(wGbqij ).ǍQ3.3.Braidingsofdiagonaltyp`ebutnotCartan.QInthenextChapter, lwreshallconcentrateonbraidingsofCartantypSe.ThereareafewexamplesQof\NicrholsalgebrasB(Vp)of nitegrouptypSeandrank2,y?whicharenotofCartantypSe,y?butwhereQwre&knowthatthedimensionis nite.WVenowlistthemall,Mfollowing[N,GS~vn3 ].ThebraidedvectorQspaceGisnecessarilyofdiagonaltrypSe;v?weGshallgivrethematrixQofthebraiding,^theconstraintsQon.{theirenrtriesandthedimensiondofB(Vp).ZBelow,?p!n9,resp.Z,denotes.{anarbitraryprimitiveQthirdroSotof1,resp.8di erenrtfrom1.gXqʍ-q111q12-q211c1@qKnY; q n91ڍ11 =URq12 q21 UZ6=1;Tde=UR4ord(q12 q21):Q(3.6)3:Xqʍ-q111cq12-q214!?eqJn; q n91ڍ11 =URq12 q21 UZ6=1;!n9 1 ʵ;Tde=UR9ord(q11 )ord(q12q21!n9):Q(3.7)Xqʍ-12cNq12q215!@qKnY; q12 q21 UZ=UR1;Tde=UR108:Q(3.8)Xqʍ-12cNq12q215!@qKnY; q12 q21 UZ=UR!n9;Tde=UR72:Q(3.9)Xqʍ-12cNq12q215!@qKnY; q12 q21 UZ=UR!n9;Tde=UR36:Q(3.10)rjXQ26ANDR9USKIEWITSCH!ANDSCHNEIDERc.}Xqʍ-12cNq12q216yF@qKnY; q12 q21 UZ=UR 2 D};Tde=UR4ord()ord( 1 D}):Q(3.11)ꦟ"'Q3.4.Braidingsof nitenon-ab`eliangrouptype.QWVebSeginwithaclassofexamplesstudiedin[MiS].QLetpbSeagroupandTUasubsetsucrhthatforallg02;t2T;gn9tg21 2Tƹ.7ThruspT#6isaQunionofconjugacyclassesof.äLet:&TE!|nf0gbSeafunctionsucrhthatforallgn9;h2QandtUR2Tƹ,J(1;t)8=UR1;Q(3.12)OpX(gn9h;t)8=UR(gn9;hth 1 \|)(h;t):Q(3.13)܍QWVeXcanthende neaYetter-DrinfeldmoSduleVy= Vp(;T;)Xorverwith|-basisxtʹ,tt 2T;andQactionandcoactionofWngivrenbygn9xt* =UR(gn9;t)xgI{tg1-;Q(3.14)36Xs2(xtʹ)* =URt xtQ(3.15)0QforallgË2UR;t2Tƹ.QConrverselyV,ӿifthefunctionde nesaYetter-DrinfeldmoSduleonthevrectorspaceVjtby(3.14)!,Q(3.15)X,thensatis es(3.12)!,(3.13).QNotethatthebraidingcofVp(;T;)isdeterminedbrypc(xs xtʹ)UR=(s;t)xstsq1 xt|rforall)s;tUR2T;QhencebrythevXaluesofonTLnTƹ.QThe~ mainexamplescomefromthetheoryofCorxetergroups([BL,ChapitreIV]).LetS0bSeaQsubsetofthegroupWOofelemenrtsoforder2.DFVoralls;s20o2tS`letm(s;s209)bSetheorderofss20.Q(Wr;S׹)_iscalledaCoxetersystemandW-%aCoxetergrffoupifW-%isgeneratedbryS>6withde ningꍑQrelations(ss209)2m(s;s-:0)A=UR1foralls;s20#2Ssucrhthatm(s;s209)is nite.QLet (Wr;S׹)bSeaCoxetersystem.FVoranyg2Wthereisasequence(s1;:::ʜ;sq)ofelementsinQSwithg̾=^s1nZV\Vsq.IIfq^Iisminimalamongallsucrhrepresentations,jthenq̾=^lC(gn9)iscalledtheQlength꨹ofgn9,and(s1;:::ʜ;sq)isarffeduced35representation꨹ofg.DQDe nition3.11.J_Let((Wr;S׹)bSeaCoxetersystem,ZandT_=Jfgn9sg21Njg2Wr;s2Sg.aDe neʍQUR:WLnT!|nf0g꨹bryp!(gn9;t)UR=(1) lK(gI{)forall4C$gË2Wr;t2T:Q(3.16)+$QThis>satis es(3.12)&and(3.13)"E.5!ThruswehaveassoSciatedtoeachCoxetergrouptheYVetter-QDrinfeldmoSduleVp(Wr;T;)UR22WbW YDS.\rjXIPOINTED!HOPFALGEBRAS27c.}QTheJfunctionssatisfyingt(3.12)!{,j(3.13)$canbSeconstructeduptoadiagonalcrhangeofthebasisQfrom<4crharactersofthecentralizersofelementsintheconjugacyclasses.-ThisisaspSecialcaseofQthePdescriptionofthesimplemoSdulesin2b XDYDzY(see[W u]andalso[L4 7o]);theequivXalenrtclassi cationQofXthesimpleHopfbimoSdulesorverXwrasobtainedin[DPRs](over|)andthenin[Ci ](overanyQ eld).𰍑QLetUtbSeanelemenrtin.GWVedenotebyOt秹and2ttheconjugacyclassandthecenrtralizeroftinQ.!Let UbSeanryleft|2tʹ-module.!ItiseasytoseethattheinducedrepresenrtationV¹=UR|x ս|tUQisaYVetter-DrinfeldmoSduleorverwiththeinducedactionofandthecoaction>:Ȅ:URV!| V; s2(g u)UR=gn9tg 1u] g u꨹forall&/gË2UR;u2U:QWVewilldenotethisYetter-DrinfeldmoSduleorverbryM@(t;U).QAssumethatis nite.4ThenV¹=URM@(t;U)isasimpleYVetter-DrinfeldmoSduleifUisasimpleQrepresenrtation of2tʹ,&,andeachsimplemoSdulein2bYDQӹhasthisform.IfwetakefromeachconjugacyQclassdoneelemenrttandnon-isomorphicsimple2tʹ-moSdules,:ranytwoofthesesimpleYVetter-DrinfeldQmoSdulesarenon-isomorphic.QLet~sid;1URiS,"beacompletesystemofrepresenrtativesoftheresidueclassesof2tʹ.WVede neQti,=URsidts1 iG$forall1iS.8Thrus= t!UROt; sidڹ t7!tid;1iS;Qissbijectivre,andasavectorspace,Vչ==eL1i,silV |U@.FVorallg2=eand1iS,thereisaQuniquelydetermined1URj%#withs1 j \|gn9si,22tʹ,Sandtheactionofg>7onsi t1u;u2U@,Sisgivrenby|lgn9si uUR=sj (s1 j \|gn9sidڹ)u:حQInUparticular,M@ifUG9isaone-dimensional2tʹ-moSdulewithbasisuandactionhu8+=(gn9)uUforallQh`22tde ned'brythecharacter`:2tO*!|Gnf0g,7then'V:hasabasisxi":=`si9! Gu;1iS,andQtheactionandcoactionofaregivrenbygn9xi,=UR(s1 j \|gsidڹ)xjand*~s2(xi)=xjf ;|Qifs1 j \|gn9si,2UR2tʹ..Notethatgtidg21 =URtjf ..HencethemoSdulewrehaveconstructedisVp(;T;), where̍QTnistheconjugacyclassoft,andisgivrenby(gn9;tidڹ)UR=(s1 j \|gsidڹ).0QWVenorwconstructanotherexampleofafunctionsatisfyingo(3.12)"w,(3.13)!.RQDe nition3.12.J_Let`TE&bSethesetofalltranspositionsinthesymmetricgroupSnP.c De ne:ʍQSnRT!UR|nf0g꨹forallgË2SnP;1ig(jӹ):Q(3.17)&f0QLetG|t=(12). O[ThecenrtralizeroftinSn ̹ish(34);(45);:::ʜ;(n(1;n)i[h(34);(45);:::ʜ;(nQ1;n)i(12).BLetbSethecrharacterof(SnP)2t]with((ijӹ))UR=1forall3URi6leti,:Rn!RbSethealgebraautomorphismgivrenbytheactionofgidڹ.WQProp`osition3.13.T?1)Forall1_iS,5therffeexistsauniquelydetermined(id;idڹ)-derivationʍQDi,:URRn!RLwith35Didڹ(xjf )=i;j ڍ(Krffoneckers2)forallj.Q2)35IFչ=URI(Vp),thatisRn=B(Vp),ifandonlyifT*37 U_37i=1tker.4(Didڹ)=|1:VffLʼnff3<;ff3<3<ff덍QTheseskrewderivXationsareveryusefulto ndrelationsofB(Vp).WVeconsidersomespSecialQcases.QExample3.14.CLetW̹=WSnP;n2,andT̹=f(ijӹ)j1i5itisnotknorwnwhetherEn is nite-dimensional.QIn4[GS~vn3<,Y?5.3.2]anotherexampleofa nite-dimensionalNicrholsalgebraofabraidedvectorspaceQ(V;c)of nitegrouptrypSeisgivenwithdim(Vp)=4anddim(B(V))=72:Thede ningrelationsQofB(Vp)arequadraticandoforder6.`QByaresultofMonrtgomery[M29],anypSointedHopfalgebraBcanbSedecomposedasacrossedQproSductzV BX'URA#|G; Xa2-coScycleōQofA,itslink-indecompSosablecomponenrtcontaining1(aHopfsubalgebra)andagroupalgebraQ|G.GHorwever,e:theCstructureofsucrhlink-indecompSosableHopfalgebrasA,inparticularinthecaseQwhenAis nite-dimensionalandthegroupofitsgroup-likreelementsG(A)isnon-abSelian,|isnotQknorwn.+TVoùde nelink-indeffcomposablelpointedHopfalgebras,˃wreùrecallthede nitionofthequiverQofAin[M29].+ThevrerticesofthequiverofAaretheelementsofthegroupG(A);>forgn9;hUR2G(A),Qtherekexistsanarrorwfromhtog$ifPgI{;h c(A)isnon-trivial,thatisif|(g.?h)UR$PgI{;h(A).'vThekHopfQalgebraAiscalledlink-indecompSosable,ifitsquivrerisconnectedasanundirectedgraph.;1.$Inthiscase,!strongrestrictionsareQknorwnforB(Vp)tobSe nite-dimensional.8BySchur'slemma,tactsasascalarqXonU@.QProp`osition3.19.T?[GS~vn3<,03.1]fAssumethatdimB(Vp)is nite.,IfdimUٹ3,sthenq#=1;andʍQif35dim{U6=UR2,35thenqË=1orqnisarffoot35ofunityoforffderthree.ffLʼnff3<;ff3<3<ffrQIn(theproSofofProposition3.19,xaresultofLusztigonbraidingsofCartantrype(see[AS25,QTheorem3.1])isused.8InasimilarwrayGraS~vnashorwedQProp`osition3.20.T?[GS~vn3<,/Y3.2]eLffetbea nitegroupofoddorder,rXandVOK22b YD.AssumethatʍQB(Vp)dis nite-dimensional.O#ThenthemultiplicityofanysimpleYetter-DrinfeldmoffduleoverasQa35dirffectsummandinVϥisatmost2.\rjXIPOINTED!HOPFALGEBRAS31c.}QInpffarticular, uptoisomorphismthereareonly nitelymanyYetter-DrinfeldmodulesV22URbUR $FYDʍQsuch35thatB(Vp)is nite-dimensional.)rffLʼnff3<;ff3<3<ff+HQThesecondstatemenrtinPropSosition3.20wasaconjectureinapreliminaryversionof[AS25].CȍQ3.5.Braidingsof(in nite)grouptyp`e.QWVebrie ymenrtionNicholsalgebrasoverafreeabSeliangroupof niterankwithabraidingQwhicrhisnotdiagonal.QExample3.21.CLetuUR=hgn9ibSeafreegroupinonegenerator.5LetV(t;2)betheYVetter-DrinfeldQmoSdule[ofdimension2sucrhthatV(t;2)=V(t;2)g ~Yand[theactionofgonV(t;2)isgivren,inaQbasisx1;x2,bry덒gx1V=URtx1;gx2V=URtx2j+x1:_QHeretUR2|2x.8Then:Q(a).8IftisnotaroSotof1,thenB(V(t;2))UR=Tƹ(V(t;2)).Q(b).If?itUR=1,athenB(V(1;2))=|;x~this?iisthewrell-known?iJordanianQquanrtumplane.QExample3.22.CMoreYgenerallyV,iftUR2|2x,letYV(t;S)betheYVetter-DrinfeldmoduleofdimensionQUR2sucrhthatV(t;S)=V(t;S)gandtheactionofgXonV(t;S)isgivren,inabasisx1;:::ʜ;x,bryPfgx1V=URtx1;gxj\=URtxj+xjv1B; 2j%S:QNotethereisaninclusionofYVetter-DrinfeldmoSdulesV(t;2)UR,!V(t;S);ehence,iftisnotarootofQ1,B(V(t;S))hasexponenrtialgrowth.QQuestion3.23.DQLetWI ޹bSetheWVeylgroupcorrespondingtotheCartanmatrix(aijJ)i;jv2IU;+wreidentifyitwithQasubgroupoftheWVeylgroupWdcorrespSondingtotheCartanmatrix(aijJ).?QWe xareducedQdecompSosition%ofthelongestelemenrt!0;IVofWIrintermsofsimplere ections.:VThenweobtainaQreducedY0decompSositionofthelongestelemenrt!0z=vsiq1 :::ZsiX.P ofWUƹfromtheexpressionof!0 4asQproSduct}ofthe!0;I kŹ'sinsome xedorderofthecomponenrts,say}theorderarisingfromtheorderQofthevrertices.8Therefore j\:=URsiq1 :::Zsi8:jY1u( i8:jO)isanumerationof2+x.QExample4.1.jz[L3 7o]ZLffet(V;c)beabraidedvectorspacewithbraidingmatrix-(4.1)9.IfqTisnotaQrffoot35of1,theneB(V)UR=|hx1;:::ʜ;xjad\c(xidڹ) 1a8:ijwxj\=0;i6=ji:Սӓ?ffLʼnff3<;ff3<3<ffQThe,TheoremsarysthatB(V)isthewell-known"pSositivepart"U2@+RAq]\(gn9)ofthequantumenvelopingQalgebraofgn9.>QTVo]statethefollorwingimpSortantTheorem,|werecallthede nitionofbraidedcommutatorsQ(1.20)X.*4Lusztigde nedroSotvrectorsX J2URB(V),q h22+ 2 7o].Onecanseefrom[L1 7o,L2]that,qupQto3anon-zeroscalar, eacrhroSotvectorcanbSewrittenasaniteratedbraidedcommutatorinsomeQsequenceX`q1#;:::ʜ;X`a p-ofsimpleroSotvrectorssuchas[[X`q1#;[X`q2;X`q3]c]c;[X`q4#;X`q5]c]c.8Seealso[Ri 7].QTheorem4.3.>jz[L1 7o,EL2}%,EL3,ERo1Y,EMu ].a~Lffet(V;c)beabraidedvectorspacewithbraidingmatrixQ(4.1)x. Assumethatqisarffootof1ofoffddorderN@;andthat3doesnotdivideNifthereexistsQIF2URXof35typffeG2.QThe35algebrffaB(V)is nitedimensionalifandonlyif(aijJ)isa niteCartanmatrix.QIf35thishappffens,thenB(V)canbepresentedbygeneratorsXid,1URiS,35andrelationszHadV˟cD(Xidڹ) 1a8:ijw(Xjf )=UR0;i6=j;Q(4.2)LʍuX Nڍ =UR0; h2 +x:Q(4.3)QMorffeover,35thefollowingelementscffonstituteabasisofB(V):`TXnhq1* q1 Xnhq2* q2 :::LXhX.P X.P ~`;0URhj\N1;1j%PS:ӓ?ffLʼnff3<;ff3<3<ffQThe TheoremsarysthatB(V)isthewell-known"pSositivepart"u2+RAqx(gn9)oftheso-calledFVrobSenius-QLusztigkrernelofgn9.! PrjXIPOINTED!HOPFALGEBRAS33c.}QMotivXatedbrytheprecedingTheoremsandresults,weintroSducedthefollowingnotionin[AS25]Q(seealso[FrG]).\QDe nition4.4.D3_Let(V;c)abraidedvrectorspaceofdiagonaltypSewithbasisx1;:::ʜ;x,andʍQmatrix(qijJ),thatis[to\c(xi xjf )UR=qi;j Xxj xid;ꦹforall(/1i;j%S:{QWVeb shallsarythat(V;c)isofCartantypffeb ifforalli;jӹ,}_qij isaroSotofunitryV,qii 6=UR1andthereexistsQaij 62URZ꨹sucrhthatqijJqjvi 6=URqzn9a8:ij Mii U;ꦹforall(/1i;j%S:Q(4.4)QTheinrtegersaij 5areuniquelydeterminedwhenchoseninthefollowingway:xaii =UR2;Q(4.5)Lʍ #Qord~qii <URaij 60; i6=j:Q(4.6)QIt^follorwsthat(aijJ)isageneralizedCartanmatrix(GCM)VinthesenseofthebSook^[K "w]./rWVeshallʍQadapt StheterminologyfromgeneralizedCartanmatricesandDynkindiagramstobraidingsofQCartan/trypSe.FVorinstance,@weshallsaythat(V;c)isof niterCartantypffe/ifitisofCartantypSeQand thecorrespSondingGCMisactuallyof nitetrype,i.Te.%aCartanmatrixassociatedtoa niteQdimensionalZpsemisimpleLiealgebra.7WVeshallsarythataYetter-DrinfeldmoSduleVisofCartanQtypffe꨹ifthematrix(qijJ)asabSorveisofCartantrype.QTVoformrulateour rstmainresult,weneedonemorede nitionfrom[AS25]forbraidingswhichQareclosetothebraidingsoftheFVrobSenius-Lusztigkrernels.QDe nition4.5.D3_Let(V;c)bSeabraidedvrectorspaceofCartantypSewithCartanmatrix(aijJ).QWVesarythat(V;c)isofFL-typffeifthereexistpSositiveintegersd1;:::ʜ;d9suchthatXݹ2+ UsucharepresentationofX itasaniteratedbraidedcommutator.QFVor2ageneralbraidedvrectorspace(V;c)of niteCartantypSe,wede neroSotvectorsx ɹintheQtensorvalgebraTƹ(Vp),A h2UR2+x,asvthesameformaliterationofbraidedcommrutatorsintheelementsQx1;:::ʜ;x `"insteadofX1;:::ʜ;X `"butwithrespSecttothebraidingcgivrenbythegeneralmatrixQ(`qijJ).QTheorem4.6.>jz[AS25,AAS49>].>LffetIѹ(V;c)beabraidedvectorspaceofCartantype.>WealsoassumeʍQthat35qij ~hasoffddorderforalli;j."ԠrjXQ34ANDR9USKIEWITSCH!ANDSCHNEIDERc.}Q(i).2WAssumethat(V;c)isloffcallyofFL-typffeandthat,;foralli,;theorffderofqii isrelativelyprimeʍQto3wheneveraij =c3forsomej,Landisdi erffentfrom3,L5,7,11,13,17.lIfB(Vp)is niteQdimensional,35then(V;c)isof niteCartantypffe.Q(ii).NIfb,(V;c)isof niteCartantypffe,mthenB(Vp)is nitedimensional,andifmorffeoverb,3doffesQnotidividetheorffderofqii foralliinaconnectedcomponentoftheDynkindiagramoftypeG2,Qthen0dim5B(Vp)UR=DYI2X6N@dimHnX.I%I;)QwherffeȝNI$=URordN*(qiiI)foralliUR2I andȝIF2X.BTheNicholsalgebrffaB(Vp)ispresentedbygeneratorsQxid,351URiS,andrffelationsxad cO(xidڹ) 1a8:ijw(xjf )̈=UR0;i6=j;Q(4.9)Lʍx NX.Iڍ ̈=UR0; h2+8OIx;33IF2X:Q(4.10)QMorffeover,35thefollowingelementscffonstituteabasisofB(Vp):I<,xnhq1* q1 xnhq2* q2 :::hX.P X.P ;0URhj\NIy1;33if j2I;1j%PS:ӓ?ffLʼnff3<;ff3<3<ff򍍑QLetުb!V$B(Vp)$bSethebraidedHopfalgebrain2b YD"generatedbryx1;:::ʜ;x ʵwithrelations(4.9)>0,*Qwherethexidڹ'sareprimitivre.LetK,`(Vp)bSethesubalgebraofުyb!VBU(V)generatedbryx2NX.IRA  , 2J+8OIx, ׍QIF2URXӹ;itisaYVetter-DrinfeldsubmoSduleofުb!VB9(Vp).WQTheorem4.7.>jz[AS45]35K,`(Vp)isabrffaided35Hopfsubffalgebra35in2b )YD #ofު+b!VBƹ(V).h*ffLʼnff3<;ff3<3<ffᓍaѹ5.''wClassifica32tionofpointedHopfalgebrasbytheliftingmethodQ5.1.LiftingofCartantyp`e.QWVenJpropSosetosubdividetheclassi cationproblemfor nite-dimensionalpoinrtedHopfalgebrasQinrtothefollowingproblems:e.(a).u$DetermineallbraidedvrectorspacesVofgrouptypSesuchthatB(Vp)is nitedimensional.ˍ(b).u$Givrena nitegroup,]determineallrealizationsofbraidedvectorspacesVasin(a)asu$YVetter-DrinfeldmoSdulesorver.0(c).u$Theqliftingproblem:rFVorB(Vp)asin(a),kccomputeallHopfalgebrasAsucrhthatgrAa5'u$B(Vp)#H.(d).u$InrvestigateQwhetheranry nitedimensionalpSointedHopfalgebraisgeneratedasanalgebrau$bryitsgroup-likeandskew-primitiveelements.QProblem<8(a)wrasdiscussedinChapters3and4. -WVehaveseentheveryimpSortantclassofQbraidingsհof niteCartantrypSeandsomeisolatedexampleswheretheNicholsalgebrais nite-Qdimensional.8Butthegeneralcaseofproblem(a)seemstorequirecompletelynewideas.#.rjXIPOINTED!HOPFALGEBRAS35c.}QProblemP(b)isofcomputationalnature.FVorbraidingsof niteCartantrypSewithCartanmatrixQ(aijJ)1i;jv$and8anabSeliangroupwrehavetocomputeelementsg1;:::ʜ;g k2Qandcharacters9Q1;:::ʜ;2wbURmsucrhthatۍtYidڹ(gjf )j(gi)UR=i(gi) a8:ij ;ꦹforall(/1i;j%S:Q(5.1)eQTVoP ndtheseelemenrtsonehastosolveasystemofquadraticcongruencesinseveralunknowns.InQmanrycasestheydonotexist.*YInparticular,if>UR2(ord)22,thenthebraidingcannotbSerealizedQorverthegroup.8WVereferto[AS25,Section8]fordetails.QProblem(d)isthesubjectofSection5.4.QWVewillnorwdiscusstheliftingproblem(c).QTheR+coradical ltration|@=A0 yDA1::: ofR+apSoinrtedHopfalgebraAisstableunderQthepOadjoinrtactionofthegroup.FVorabSeliangroupsand nite-dimensionalHopfalgebras,theQfollorwingstrongerresultholds.4ZItisthestartingpSointoftheliftingproScedure,andwewilluseitQsevreraltimes.QIfu0Misa|-moSdule,wredenotebyM@2 I#=Afm2Mjgn9m=(g)m꨹forall&/gX2g,2wb ,theQisotrypiccompSonentoftypSe.[QLemma5.1.5LffetNAbea nite-dimensionalHopfalgebrawithabeliangroupG(A)UR=NanddiagramʍQRJ.fiLffet35V¹=URR(1)22b $FYD@with35bffasisxi,2V2p8:iRAg8:i ;gi2;i2wb ;1iS:Q(a).NLTheisotypiccffomponentoftrivialtypffeofA1isA0.Therffefore,UA1V=URA0 (6="Ph(A1)2 )andl>6="Ph(A1) 2 'p r D!#A1=XA02 >'pV xVp#|:Q(5.2)EQ(b).fiFor35allgË2UR,2wbwith6=",GzXPgI{;1 qv(A) vSs6=UR0() therffe35issomeR1`:gË=g`;=`;Q(5.3)LʍHԨPgI{;1 qv(A) "vSs=UR|(1gn9):Q(5.4)QPrffoof.:(a)follorwsfrom[AS15,Lemma3.1]andimplies(b).8See[AS1,Lemma5.3].Ecffxff ̟ff ̎ ̄cffQWVe&passumethatAisa nite-dimensionalpSoinrtedHopfalgebrawithabeliangroupG(A)=,QandthatvgrAUR'B(Vp)#|;,Qwhere҃V22URbUR $FYDisagivrenYVetter-DrinfeldmoSdulewithbasisxi,2URV2p8:iRAg8:i ;g1:::;g2;1;:::ʜ;28wbQ;1URiS:QWVe+ rstliftthebasiselemenrtsxidڹ./Using(5.2),ԪwechoSoseai,2URP(A)O8:ig8:i,r;1#suchthatthecanonicalkQimage!Mofai'inA1=XA0Qisxi(whicrhweidentifywithxidڹ#1),I1URiS.Since!Mtheelementsxi'togetherQwithxgenerateB(Vp)#|,յitfollorwsfromastandardargumentthata1;:::ʜ;a andtheelementsQingenerateAasanalgebra.ۍQOur}kaimisto ndrelationsbSetrween}kthea20RAidsandtheelemenrtsinwhichde neaquotientHopfQalgebraofthecorrectdimensiondim&B(Vp)ord().0Theideaisto"lift\therelationsbSetrweentheQx20RAids꨹andtheelemenrtsininB(Vp)#|.$=rjXQ36ANDR9USKIEWITSCH!ANDSCHNEIDERc.}QWVemnorwassumemoreoverthatV<ݹisof niteCartantypSewithCartanmatrix(aijJ)withrespectQtothebasisx1;:::ʜ;x,thatis(5.1)\holds.8WVealsoassumexordq(jf (gidڹ))isoSddforallI,i;j;Q(5.5)36xNi,=URordN*(idڹ(gi))isprimeto3I,forallkr4iUR2I;IF2X{oftrypSe-~'G2:Q(5.6)gߍQWVe xapresenrtation=hy1i_bhyi,7 anddenotebyM`utheorderofy`,7 1`n9.QThenTheorem4.6andformrulas(1.17)& implythatB(Vp)#|canbSepresentedbygeneratorsh`,Q1UR`n9,andxidڹ,1URi>6withde ningrelationsXhtMi?``=UR1; 1`n9;Q(5.7)36Xh`ht=URhth`; 1t<`n9;Q(5.8)Xh`xi,=URidڹ(y`)xih`; 1UR`;1iS;Q(5.9)Xx NX.Iڍ r=UR0; h2+8OIx;IF2Xӹ;Q(5.10)OpXad((xidڹ) 1a8:ijw(xjf )UR=0;i6=j;Q(5.11)3QandwheretheHopfalgebrastructureisdeterminedbryX(h`)UR=h` h`; 1`n9;Q(5.12)X(xidڹ)UR=xi 1+gi xid; 1URiS:Q(5.13)3QThrus3Aisgeneratedbytheelementsaid;1piS,Eand3hl!;1ln9.By3ourpreviouscrhoice,Qrelations(5.7),(5.8),(5.9)\and(5.12)!,(5.13)%XallholdinAwiththex20RAidsreplacedbrythea20RAis.QTheremainingproblemistoliftthequanrtumSerrerelations(5.11)$7>andtheroSotvectorrelationsQ(5.10)X.8WVewilldothisinthenexttrwoSections."Q0Q5.2.LiftingthequantumSerrerelations.QWVedividetheprobleminrtotwocases.Tu#u$LiftingeRofthe\quanrtumSerrerelations"xidxjd6,jf (gi)xjxi=&0,wheneRi6=j%areindi erenrtu$compSonenrtsoftheDynkindiagram.u#u$Lifting~ofthe\quanrtumSerrerelations"adcz(xidڹ)21a8:ijw(xjf )qL=0,4when~i6=j>Qareinthesameu$compSonenrtoftheDynkindiagram.TQThe rstcaseissettledinthenextLemmafrom[AS45].QLemma5.2.5Assume35that1URi;j%S;ijz[AS45] Lffetbea niteabeliangroupandDbalinkingdatumof niteCartantypeʍQforǹwithYetter-DrinfeldmoffduleVp.=Thenu(DUV)isa nite-dimensionalpointedHopfalgebrawithQgr=u(DUV)UR'B(Vp)#|.}]ffLʼnff3<;ff3<3<ff~QTheproSofoftheTheoremisbryinductiononthenumbSerofirreduciblecomponenrtsoftheDynkinQdiagram.In(>theinductionstepanewHopfalgebraisconstructedbrytwistingthemultiplicationQofthetensorproSductoftrwoHopfalgebrasbrya2-cocycle.The2-cocycleisde nedintermsofQthelinkingdatum.eQNote-thattheFVrobSenius-Lusztigkrerneluq(g)ofasemisimpleLiealgebragisaspecialcaseofQu(DUV).!Here`theDynkindiagramofDisthedisjoinrtunionoftwocopiesoftheDynkindiagramofQg,LFand$correspSondingpoinrtsarelinkedpairwise.ButmanyotherlinkingsarepSossible,LFforexampleQ4copiesofA3?linkredinacircle[AS45,5.13].See[D9]foracombinatoricaldescriptionofalllinkingsQofDynkindiagrams.QLetusnorwturntothesecondcase.*Luckilyitturnsoutthat(uptosomesmallorderexceptions)QinthesecondcasetheSerrerelationssimplyholdintheliftedsituationwithoutanrychange.HQTheorem5.6.>jz[AS45,=Theorem-6.8].LffetpII2lX.AssumethatNI6=l3.IfIaisoftypffeBnP,Cn orʍQF4,srffesp.G2,assumegfurtherthatNI6=<5,rffesp.NI6=<7.ThengthequantumSerrfferelationsholdQfor35alli;j%2URI;i6=j;ij.apffLʼnff3<;ff3<3<ff~Q5.3.Liftingthero`otvectorrelations.QAssume< rstthattheroSot PUR3isodd[AS35]. (c)u$TheDynkindiagramisarbitraryV,butwreassumegn9N8:imi/=UR1foralli[AS45].ʄ(d)u$TheDynkindiagramisoftrypSeAnP,anynUR2,andN6>UR3,seeSection6ofthispapSer.QThecasesA2;Np=3andB2;N+oSddand6N6=5,Gwrererecentlydonein[BDR].*HereNIdenotestheQcommonorderofidڹ(gi)foralliwhentheDynkindiagramisconnected.ፑQ5.4.Generationindegreeone.QLetusnorwdiscussstep(d)oftheLiftingmethoSd.QItAisnotdiculttoshorwthatourconjecture2.7abSoutNicholsalgebras,(inthesettingofQHB=UR|,isequivXalenrtto䪍QConjecture5.7.Jf[AS35]. FA2nypffointed nitedimensionalHopfalgebraover|isgeneratedbyʍQgrffoup-like35andskew-primitiveelements.QWVe3harveseeninSection2.1thatthecorrespSondingconjectureisfalsewhentheHopfalgebraisQin nite-dimensional{YorwhentheHopfalgebrais nite-dimensionalandthecrharacteristicoftheQ eldis>UR0.8Astrongindicationthattheconjectureistrueisgivrenby:QTheorem5.8.>jz[AS45].0LffetvAbea nite-dimensionalpointedHopfalgebrawithcoradical|andʍQdiagrffam35RJ,thatisϞ<.grʣ]AUR'RJ#|:BЍQAssume%thatRJ(1)isaYetter-Drinfeldmoffduleof niteCartantypewithbraiding(qijJ)1i;jvS.[dForQallsi,letqiw=qiiI;Ni=orffdlJ(qidڹ)."Assumesthatorffd(qijJ)isoffddandNiMisnotdivisibleby3and>7Qfor35all1URi;j%S.fQ1.u$For^any1=imcffontained^inacffonnected^componentoftypeBnP,Cn /orF4 resp.gG2,u$assume35thatNiisnotdivisibleby5rffesp.fiby5or7.Q2.u$If35iandjbffelongtodi erentcomponents,assumeqidqj\=UR1orord, (qiqjf )UR=Ni.QThenRBisgenerffatedasanalgebrffabyRJ(1),thatisAisgeneratedbyskew-primitiveandgroup-likeQelements.PffLʼnff3<;ff3<3<ffQLet"usdiscusstheideaoftheproSofofTheorem5.8.-^ArtonedecisivepSoint, weuseourpreviousQresultsabSoutbraidingsofCartantrypeofrank2.QLet9S<bSethegradeddualofRJ.fBythedualitryprincipleinLemma2.4,SisgeneratedindegreeQonesincePƹ(RJ)h=R(1).ZOurproblemistoshorwthatRGisgeneratedindegreeone,thatisSԹisaQNicrholsalgebra.QSince7TS+isgeneratedindegreeone,JthereisasurjectionofgradedbraidedHopfalgebrasS!QB(Vp),whereVv =ٙS׹(1)hasthesamebraidingasRJ(1).8Butwreknowthede ningrelationsofQB(Vp),sinceitisof niteCartantrypSe.8SohavetoshowthattheserelationsalsoholdinS׹.(rjXQ40ANDR9USKIEWITSCH!ANDSCHNEIDERc.}QInthecaseofaquanrtumSerrerelationad9ycg(xidڹ)21a8:ijw(xjf )=0,i6=jӹ,wreconsidertheYVetter-QDrinfeld3KsubmoSduleWofS"generatedbryxi%andadΟcG(xidڹ)21a8:ijw(xjf )andassumethatadΟc(xidڹ)21a8:ijw(xjf )Q6=0.Theassumptions(1)and(2)oftheTheoremguaranrteethatWalsoisofCartantypSe,-KbutQnotof niteCartantrypSe.8ThusadG+cu(xidڹ)21a8:ijw(xjf )UR=0inS׹.ʍQSince`!thequanrtumSerrerelationsholdinS׹,{theroSotvectorrelationsfollowautomaticallyfromQthenextLemmawhicrhisaconsequenceofTheorem4.7.uQLemma5.9.5[AS45,Lemma7.5]LffetS=]n0S׹(n)bea nite-dimensionalgradedHopfalgebraQin2b YD!suchthatS׹(0)Y=|1. AssumethatVQ=YS(1)isofCartantypffewithbasis(xidڹ)1i;jv$asQdescribffed35inthebffeginning35ofthisSeffction.fiAssumetheSerrerelationsX;(ad\cxidڹ) 1a8:ijwxj\=UR035forall'1i;j%S;i6=jandij:QThen35therffoot35vectorrelations x NX.Iڍ r=UR0; h2+8OIx;IF2X;Qhold35inS.!ffLʼnff3<;ff3<3<ffDQ5.5.Applications.ʍQAs'laspSecialcaseofthetheoryexplainedaborve'lweobtainacompleteanswertotheclassi cationQprobleminasigni canrtcase.QTheorem5.10.E*z[AS45]4Lffetpbeaprime>UR17,՛s1,and4=(Z=(p))2sn<.?iUptoisomorphismtherffeQarffesonly nitelymany nite-dimensionalpointedHopfalgebrasAwithG(A)B'.(TheysallhaveQthe35form18AUR'u(DUV);33wherffe'YDis35alinkingdatumof niteCartantypffeforudimԺAURn()6pQfor35any nite-dimensionalpffointed35HopfalgebrffaAwithG(A)UR=|.dffLʼnff3<;ff3<3<ffQRemark5.12.?ҹAsacorollaryoftheTheoremanditsproSof,Hwregetthecompleteclassi cationofQall5S nitedimensionalpSoinrtedHopfalgebraswithcoradicalofprimedimensionp,Gpj6=2;5;7.ByQ[AS25,8Theorem)91.3],theonlypSossibilitiesfortheCartanmatrixofD~withofoddprimeorderQp꨹are6q(a)u$A1andA1jA1,ʄ(b)u$A2,ifpUR=3orpUR1moSd":3,(c)u$B2,ifpUR1moSd":4,ʄ(d)u$G2,ifpUR1moSd":3,(e)u$A2jA1andA2A1,ifpUR=3.QThe9vNicrholsalgebrasoverZ=(p)fortheseCartanmatricesarelistedin[AS25,\Theorem1.3].HenceʍQwreOobtainfromTheorem5.10forpUR6=2;5;7OthebSosonizationsoftheNicholsalgebras,theliftingsQinicase(a),Cthatisquanrtumlinesandquantumplanes[AS15],CandtheliftingsoftypSeA2m[AS35]inQcase(b).QThisresultwrasalsoobtainedbyMusson[Mus],usingtheliftingmethoSdand[AS25].QThe\+casepUR=2\+wrasalreadydonein[N]. aInthiscasethedimensionofthepSointedHopfalgebrasQwith2-dimensionalcoradicalisnotbSounded.nxQLetfRusmenrtionbrie ysomeclassi cationresultsforHopfalgebrasofspSecialorderwhichcanQbSeEobtainedbrythemethodswrehavedescribSed.Letp>2Ebeaprime.ThenallpoinrtedHopfQalgebras!Aofdimensionp2nP;1*n5!areknorwn.FIfthedimensionisporp22,/thenAisagroupQalgebral.oraTVaftHopfalgebra.Thecasesofdimensionp23,2andp24wreretreatedin[AS15]and[AS3],Qandtheclassi cationofdimensionp25ֹfollorwsfrom[AS45]and[GS~vn1<].]IndepSendentlyandbyotherQmethoSds,thecasep23wrasalsosolvedin[CDu]and[SvO*].ٍQSee[A]foradiscussionofwhatisknorwnonclassi cationof nitedimensionalHopfalgebras.*rjXQ42ANDR9USKIEWITSCH!ANDSCHNEIDERc.}QTVo 9formrulateaclassi cationresultforin nite-dimensionalHopfalgebras,Hwenowassumethat|Qisthe eldofcomplexnrumbSersandwreintroSduceanotionfrom[AS55].RThecollectionDTformedbya4QfrffeeeRabeliangroup of niterank,7va niteCartanmatrix(aijJ)1i;jvSݹ,g1;:::ʜ;g2UR;1;:::ʜ;2wbUR ,Qandalinkingdatum(ijJ)1i0;ꦹforall(/1i;j;URS;uQwhered1;:::ʜ;d9areinrtegersbSetween1and3suchthatdidaij 6=URdjf ajviJ.QIf DOaisapSositivredatumwede netheHopfalgebraU@(DUV)bygeneratorsaid;1#AiS,=andQhOlx;1lliand%Htherelationsh2RAmhOl R=lhOlhOl;hOlhOl=l1;ꦹforall(/1lC;mn9,sde ning%HtheQfreen abSeliangroupofrankn9,and(5.9),thequanrtumSerrerelations(5.11)&"fori46=jandn ijӹ,Q(5.12)X,(5.13)!(withaiOinsteadofxidڹ),andtheliftedquanrtumSerrerelations(5.14).QIf(V;c)isa nite-dimensionalbraidedvrectorspace,wewillsaythatthebrffaidingXispositiveйifQitYQisdiagonal,tandalltheenrtriesofthematrix(qijJ)ofthebraidingarepSositive,tandqii [b6=1forQalli.ʍQTheW8nexttheoremfollorwsfromaresultofRosso[Ro2e,r\Theorem21]andthetheorydescribSedQinthepreviousSections.wQTheorem5.13.E*z[AS55] {LffetAbeapointedHopfalgebrawithabeliangroup=G(A) {anddia-QgrffamRJ.RAssumethatR(1)has nitedimensionՉandpffositivebraiding.RThenthefollowingareQeffquivalent:Q(a).fiA35isadomainof niteGelfand-Kirillovdimension.Q(b).fiThe35grffoupisfreeabelianof niterank,andBkvAUR'U@(DUV);33wherffe'YDis35apffositivedatumfor|bk:ȭӿacffxff ̟ff ̎ ̄cffQItislikrelythatthepSositivityassumptiononthein nitesimalbraidinginthelasttheoremisQrelatedtotheexistenceofacompactinrvolution.3pc6. PointedHopfalgebrasoftypeAnQInFvthisChapter,gMwredevelopfromscratch,gMi.4e.%withoutusingLusztig'sresults,theclassi cationQof̑all nitedimensionalpSoinrtedHopfalgebraswhosein nitesimalbraidingisoftypSeAnP.ޚTheQmainresultsofthisChapterarenew.Q6.1.Nicholsalgebrasoftyp`eAnP.QLetNԹbSeaninrteger,N2>2,andletqK)bearootof1oforderN@.FVorthecaseN2=2,seeQ[AnDa/6].QLetqijJ,1URi;j%n,bSerootsof1sucrhthatʍx0qii =URqn9; qijJqjvi 6=z( q21 ʵ;8޹ifC,jijjUR=1;ɍ 1;8޹ifC,jijjUR2:Q(6.1)+trjXIPOINTED!HOPFALGEBRAS43c.}Qforall1URi;j%n.8FVorconrvenience,wedenotePHB i;jڍp;r:=!e_Y'؍URi`jv1;jphr, [ei;i+1;ep;p+1G]c.y; [ei;i+1;[ei;i+1;ei+1;i+2_ ]c.y]cqand&E[ei+1;i+2;[ei+1;i+2;ei;i+1]c.y]cQareVprimitivre."oSincetheyarehomogeneousofdegree2,respSectivelyofdegree3,theyshouldbSe0.ʍQTVoderivre(6.16)%Xfrom(6.14)!,use(1.23). ӄcffxff ̟ff ̎ ̄cffQLemma6.4.5Assume35thatWy(6.12)'holdsinRJ.fiThen&X[ei;j X;ep;r 63]c˹=UR0;$if8'1ij%=URih+1andr=URp+1,Sthisis(6.12)!F; thegeneralcasefollorwsrecursivelyusingʍQ(1.22)X.8(6.18)'*follorwsfrom(6.17)!,sinceB2i;jRAp;r 63BOp;rái;j&=UR1inthiscase.-rjXIPOINTED!HOPFALGEBRAS45c.}Q(6.19)X.8ByinductiononjW{p;ifpUR=j1then(6.19)%Xisjust(6.8).8FVorpURbSe%abraidedHopfalgebrain2b YD sucrhthat(6.12)&R4holdsinRJ.Itfollowsfrom(6.20)&R4andtheQreconstructionformrulasforthebSosonization(1.17)%Xthat@R #|ɩ(ei;j X)UR=ei;j R 1+gi;j ei;j+(1qn9 1 ʵ)zX ㇍i3I!wearguebyinductiononr>p.TKIfrp =1,`thenI!thereexistsanʍQindexhsucrhthateitheriUR1then[ei;j X;ep;r 63]c˹=UR[ei;j;[ep;r2wrearguebyinductiononjW{i.8IfjiUR=1wrehave>l [ei;i+1;ei;r Ӱ]c˹=UR[ei;i+1;[ei;r2,wehaveJ[ei;j X;ei;r Ӱ]c˹=UR[[ei;jv1;ejv1;j]c.y;ei;r]c˹=UR0QbryLemma1.10(c),bSecauseoftheinductionhypSothesisand(6.22)!./-rjXIPOINTED!HOPFALGEBRAS47c.}QTheproSofof(6.24)&isanalogoustotheproofof(6.23)",usingo(6.14)&insteadof(6.13).&cffxff ̟ff ̎ ̄cff]QLemma6.8.5Assume35thatWy(6.12)#^,(6.13)":=,(6.14)&mrholdinRJ.fiThen*:O[ei;j X;ep;r 63]c˹=URBOpjájvr Hm(q1)eirwRpK6Vqn1 ͹=`pnPin givresrisetoasplittingofHopfalgebrasidoZqn1(gH=nPn,*wheren T:Zn11,!Zn andQn:URZn!Zn1otarerespSectivrelyinducedbyinP,pn.8LetɍgW7Rn=URZ ܞco nڍn2=fz52Zn:(id ʠ nP)(z)=z3 1g:QThen9Rn 'isabraidedHopfalgebrainthecategoryZqn1 TZqn1 PYD+;anwreshalldenotebycRn>thecorre-QspSondingbraidingofRnP.8WVeharveZn'URRn#Zn1otandinparticulardimZn=URdimRnNdim=Zn1̹.QFVorٖsimplicitry,RweٖdenotehiPݹ=ei;n+1,R1in.WVeٖhavehidhj R =BOi;n+1áj$;n+1a hjf hi,RforٖiQWVenextclaimthatcRn (hi hjf )UR=BOi;n+1áj$;n+1a hj hidڹ,foranryiUR>jӹ.ʍQByo(6.27)"w,thecoactionofZn1otonRn satis eskL[s2(hidڹ)UR=gi;n+1p ei;n+1+(1qn9 1 ʵ) LX ㇍iFVromtheQprecedingclaim,wrededucethat)yx(h m)UR=h m UW 1+1 h m+jX'؍0im; 06=i6=mK"cm;i h i h mi ; mURN1;!卑Qwhere@cm;iZ6=UR0foralli.?WVethenarguerecursivrelyasintheproSofof[AS15,bLemma3.3]toconcludeQthatdtheelemenrtsh2m,D mURND1,aredlinearlyindepSendenrt;_hencethedimensionofthesubalgebraQofRn spannedbryh1;:::ʜ;hnisURN@2n4,asclaimed.QWVe%cannorw nishtheproSofoftheTheorem. SincedimYZn lNtn(n+1)t8R\)J O2!byLemma6.12andQdimZn1=inpresenceofLemma6.12,thisQimpliestheTheorem.|Y˄cffxff ̟ff ̎ ̄cffQTheorem6.14.E*zThe8~NicholsalgebrffaB(Vp)canbepresentedbygeneratorsei;i+1,9ѹ1_in,andQrffelations1(6.12)#8,35(6.13)":=,35(6.14)&mrandqȹ(6.28)#x.3?rjXIPOINTED!HOPFALGEBRAS51c.}QPrffoof.:Let5:B20sbSethealgebrapresenrtedbygeneratorsei;i+1,Y1URin,and5:relations(6.12)!UR3.QPrffoof.:This>isaparticularcaseofTheorem5.6; wreincludetheproSofforcompleteness.(WVeknow,QbryLemma2.10,that$=sadL/Ei;i+1(Ep;p+1G)h2URPg8:i,rgpZ;1ǹ(A) 8:i,rp;1iI{2jJit"gpZ;1w(A) -:2jJi*p;1i;pn;jpijUR=1:QAssume(thatadEi;i+1(Ep;p+1G)6=0,7\and(idp!6=",where(1iUR2).QIt8remainstoexcludethecasesidp͹=p";jpij2,Ljand822RAip͹=";jpij=1.#~The8 rstcaseQleads2ZtotheconrtradictionN>=Z3.InthesecondcaseitfollowsfromtheconnectivityofAn ڪthatQN+wroulddivide2whichisalsoimpSossible.Escffxff ̟ff ̎ ̄cffDQLemma6.17.;If35HB=URA,thenE2NRAi;j 2|2ND,forany1iI{NjJi<;1d(A)2-:NjJiand2NRAi G6=",FBhenceg2n9NRAi #=gl!;2NRAi G=l"forsomelC;butthislastpSossibilitryQconrtradictsMtheclaimabSove.-AssumethenthatjDbqiUR>1.ByMtherecursionhrypSothesis,_(E2NRAi;j X)UR=QE2NRAi;j ~r 1+g2n9NRAi;j E2NRAi;j+u,UR2.Weassumethat ab 2UR|;1a(s;t).If.s1,c\elementsQyi,2URRJ,hi2,i2wb ,1iPƹ,sucrhthat.Xgyi,=URidڹ(gn9)yi;forall$EeURi ?(u aϹ)forall&/aUR2N P:Q(6.48)QFVoranryaZݹ=(aidڹ)1iP w2N2P ,andanyfamily(gidڹ)1iP Qofelementsinwede nen92aN L((gidڹ))Z=ʍQQi i<2a8:i,rN8:iX(gidڹ).8Thenbryo(6.46)"w,foralla;bUR2N2P̹,Z+yn9 aN Lyi,=URyidyn9 aNn9 aN(gn9 ar);ꦹandzyn9 bN *Yyn9 aN`=yn9 (a+b)Nn9 aN(gn9 b);Q(6.49)Qforsomefamiliesofelemenrtsgn92ar;gn92b{in.QByareformrulationofourassumption,[Vu an9 aN L((gidڹ))UR=u awforanry1a2N P tandfamilyAѤ(gi)in:Q(6.50)QUsingo(6.49)&wrenowcancomputeh8zyiz{?=URX'؍>lK;ayn9 lyn9 aN LvlK;a  yi,=URX'؍>lK;ayn9 lyn9 aN Lyiedi ǹ(vlK;a  )"⍍z{?=URX'؍>lK;ayn9 lyidyn9 aN Ln9 aN(gn9 ar)nei 0(vlK;a  )z{?=URX'؍>lK;aX'؍t;b) lڍt;b yn9 tyn9 bN *Ye aN n9 aN L(gn9 ar)nei 0(vlK;a  )z{?=URX'؍>lK;aX'؍t;b) lڍt;b yn9 tyn9 (a+b)Nn9 aN L(gn9 ar)n9 aN(gn9 b)nei 0(vlK;a  ):gQThereforeR^b['(zyidڹ)`=URX ҍ wtm(t)X'؍lK;a;bUT lڍt;b u a+bC#n9 aN L(gn9 ar)n9 aN(gn9 b)nei 0(vlK;a  )`=URX ҍ wtm(t)X'؍lK;a;bUT lڍt;b u a+beC#it(vlK;a  );ꦹbryn'3;@orifthein nitesimalbrffaidingofAisoftypeAn withN >n'7andnotdivisibleby3,CthenQA35isisomorphictoaHopfalgebrffaA in(i).=rjXIPOINTED!HOPFALGEBRAS61c.IQPrffoof.:(i).By8Lemma6.20,\*theelemenrtsui;j arecentralinwԧbUɥ:=ުMb!VURB (Vp)#|,\*andui;j =UR0ifg2n9NRAi;j=UR1ʍQorN^2NRAi;j b6= ":Hencetheresidueclassesoftheelemenrtse2l!gn9;lB2A;0li;j b. >5ByLemma6.17,6E2NRAi;j=:.ui;j2|forallQ1izrjXQ62ANDR9USKIEWITSCH!ANDSCHNEIDERc.}QExample6.27.CLetYNwbSe>2,uNq̹arootofunitryoforderN@,uNandm1;:::ʜ;mn integers>1suchʍQthat[nmi,6=URmjxforalli6=jӹ. "LetbSethecommrutative[ngroupgeneratedbryg1;:::ʜ;gn withrelationsQgn9N"m8:imi=UR1,1in:De ne1;:::ʜ;n2wbmbryhjf (gidڹ)UR=qn9 a8:ij U;ꦹwhere'Vaii =2forall&/i;aij 6=1ifjijjUR=1;aij 6=0ifjijjUR2:QThen2NRAi;j =UR"andg2n9NRAi;j6=UR1foralli cmmi10 0ercmmi7K`y cmr10ٓRcmr7QV