On Lie Algebras in the Category of Yetter-Drinfeld

arXiv:q-alg/9612023v117Dec1996ONLIEALGEBRASINTHECATEGORYOFYETTER-DRINFELDMODULESBODOPAREIGISAbstract.ThecategoryofYetter-DrinfeldmodulesYDKKoveraHopfalgebraK(withbijektiveantipodeoverafieldk)isabraidedmonoidalcategory.IfHisaHopfalgebrainthiscategorythentheprimitiveelementsofHdonotformanordinaryLiealgebraanymore.Weintroducethenotionofa(generalized)LiealgebrainYDKKsuchthatthesetofprimitiveelementsP(H)isaLiealgebrainthissense.AlsotheYetter-DrinfeldmoduleofderivationsofanalgebraAinYDKKisaLiealgebra.FurthermoreforeachLiealgebrainYDKKthereisauniversalenvelopingalgebrawhichturnsouttobeaHopfalgebrainYDKK.Keywords:braidedcategory,Yetter-Drinfeldmodule,Liealge-bra,universalenvelopingalgebra.1991MathematicsSubjectClassification16W30,17B70,16W55,16S30,16S401.IntroductionTheconceptofHopfalgebrasinbraidedcategorieshasturnedouttobeveryimportantinthecontextofunderstandingthestructureofquantumgroupsandnoncommutativenoncocommutativeHopfal-gebras.InparticulartheworkofRadford[7],Majid[3],Lusztig[2],andSommerh¨auser[8]showtheimportanceofthedecompositionofquantumgroupsintoaproductofordinaryHopfalgebrasandofHopfalgebrasinbraidedcategories.SincebytheworkofYetter[9]Hopfalgebrasinbraidedcategoriesthataredefinedonanunderlying(finite-dimensional)vectorspacecanbeconsideredasHopfalgebrasinsomecategoryofYetter-Drinfeldmodules,wewillrestrictourattentiontoHopfalgebrasHinacategoryofYetter-DrinfeldmodulesYDKKoveraHopfalgebraKwithbijectiveantipode.Therearetwostructurallyinterestingandimportantconceptsthatsurviveinthisgeneralizedsituation,theconceptofgroup-likeelementsDate:February9,2008.1991MathematicsSubjectClassification.Primary16W30,17B70,16W55,16S30,16S40.12BODOPAREIGIS(Δ(g)=g⊗g,ε(g)=1)andtheconceptofprimitiveelements(Δ(x)=x⊗1+1⊗x,ε(x)=0).ForordinaryHopfalgebrasHthesetofprimitiveelementsP(H)ofHformsaLiealgebra.Thisresult(inasomewhatgeneralizedform)stillholdsforHopfalgebrasinasymmetricmonoidalcategory.Thisis,however,nottrueforbraidedmonoidalcategories.TherehavebeenvariousattemptstogeneralizethenotionofLiealgebrastobraidedmonoidalcategories.Themainobstructionforsuchageneralizationistheassumptionthatthecategoryisonlybraidedandnotsymmetric.OneofthemostimportantexamplesofsuchbraidedcategoriesisgivenbythecategoryofYetter-DrinfeldmodulesYDKKoveraHopfalgebraKwithbijectiveantipodewhichisalwaysproperlybraided(exceptforK=k,thebasefield)[6].WeintroduceaconceptofLiealgebrasinYDKKthatgeneralizestheconceptsofordinaryLiealgebras,Liesuperalgebras,Liecoloralgebras,and(G,χ)-Liealgebrasasgivenin[5].TheLiealgebrasdefinedonYetter-Drinfeldmoduleshavepartiallydefinedn-arybracketoperationsforeveryn∈Nandeveryprimitiven-throotofunity.Theysatisfygeneralizationsofthe(anti-)symmetryandJacabiidentities.OurmainaimistoshowthattheseLiealgebrashaveuniversalen-velopingalgebraswhichturnouttobeHopfalgebrasinYDKK.Con-verselythesetofprimitiveelementsofaHopfalgebrainYDKKissuchageneralizedLiealgebra.WealsogiveanexamplethatgeneralizestheconceptoforthogonalorsymplecticLiealgebras.2.BraidSymmetrizationWebeginwithtwosimplemoduletheoreticobservations.Thefol-lowingiswellknown:ifA,BarealgebrasandMisanA-B-bimodule,thenHomA(.P,.M)isarightB-moduleforeveryA-moduleP.Weneedacomoduleanalogueofthis.LetAbeanalgebra,Cbeacoalgebra,andAMCbeanA-C-dimodule,i.e.aleftA-moduleandarightC-comodulesuchthatδ(am)=(a⊗1)δ(m).Proposition2.1.LetPbeafinitelygeneratedleftA-module.ThenHomA(.P,.M)isarightC-comodulewiththecanonicalcomodulestruc-turesuchthatHomA(P,M)δ−→HomA(P,M)⊗C−→HomA(P,M⊗C)=HomA(P,δ).Proof.Letp1,...,pnbeageneratingsetofPandletf∈HomA(.P,.M).Letmi:=f(pi).ThenbythestructuretheoremoncomodulesthemiONLIEALGEBRASINTHECATEGORYOFYETTER-DRINFELDMODULES3arecontainedinafinitedimensionalsubcomoduleM0⊆MwhichisevenacomoduleoverafinitedimensionalsubcoalgebraC0⊆C,i.e.thediagramM0M0⊗C0-δMM⊗C-δ??commutes.FurthermoreM1:=AM0isaC0-comodulecontainedinM,sinceMisadimodule,andf:P−→MobviouslyfactorsthroughM1.SinceMandM1aredimodulesthediagramHomA(.P,.M1)HomA(.P,.M1⊗C0)-δ∗HomA(.P,.M)HomA(.P,.M⊗C)-δ∗HomA(.P,.M1)⊗C0∼=HomA(.P,.M)⊗C???commutes,soeachfhasauniquelydefinedimageδ∗(f)∈HomA(.P,.M)⊗C.NowitiseasytocheckthatthismapinducesacomodulestructureonHomA(P,M).Thesecondobservationisthefollowing.Weconsiderk-algebrasAandB.Letα:B−→Abeanalgebrahomomorphism.αin-ducesanunderlyingfunctorVα:A-Mod−→B-ModwithrightadjointHomB(A,-):B-Mod−→A-Mod.Ifα:B−→AissurjectivethenHomB(A,M)−→HomB(B,M)∼=Misinjective,sothatwecaniden-tifyHomB(A,M)={m∈M|Ker(α)m=0}.LetBnbetheArtinbraidgroupwithgeneratorsτi,i=1,...,n−1andrelationsτiτj=τjτiif|i−j|≥2;τiτi+1τi=τi+1τiτi+1.(1)Letζ∈kbeinvertible.ThenkBn∋τi7→ζτi∈kBn(forthegeneratorsτiofBn)isanalgebraautomorphismdenotedagainbyζ:kBn−→kBn.ThisholdstruesincetherelationsforBnarehomogeneous.(Observethatthisconstructioncanbeperformedforeverygroupalgebraifthegroupisgivenbygeneratorsandhomogeneousrelations.Thegivenconstructionofanautomorphismforeveryζ∈U(k)definesagrouphomomorphismU(k)−→Aut(kBn)−→Aut(kBn-Mod).)NowconsiderthecanonicalquotienthomomorphismBn−→Snfromthebraidgroupontothesymmetricgroup.Itinducesasurjectivehomomorphismγ:kBn−→kSn