On some universal algebras associated to the categ

arXiv:math/0101059v2[math.QA]28Jan2001ONSOMEUNIVERSALALGEBRASASSOCIATEDTOTHECATEGORYOFLIEBIALGEBRASB.ENRIQUEZAbstract.Inourpreviouswork(math/0008128),westudiedthesetQuant(K)ofalluniversalquantizationfunctorsofLiebialgebrasoverafieldKofcharac-teristiczero,compatiblewithdualsanddoubles.WeshowedthatQuant(K)iscanonicallyisomorphictoaproductG0(K)×∐∐(K),whereG0(K)isauniversalgroupand∐∐(K)isaquotientsetofasetB(K)offamiliesofLiepolynomialsbytheactionofagroupG(K).WeproveherethatG0(K)isequaltothemulti-plicativegroup1+~K[[~]].SoQuant(K)is‘ascloseasitcanbe’to∐∐(K).WealsoprovethattheonlyuniversalderivationsofLiebialgebrasaremultiplesofthecompositionofthebracketwiththecobracket.Finally,weprovethatthestabilizerofanyelementofB(K)isreducedtothe1-parametersubgroupgeneratedbythecorresponding‘squareoftheantipode’.1.Mainresults1.1.ResultsonQuant(K).LetKbeafieldofcharacteristiczero.In[2],weintroducedthegroupG0(K)ofalluniversalautomorphismsoftheadjointrepre-sentationsofK[[~]]-Liebialgebras.LetusrecallthedefinitionofG0(K)moreexplicitly.Let~beaformalvariableandletLBA~bethecategoryofLiebialgebrasoverK[[~]],whicharetopologicallyfreeK[[~]]-modules.AnelementofG0(K)isafunctorialassignment(a,[,],δa)7→ρa,whereforeachobject(a,[,],δa)ofLBA~,ρaisanelementofEndK[[~]](a),suchthat(ρamod~)=ida,ρa∗=(ρa)t,andforanyx,yina,[ρa(x),y]=ρa([x,y]),andρaisgivenbyacompositionoftensorproductsofthebracketandcobracketofa,thiscompositionbeingthesameforallLiebialgebras(weexpressthelatterconditionbysayingthata7→ρaisuniversal).View1+~K[[~]]asamultiplicativesubgroupofK[[~]]×.Thereisauniquemapα:1+~K[[~]]→G0(K),suchthatforanyLiebialgebraa,(α(λ))a=λida.Thismapmakes1+~K[[~]]asubgroupofG0(K).WewillshowTheorem1.1.G0(K)isequaltoitssubgroup1+~K[[~]].In[2],wedefinedQuant(K)asthesetofallisomorphismclassesofuniversalquantizationfunctorsofLiebialgebras,compatiblewithdualsanddoubles.IfweDate:January2001.12B.ENRIQUEZdenotebyLBAandQUEthecategoriesofLiebialgebrasandofquantizeduni-versalenvelopingalgebrasoverK,andbyclass:QUE→LBAthesemiclassicallimitfunctor,thenauniversalquantizationfunctorofLiebialgebras,compatiblewithdualsanddoubles,isafunctorQ:LBA→QUE,suchthat1)class◦Qisisomorphictotheidentity;2)(universality)thereexistsanisomorphismoffunctorsbetweena7→Q(a)anda7→U(a)[[~]](theseareviewedasfunctorsfromLBAtothecategoryofK[[~]]-modules)withthefollowingproperties:ifwecomposethisisomorphismwiththesymmetrisationmapU(a)[[~]]→S(a)[[~]],andifwetransporttheop-erationsofQ(a)onS(a)[[~]],thentheexpansionin~oftheseoperationsyieldsmapsSi(a)⊗Sj(a)→Sk(a)andSi(a)→Sj(a)⊗Sk(a);werequirethatthesemapsbecompositionsoftensorproductsofthebracketandcobracketofa,thesecompositionsbeingindependentofa(see[1])3)ifQ∨(resp.,D(Q))denotestheQUE-dual(resp.,Drinfelddouble)ofanobjectQofQUE,andD(a)denotesthedoubleLiebialgebraofanobjectaofLBA,thentherearecanonicalisomorphismsQ(a∗)→Q(a)∨andQ(D(a))→D(Q(a));moreover,theuniversalR-matrixofD(Q(a))shouldbefunctorial(see[2]).In[2],wealsointroducedanexplicitset∐∐(K)ofequivalenceclassesoffam-iliesofLiepolynomials,satisfyingassociativityrelations,andweconstructedacanonicalinjectionof∐∐(K)inQuant(K).Moreover,weconstructedanactionofG0(K)onQuant(K)andshowedthatthemapG0(K)×∐∐(K)→Quant(K)givenbythecompositionG0(K)×∐∐(K)⊂G0(K)×Quant(K)→Quant(K)(inwhichthesecondmapistheactionmapofG0(K)onQuant(K))isabijection.Theorem1.1thereforeimpliesCorollary1.1.Ifa=(a,[,],δa)isanobjectofLBAandλ∈(1+~K[[~]]),letaλbetheobjectofLBA~isomorphicto(a,[,],λδ).Thegroup1+~K[[~]]actsfreelyonQuant(K)bytherule(λ,Q)7→Qλ,whereQλisthefunctora7→bQ(aλ)andbQisthenaturalextensionofQtoafunctorfromLBA~toQUE.Thenthemap1+~K[[~]]
×∐∐(K)→Quant(K)givenbythecomposition1+~K[[~]]
×∐∐(K)⊂1+~K[[~]]
×Quant(K)→Quant(K)isabijection.ThereforeQuant(K)is‘ascloseasitcanbe’to∐∐(K).1.2.Universal(co)derivationsofLiebialgebras.LetD(resp.,C)bethespaceofalluniversalderivations(resp.,coderivations)ofLiebialgebras.Moreexplicitly,D(resp.,C)isthelinearspaceofallfunctorialassignmentsa7→λa,whereforeachobjectaofLBA,λabelongstoEnd(a),isuniversalintheabovesense,andisaderivation(resp.,coderivation)oftheLiealgebrastructureofUNIVERSALALGEBRASASSOCIATEDTOLIEBIALGEBRAS3a.Itiswell-knownthatif[,]aandδaarethebracketandcobracketmapsofa,then[,]a◦δaisaderivationofa;e.g.,ifaisfinite-dimensional,ifPi∈Iai⊗biisthecanonicalelementofa⊗a∗andifwesetu=Pi∈I[ai,bi],thenwehavetheidentity([,]a◦δa)(x)=[u,x]inthedoubleLiealgebraofa;andsince[,]a∗◦δa∗isaderivationofa∗,itstranspose[,]a◦δaisacoderivationofa.ThenTheorem1.2.DandCbothcoincidewiththeone-dimensionalvectorspacespannedbytheassignmenta7→[,]a◦δa.IfVisavectorspace,wedenotebyF(V)thefreeLiealgebrageneratedbyV.Thentheassignmentc7→F(c)isafunctorfromthecategoryLCAofLiecoalgebrastoLBA.TheproofofTheorem1.2impliesthefollowinganalogousstatementforthesubcategoryofLBAoffreeLiealgebrasofLiecoalgebras.Proposition1.1.Letc7→λcbeafunctorialassignment,whereforeachobjectcofLCA,λcisabothaderivationandacoderivationofF(c).Thenthereexistsascalarλ,suchthatforanyobjectcofLBA,λc=λ[,]F(c)◦δF(c).1.3.IsotropyoftheactionofG(K)onB(K).Werecordherethedefinition