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arXiv:math/0102127v1[math.QA]16Feb2001ContemporaryMathematicsVertexLiealgebras,vertexPoissonalgebrasandvertexalgebrasChongyingDong,HaishengLi,andGeoffreyMasonAbstract.ThenotionsofvertexLiealgebraandvertexPoissonalgebraarepresentedandconnectionsamongvertexLiealgebras,vertexPoissonalgebrasandvertexalgebrasarediscussed.1.IntroductionVertex(operator)algebras(see[B],[FLM])whichareafamilyofnew“algebras”areknownessentiallytobechiralalgebrasintwo-dimensionalquantumfieldtheory.ThenewalgebrasarecloselyrelatedtoclassicalLiealgebrasofcertaintypes.Ononehand,from[B]onecanobtainaLiealgebraV/DVfromanyvertexalgebraVwhereDisacanonicalendomorphismofV.(ThefamousMonsterLiealgebraisasubalgebraofV/DVforasuitableV.)Ontheotherhand,onecanconstructvertex(operator)algebrasfromcertainhighestweightrepresentationsoffamiliarinfinitedimensionalLiealgebrassuchastheVirasoroalgebra,affineKac-Moodyalgebras,Heisenbergalgebras(cf.[DL],[FF1],[FZ],[Li2],[MP]).InthispaperwedefineanotionofwhatwecallvertexLiealgebratounifythefamiliarinfinitedimensionalLiealgebras.ThedefinitionofavertexLiealgebraismotivatedbytheVirasoroalgebra,affinealgebrasandthenotionofvertexalgebra.Roughlyspeaking,thenotionofvertexLiealgebraisa“stringy”analogueofthenotionofLiealgebra,whichgeneralizestheVirasoroalgebraandaffineLiealgebras.AvertexLiealgebradefinedinthispaperisaLiealgebrawhoseunderlyingvectorspaceisessentiallytheloopspaceofacertainvectorspaceUanditisakindof“affinealgebra”basedonvectorspaceUinsteadofafinitedimensionalLiealgebra.Soforeveryvectoruinthebasevectorspace,onecanformthefieldorvertexoperatoru(z)=Pn∈Zu(n)z−n−1.InthecaseoftheVirasoroalgebra,Uiseitheronedimensionalortwodimensionaldependingonthecenterbeingzeroornot.(TheVirasoroalgebraisnotaclassicalaffinealgebrainanysense.)Wealso1991MathematicsSubjectClassification.Primary17B69;Secondary16A68,81T40.C.DongwassupportedbyNSFgrantDMS-9303374,DMS-9987656andaresearchgrantfromtheCommitteeonResearch,UCSantaCruz.H.LiwassupportedbyNSFgrantDMS-9616630.G.MasonwassupportedbyNSFgrantDMS-9401272,DMS-9700909andaresearchgrantfromtheCommitteeonResearch,UCSantaCruz.c0000(copyrightholder)12CHONGYINGDONG,HAISHENGLI,ANDGEOFFREYMASONdefineanotionofvertexPoissonalgebra.WestudytheconnectionamongvertexLiealgebras,vertexPoissonalgebrasandvertexalgebras.ThenotionofvertexLiealgebrasnotonlyunifiestheVirasoroalgebra,affineLiealgebras,loopalgebrasandotherimportantinfinitedimensionalLiealgebras,butalsoprovidesnewexamplesofvertexalgebrasviatheir“highestweightrep-resentations.”Weprovethatforeachcomplexnumberthereisa“highestweightmodule”(basedapolardecomposition)forthevertexLiealgebrawhichhasthestructureofavertexalgebra.WealsoshowhowarestrictedmoduleforthevertexLiealgebrahasanaturalmodulestructureforcertainvertexalgebrasconstructedfromthevertexLiealgebra.SoitisveryimportanttoconstructnewvertexLiealgebras.VertexPoissonLiealgebrasareaspecialclassofvertexLiealgebraswhosebasevectorspacesarePoissonalgebrassatisfyingadditionalaxioms.CloselyrelatedtovertexPoissonalgebrasarePossionbracketsstudiedin[DFN],[DN]and[GD].WewereamazedtofindthatlocalPoissonbracketsintroducedin[DFN]wassoclosetoBorcherds’commutatorformulainthetheoryofvertexalgebras.OurfirstexerciseistomakelocalPoissonbracketspreciseintermsofformalvariableswheredeltafunctionsareformalseriesinsteadofdistributions.DifferentialgeometricPois-sonbracketsintroducedin[DN],whereinterestingconnectionsbetweendifferentialgeometricobjects(connection,curvatureandsymplecticstructure)andalgebraicobjects(Liealgebra,commutativeassociativealgebra)havebeenestablished,pro-videalotofexamplesofvertexPoissonLiealgebras.AttheendofSection3,wequoteseveralinterestingresultsfrom[DFN],[DN]and[Po]regardingtodifferentialgeometricPoissonbrackets.In[BD],anotionofcossionalgebrawasdefinedintermsofalgebraicgeometrywhereasdesignatedbytheauthors,cossionisthecombinationofthetwowordschiralandpoission.Later,anotionofvertexPoissonalgebrawasdefinedin[EF]intermsofassociativerings.(TherelationbetweencoissionalgebrasandvertexPoissonalgebraswasdiscussedin[EF].)ThenotionofvertexPoissonalgebrapre-sentedhereisdefinedintermsofformalcalculusandclassical(Lieandassociative)algebras.Presumably,ournotionisessentiallythesameasthatof[EF].Mostofthisworkwascarriedoutinthelate96andearly97.Duringthistime,Kac’sbook[K2]appearedwheresimilarresultstoourshadbeenobtained.Inparticular,anotionofconformalalgebrawasintroduced.AconformalalgebrasatisfiesasetofaxiomswhicharecertainmodificationsofthoseforavertexalgebraandaconformalalgebranaturallygivestoavertexLiealgebra.AnotionofvertexLiealgebrawasalsointroducedin[Pr].ItseemsthatthenotionofvertexLiealgebrain[Pr]thenotionofconformalalgebra[K2]arethesame.Therearecertainoverlapsbetweenthisworkand[Pr].Inparticular,Theorem4.8andLemma5.3werealsoobtainedin[Pr].Thepaperisorganizedasfollows.Section2isaboutformalcalculus.InSection3weformulatethenotionsoflocalvertexLiealgebraandlocalvertexPoissonalgebraandpresentsomeresults.WealsogiveseveralexamplesincludingtheWittalgebra,Virasoroalgebra,loopalgebras,affinealgebrastoillustratetheconcepts.Inparticular,wediscussvertexPoisson