Lie superalgebras and irreducibility of A_1^(1)-mo

arXiv:math/0602181v1[math.QA]9Feb2006LIESUPERALGEBRASANDIRREDUCIBILITYOFA(1)1–MODULESATTHECRITICALLEVELDRAˇZENADAMOVI´CAbstract.Weintroducetheinfinite-dimensionalLiesuperalgebraAandcon-structafamilyofmappingsfromcertaincategoryofA–modulestothecat-egoryofA(1)1–modulesofcriticallevel.Usingthisapproach,weprovetheirreducibilityofalargefamilyofA(1)1–modulesatthecriticallevelparame-terizedbyχ(z)∈C((z)).Asaconsequence,wepresentanewproofofirre-ducibilityofcertainWakimotomodules.WealsogiveanaturalrealizationsofirreduciblequotientsofrelaxedVermamodulesandcalculatecharactersoftheserepresentations.1.IntroductionIntheanalysisofcertainFockspacerepresentationsofinfinite-dimensionalLie(super)algebras,oneofthemainproblemistoproveirreducibilityoftheserepresen-tations.IrreduciblehighestweightrepresentationsofaffineLiealgebrasofcriticallevelcanberealizedbyusingcertainbosonicFockrepresentation,calledtheWaki-motomodules(cf.[W],[FF],[FB],[F],[S]).IrreducibilityofcertainWakimotomodulesgaveaverynaturalproofoftheKac-Kazhdanconjectureoncharactersofirreduciblerepresentationsofcriticallevel(cf.[KK]).Ontheotherhand,thecategoryofrepresentationsofcriticallevelismuchricherthanthecategoryO.SoonecaninvestigatethemodulesoutsidethecategoryOandtrytounderstandtheirstructure.InparticularonecaninvestigatetherelaxedVermamodules,theirirreduciblequotientsandthecorrespondingcharacters.SuchkindofrepresentationsappearedinthecontextofrepresentationtheoryoftheaffineLiealgebracsl2onnon-criticallevels(cf.[FST],[AM]).Inthepresentpaperweshalldemonstratethattheserelaxedrepresentationsappearnaturallyatthecriticallevelandshouldbeincludedintherepresentationtheoryatthislevel.WewillgiveafreefieldrealizationoftheirreduciblequotientsofrelaxedVermamodulesofcriticallevelandcalculatetheircharactersinthecaseofaffineLiealgebracsl2.Outsidethecriticallevel,therepresentationtheoryofaffineLiealgebraA(1)1isrelatedtotherepresentationtheoryoftheN=2superconformalalgebra.In[FST],2000MathematicsSubjectClassification.Primary17B69,Secondary17B67,17B68,81R10.Keywordsandphrases.vertexsuperalgebras,affineLiealgebras,Cliffordalgebra,Weylalge-bra,latticevertexalgebras,criticallevel.PartiallysupportedbytheMZOSgrant0037125oftheRepublicofCroatia.12DRAˇZENADAMOVI´Ctheauthorsconstructedmappingsbetweencertaincategoriesofrepresentationsofcsl2andN=2superconformalalgebra.Inthecontextofvertexalgebrasthesemappingswasconsideredin[A1].Butatthecriticalleveltherepresentationtheoryofcsl2isverydifferenttotherepresentationtheoryoutsidethecriticallevel.Inparticular,theassociatedvertexalgebraN(−2Λ0)containsaninfinite-dimensionalcenter(cf.[F]).Inthepresentpaperwefindaninfinite-dimensionalLiesuperalgebraAwiththeimportantpropertythatitsrepresentationtheoryisrelatedtothoseofcsl2atthecriticallevel.ThisalgebrahasgeneratorsG±(r),T(n),S(n),r∈12+Z,n∈Z,whichsatisfythefollowingrelations[S(n),A]=[T(n),A]=0{G+(r),G−(s)}=2S(r+s)+(r−s)T(r+s)−(r2−14)δr+s,0{G+(r),G+(s)}={G−(r),G−(s)}=0foralln∈Z,r,s∈12+Z.ThemaindifferencebetweenouralgebraAandtheN=2superconformalalgebraisinthefactthatAcontainslargecenterandthatitdoesn’tcontaintheVirasoroandHeisenbergsubalgebra.NextweconsiderthevertexsuperalgebraVassociatedtoavacuumrepresentationforA.ThisvertexsuperalgebraisintroducedinSection4asavertexsubalgebraofF⊗M(0),whereFisaCliffordvertexsuperalgebraandM(0)acommutativevertexalgebra.Thenfollowing[A1]weshowthatthereanon-trivialvertexalgebrahomomorphismg:N(−2Λ0)→V⊗F−1,whereF−1isalatticevertexsuperalgebraassociatedtothelatticeZβ,hβ,βi=−1.ThisresultallowsustoconstructN(−2Λ0)–modulesfromV–modules.Moreover,weprovethatifUisanirreducibleV–modulesatisfyingcertaingradingcondition,thenU⊗F−1=⊕s∈ZLs(U)isacompletelyreducibleN(−2Λ0)–module.ThereforeeverycomponentLs(U)isanirreducibleA(1)1–moduleatthecriticallevel.Itisimportanttonoticethattheirreducibilityresultisprovedbyusingthetheoryofvertexalgebras(cf.Lemma6.1).InthiswaytheproblemofconstructingirreducibleA(1)1–modulesisreducedtotheconstructionofirreducibleV–modules.ButonirreducibleV–modules,theactionoftheLiesuperalgebraAcanbeexpressedbytheactionofgeneratorsofinfinite-dimensionalCLiffordalgebras.Byusingthisfact,inSection5weprovetheirreducibilityofalargefamilyofV–modules.Thesemodulesareparameterizedbyχ(z)∈C((z)).InSection6weconstructmappingsLswhichsendirreducibleV–modulestotheirreduciblecsl2modulesatthecriticallevel.Asanapplication,inSection7wepresentaproofofirreducibilityofalargefamilyoftheWakimotomodules.InSection8westudytheirreduciblehighestweightcsl2–modulesofcriticallevel.Inparticular,westudythesimplevertexalgebraL(−2Λ0).InSection9wegetrealizationofirreduciblequotientsofrelaxedVermamodules.Itturnsoutthattheseirreduciblemodulescanberealizedoncertainlatticetypevertexalgebra.32.VertexalgebraN(kΛ0)Wemaketheassumptionthatthereaderisfamiliarwiththeaxiomatictheoryofvertexsuperalgebrasandtheirrepresentations(cf.[DL],[FHL],[FLM],[LL],[K2],[Z]).InthissectionwerecallsomebasicfactsaboutvertexalgebrasassociatedtoaffineLiealgebras(cf.[FZ],[Li1],[MP]).Letgbeafinite-dimensionalsimpleLiealgebraoverCandlet(·,·)beanon-degeneratesymmetricbilinea