Vertex-algebraic structure of the principal subspa

arXiv:0710.1527v2[math.QA]9Dec2007VERTEX-ALGEBRAICSTRUCTUREOFTHEPRINCIPALSUBSPACESOFCERTAINA(1)1-MODULES,II:HIGHERLEVELCASEC.CALINESCU,J.LEPOWSKYANDA.MILASAbstract.Wegiveanaprioriproofoftheknownpresentationsof(thatis,completenessoffamiliesofrelationsfor)theprincipalsubspacesofallthestandardA(1)1-modules.ThesepresentationshadbeenusedbyCapparelli,LepowskyandMilasforthepurposeofobtainingtheclassicalRogers-Selbergrecursionsforthegradeddimensionsoftheprincipalsubspaces.Thispapergeneralizesourpreviouspaper.1.IntroductionTheaffineKac-MoodyalgebraA(1)1=dsl(2)isthesimplestinfinite-dimensionalKac-MoodyLiealgebra,andinsomesensethemostprominentone.Notonlydoesdsl(2)giveinsightintothehigher-rankaffineLiealgebras,butinfact,considerationsofstandard(=integrablehighestweight)dsl(2)-moduleshavefrequentlyledtonewideas.Forinstance,explicitconstructionsofthestandarddsl(2)-moduleshavebeenusedtoobtainvertex-operator-theoreticderivationsoftheclassicalRogers-Ramanujanidentitiesandrelatedq-seriesidentities(cf.[LW1]–[LW4],[LP1],[LP2],[MP1],[MP2]).Anotherimportantuseofstandarddsl(2)-modulesisinthe“coset”constructionofunitaryVirasoro-algebraminimalmodels[GKO].Thesedevelopmentsaredeeplyrelatedtotwo-dimensionalconformalfieldtheory.Morerecently,toeachstandard[sl(n)-moduleL(Λ),FeiginandStoyanovskyassociatedadistinguishedsubspaceW(Λ),whichtheycalledthe“principalsubspace”ofL(Λ)([FS1],[FS2]),andinterestingly,thegradeddimensionsoftheprincipalsubspacesofthestandarddsl(2)-modulesareessentiallytheGordon-Andrewsq-series([FS1],[G];cf.[A]).Theseq-serieshadpreviouslyappearedasthegradeddimensionsofthe“vacuum”subspaces,withrespecttoacertaintwistedHeisenbergsubalgebra,oftheodd-levelstandarddsl(2)-modules([LW2]–[LW4],[MP1]).Sinceeachstandard[sl(n)-moduleL(Λ)oflevelk,k≥1,isamoduleforacertainvertexoperatoralgebra([FZ];cf.[DL],[LL]),itisnaturaltoemployideasfromvertexoperatoralgebratheorytogainabetterinsightintothestructureofprincipalsubspaces.In[CLM1]–[CLM2],forthecasedsl(2),thetheoryofvertexalgebrasandrelatedalgebraicstructures,includingintertwiningoperators[FHL],hasbeenusedtodothis,viatheconstructionofcertainexactsequences,whichledtoavertex-algebra-theoreticinterpretationoftheclassicalRogers-RamanujanandRogers-Selbergrecursions.ThisinturnexplainedtheappearanceoftheGordon-Andrewsq-series,andtheseq-seriescanbeimplementedbymeansof“combinatorialbases”oftheprincipalsubspaces,C.C.gratefullyacknowledgespartialsupportfromtheCenterforDiscreteMathematicsandTheoreticalCom-puterScience(DIMACS),RutgersUniversity.J.L.gratefullyacknowledgespartialsupportfromNSFgrantDMS–0401302.12C.CALINESCU,J.LEPOWSKYANDA.MILASrevealingafundamental“difference-twocondition”thathadalreadyariseninthesettingof[LW2]–[LW4].Animportanttechnicalresultusedin[CLM1]and[CLM2]wasacertainpresentationof(thatis,thecompletenessofacertainfamilyofrelationsfor)theprincipalsubspacesofthestandarddsl(2)-modules(cf.Theorem2.1in[CLM2]).ThisresulthadbeenstatedasTheorem2.2.1′in[FS1].However,theproofsofthisresultthatweareawareofallturnouttorequireeitheraprioriknowledgeofacombinatorialbasisoftheprincipalsubspaceW(Λ)(see(2.6)below)orinformationcloselyrelatedtosuchknowledge.Butwhatoneideallywantsisratheranaprioriproofofthepresentation,whichcouldthenbeusedtoconstructtheexactsequencesmentionedabove,andtherebytoproducethebases.ThusitisanimportantproblemtotrytofindanaprioriproofofthepresentationofW(Λ),andwewereabletoachievethisforthelevelonestandarddsl(2)-modulesin[CalLM1].Ourproofin[CalLM1]wasobtainedintwosteps.WefirstarguedthatthepresentationofW(Λ1)followsfromthepresentationofW(Λ0),andthenweprovedthepresentationofW(Λ0).(Thesetwostepsareinfactinterchangeable,sowecouldhaveplacedtheproofofthepresentationofW(Λ0)first.)Inthepresentpaperwegiveanaprioriproofofthepresentationoftheprincipalsubspacesmoregenerallyforallthestandarddsl(2)-modules.Thehigher-levelcasebringsadditionalsub-tleties,andourapproachisdifferentfromthatin[CalLM1].Insteadoftryingtoreducetheproblemofprovingthepresentationofprincipalsubspacestoa“preferred”principalsubspace(e.g.,W(kΛ0)),wefounditmoreconvenientandmoreeleganttoprovethepresentationofalltheprincipalsubspacesofagivenlevelatonce.ThisisdoneintheproofsofTheorems3.1and3.2througha(necessarily)ratherdelicateargument,whichusesvariouspropertiesofprincipalsubspacesandintertwiningoperatorsamongstandardmodules.Inournewapproachalltheprincipalsubspacesareonmore-or-lessequalfooting.Thuswenotonlygeneralizethemainresultin[CalLM1]toallthestandarddsl(2)-modules,butwealsogiveanewproofofthepresen-tationoftheprincipalsubspacesinthelevelonecase,differentfromtheonein[CalLM1].Thisiswhywewritetheproofofthek=1caseseparatelyandinfulldetailbelow;wewillalsobegeneralizingthisk=1proofinadifferentdirectionelsewhere.Thispaperbringsourin-depthanalysisoftheprincipalsubspacesofthestandarddsl(2)-modulestoanend.Eventhoughthestudyoftheprincipalsubspacesofthedsl(2)-modulesisfacilitatedbythecommutativityoftheunderlyingnilpotentLiealgebrausedtodefinethesesubspaces,manymethodsinthispapercanbeappliedtomoregeneralaffineLiealgebras,bothuntwistedandtwisted.Inasequ