9905064v1 Classification of irreducible modules fo

arXiv:math/9905064v1[math.QA]11May1999ClassificationofirreduciblemodulesforthevertexoperatoralgebraM(1)+II:higherrankChongyingDong1DepartmentofMathematics,UniversityofCalifornia,SantaCruz,CA95064,U.S.AKiyokazuNagatomo2DepartmentofMathematics,GraduateSchoolofScience,OsakaUniversityOsaka,Toyonaka560-0043,JapanAbstract:ThevertexoperatoralgebraM(1)+isthefixedpointsetoffreebosonicvertexoperatoralgebraM(1)ofrankℓunderthe−1automorphism.AllirreduciblemodulesforM(1)+areclassifiedinthispaperforanyℓ.1IntroductionThisisthethirdpaperinstudyingθ-orbifoldmodelsassociatedtolatticevertexoper-atoralgebrasVLforevenintegrallatticesLwhereθisanautomorphismofVLoforder2liftedfromthe−1isometryofL.TheVLcontainstherankℓfreebosonicvertexoperatoralgebraM(1)andtheautomorphismθpreservesM(1).In[DN1]westudiedtheorbifoldmodelM(1)+whichistheθ-fixedpointsetofM(1)andclassifiedalltheinequivalentirreduciblemodulesbydeterminingassociatedZhu’salgebraA(M(1)+)explicitlyinthecaseofrankone.Theresultsandthemethoddevelopedin[DN1]wereeffectivelyusedin[DN2]togettheclassificationresultfortheinequivalentirreduciblemodulesforthechargeconjugationorbifoldmodel,whichistheθ-invariantsofalatticevertexoperatoralgebraVLforarankonelatticeL.Inthispaper,weinvestigatethe1SupportedbyNSFgrantDMS-9700923andaresearchgrantfromtheCommitteeonResearch,UCSantaCruz.2SupportedinpartbyGrant-in-AidforScientificResearch,theMinistryofEducation,ScienceandCulture.1θ-orbifoldmodelM(1)+forarbitraryrankℓfreebosonicvertexoperatoralgebraM(1)andclassifytheirreduciblemodulesforM(1)+.TheresultsinthispaperareexpectedtobeusedtostudytherepresentationtheoryforavertexoperatoralgebraV+Lwhichistheθ-invariantsofVLforalatticeLofrankℓ.ThefreebosonicvertexoperatoralgebraH=M(1)ofrankℓ(cf.[FLM])isanaffinevertexoperatoralgebraassociatedtoanℓ-dimensionalabelianLiealgebrah(seeSubsection2.2below).Themapθ:h−→hdefinedbyθ(h)=−hinducesavertexoperatoralgebraautomorphismdenotedbythesamesymbolθ.ThenthefixedpointsetH+ofθisasimplevertexoperatorsubalgebraofH.ItiswellknownthatalltheirreduciblemodulesforHareexhaustedbyFockrepresentationM(1,λ)fortheaffinealgebraˆhwiththehighestweightλ∈h.AsamoduleforH+,M(1,λ)andM(1,−λ)areisomorphicandirreducibleifλ6=0.ButM(1,0)=HdecomposesintoitsirreduciblecomponentsH=H+⊕H−whereH±aretheeigenspacesofθ.Oneofthefeaturesoforbifoldmodelsistheexistenceofextrairreduciblemoduleswhichcomefromthetwistedsectors.TheHhasexactlyoneirreducibleθ-twistedmoduleH(θ)withtheθaction,whichgivesrisetotwoinequivalentirreduciblemodulesH(θ)±forH+whereH(θ)±aretheeigenspacesofθ.ThemainresultinthispaperisthatM(1,λ)(λ6=0),H±andH(θ)±areallinequivalentirreducibleH+-modules.In[Z],ZhuintroducedanassociativealgebraA(V)foranyvertexoperatoralgebraV,whichgivesalotofinformationonVasfarastherepresentationtheoryconcerns.Forinstance,thereisaonetoonecorrespondencebetweenthesetofequivalenceclassesofirreduciblemodulesfortheassociativealgebraA(V)andthesetofequivalenceclassesofirreducibleadmissiblemodulesforV.Thisfacthasbeenusedtoclassifytheirreduciblemodulesforaffinevertexoperatoralgebras[FZ],Virasorovertexoperatoralgebras[W],latticevertexoperatoralgebras[DLM3],M(1)+inthecaseℓ=1[DN1]andthe−1orbifoldvertexoperatoralgebraassociatedtotherank1lattice[DN2].Thisideawasdevelopedfurtherin[DLM2]todealwithtwistedrepresentationsandtheθ-twistedmodulesforlatticevertexoperatoralgebrasVLwereclassifiedalongthisline[DN3].TheclassificationresultinthispaperisalsoachievedbyusingZhu’salgebra.ThestrategyistodetermineZhu’salgebraA(H+)andtofindasetofgoodgeneratorsandtheirrelations.ThedeterminationofZhu’salgebraisnotonlyrelatedtotherepresentationtheorybutalsothestructuretheoryforagivenvertexoperatoralgebra.ForexamplewefoundaPoincar´e-Birkhoff-WitttypetheoremforaH+in[DN1]inthecaseℓ=1.SoinvestigationofZhu’salgebrashedslightonthehiddenstructureof2VOA’s.Itisworthpointingoutthatthereisamaindifferencebetweenrankonecaseandtheothers.Zhu’salgebraforrankonecaseiscommutativebutisnotforhigherrankcase.ForinstancethetoplevelofthemoduleH−isℓ-dimensional.ThealgebrastructureofA(H+)andideasgivenin[DN1]intherankonecaseareveryhelpfulbutnotenoughtoattackthehigherrankcase.ToovercomethedifficultyarisingfromnoncommutativityofA(H+)weintroduceanidealIwhichisisomorphictothedirectsumoftwocopiesofmatrixalgebraMℓ(C).ThenweshowsthequotientalgebraA(H+)/Iiscommutativeandisgeneratedbytheelementsωa,JaandΛab(seeSection5).ItisfairtosaythatwedonotdeterminethealgebrastructureofA(H+)completelyintermsofgeneratorsandrelations.ButtherelationsamonggeneratorsofA(H+)foundinthispaperaregoodenoughtoclassifyallirreduciblemodulesforA(H+)andfortheVOAH+.Weorganizethepaperasfollows.InSection2,wereviewdefinitionsandstatespropertiesoftheVOAH+.Thelistofinequivalentirreduciblemodulesisgivenhere.WeexplainthenotionofZhu’salgebrasinSection3andprovesomeformulaswhichweneedlater.Section4isdevotedtofindafinitesetofgeneratorsforA(H+).InSection5,weintroducetheelementsEuab,EtabandΛabaswellasωa,Jawhichforma“nice”generatingsetofZhu’salgebra.ItwillbeshownthattheelementsEuabandEtabformsthematrixalgebraMℓ(C)respectively.Wederivemorerelationsamongthegenerato