Classification of irreducible modules of the verte

arXiv:0807.0817v1[math.QA]4Jul2008ClassificationofirreduciblemodulesofthevertexalgebraV+LwhenLisanondegenerateevenlatticeofanarbitraryrankGaywaleeYamskulna1DepartmentofMathematicalSciences,IllinoisStateUniversity,Normal,IL61790andInstituteofScience,WalailakUniversity,NakonSiThammarat,ThailandAbstractInthispaper,wefirstclassifyallirreduciblemodulesofthevertexalgebraV+LwhenLisanegativedefiniteevenlatticeofarbitraryrank.Inparticular,weshowthatanyirreducibleV+L-moduleisisomorphictoasubmoduleofanirreducibletwistedVL-module.WethenextendthisresulttoavertexalgebraV+LwhenLisanondegenerateevenlatticeoffiniterank.KeyWords:Vertexalgebra.1IntroductionItiswellknownthatforanondegenerateevenlatticeL,thereisacorrespond-ingvertexalgebra,VL,withanautomorphismθoforder2whichisinducedfromthe-1isometryofthelattice(cf.[B,FLM]).Moreover,theθ-invariantvertexsub-algebraV+Lisirreducible(cf.[DM]).ThesevertexalgebrasV+Lprovidealargeclassofconcreteandimportantexamplesofvertexalgebras.Infact,thestudyofV+Lwasinitiatedin[FLM]duringthecourseofthecon-structionofthemoonshinemodule.Toalargeextent,thestudyofvertexalgebrasisthestudyoftheirrep-resentations.Asforclassicalalgebras,thestudyofcompletereducibilityofmodulesisoffundamentalimportance.TheclassificationofirreducibleV+L-modulesandthecompletereducibilityofV+L-modulesforthecasewhenLisapositivedefiniteevenlatticeofanarbitraryrankandwhenLisarankonenegativedefiniteevenlatticehasbeendonebyAbe,Dong,Jiang,JordanandNagatomo(cf.[A,AD,DJ,DN1,J]).Inthispaper,wetakeafurtherstepinunderstandingV+Lbystudyingclas-sificationsofirreducibleV+L-moduleswhenLisanegativedefiniteevenlatticeofafiniterankandwhenLisanondegenerateevenlatticeofafiniterank.Forthesetwocases,weprovethatanyirreducibleV+L-moduleisisomorphic1E-mail:[e-mail],PartiallysupportedbyaPre-TenureFacultyInitiativegrantfromtheCollegeofArtsandSciences,ISU1toasubmoduleofanirreducibletwistedVL-module.ThemainideaoftheproofsistoclassifyirreduciblemodulesoftheZhualgebrasA(V+L).Thereisaone-to-onecorrespondencebetweenthesetofinequivalentirreducibleA(V+L)-modulesandthesetofinequivalentirreducible(admissible)V+L-modules(cf.[Z]).Thispaperisorganizedasfollows.InSection2,wereviewthedefinitionofavertexalgebra,andrecallvariousnotionsofitstwistedmodules.Inaddition,wediscussaZhualgebraanditsproperties.Also,werecalltheconstructionsofM(1)+,V+L,theirmodulesandrelatedtopicsthatwewillneedinlatersections.InSection3,westudytheZhualgebraofV+LwhenLisanegativedefiniteevenlatticeofafiniterankandweusethisinformationtoclassifyallirreducible(admissible)V+L-modules.InSection4,weclassifyallirreducible(admissible)V+L-moduleswhenLisanondegenerateevenlatticeofafiniterank.Acknowledgments:WewanttothankChongyingDongforreadingthismanuscript.Inthisresearch,wegreatlybenefitedfromstudyingtheworkofAbeandDongin[AD].Wehaveadoptedmanyoftheirimportantideasintothispaper.2Preliminaries2.1VertexalgebrasandZhualgebrasFirst,wedefinevertexalgebras,andtheirautomorphisms.Next,werecallvariousnotionsoftwistedmodulesforavertexalgebra.WealsodiscussaboutZhualgebras.Definition2.1.[LLi]AvertexalgebraVisavectorspaceequippedwithalinearmapY(·,z):V→(EndV)[[z,z−1]],v7→Y(v,z)=Pn∈Zvnz−n−1andadistinguishedvector1∈Vwhichsatisfiesthefollowingproperties:1.unv=0forn0.2.Y(1,z)=idV.3.Y(v,z)1∈V[[z]]andlimz→0Y(v,z)1=v.4.(theJacobiidentity)z−10δz1−z2z0Y(u,z1)Y(v,z2)−z−10δz2−z1−z0Y(v,z2)Y(u,z1)=z−12δz1−z0z2Y(Y(u,z0)v,z2).Wedenotethevertexalgebrajustdefinedby(V,Y,1)or,briefly,byV.2Definition2.2.AZ-gradedvertexalgebraisavertexalgebraV=⊕n∈ZVn;forv∈Vn,n=wtv,equippedwithaconformalvectorω∈V2whichsatisfiesthefollowingrelations:•[L(m),L(n)]=(m−n)L(m+n)+112(m3−m)δm+n,0cVform,n∈Z,wherecV∈C(thecentralcharge)andY(ω,z)=Xn∈ZL(n)z−n−2=Xm∈Zωmz−m−1!;•L(0)v=nv=(wtv)vforn∈Z,andv∈Vn;•Y(L(−1)v,z)=ddzY(v,z).Definition2.3.[LLi]Avertexsub-algebraofavertexalgebraVisavectorsub-spaceUofVsuchthat1∈UandsuchthatUisitselfavertexalgebra.Definition2.4.[LLi]AnautomorphismofavertexalgebraVisalinearisomorphismofVsuchthatg(1)=1,andgY(v,z)u=Y(g(v),z)g(u)foru,v∈V.LetGbeafiniteautomorphismgroupofV.WedenotetheG-fixed-pointvertexsub-algebraofVbyVG(={v∈V|gv=vforallg∈G}).Forg,anautomorphismofthevertexalgebraVoforderT,wedenotethedecompositionofVintoeigenspaceswithrespecttotheactionofgasV=⊕T−1r=0Vr,whereVr={v∈V|g(v)=e2πir/Tv}.Definition2.5.[D,FFrR,FLM,Le]Aweakg-twistedV-moduleMisavectorspaceequippedwithalinearmapYM(·,z):V→(EndM){z},v7→YM(v,z)=Pn∈Qvnz−n−1whichsatisfiesthefollowingproperties:forv∈V,u∈Vr,andw∈M1.vnw=0forn0.2.YM(u,z)=Pn∈Zun+rTz−n−rT−1.3.YM(1,z)=idM.4.(thetwistedJacobiidentity)z−10δz1−z2z0YM(u,z1)YM(v,z2)−z−10δz2−z1−z0YM(v,z2)YM(u,z1)=z−12z1−z0z2−r/Tδz1−z0z2YM(Y(u,z0)v,z2).3Ifgistheidentitymap,aweakg-twistedV-moduleiscalledaweakV-module.Remark2.6.LetGbeafiniteautomorphismgroupofVandletgbeamemberofG.Thenanyg-twistedweakV-moduleisaweakVG-module.Definition2.7.Anirreducibleweakg-twistedV-moduleisaweakg-twistedV-modulethathasnoweakg-twistedV-submoduleexcept0anditself.Here,aweakg-twistedsub-moduleisdefinedintheobviousway.Definition2.8.[DLM2]An(ordinary)g-twistedV-moduleisaweakg