Induced modules for vertex operator algebras

arXiv:hep-th/9412038v15Dec1994InducedmodulesforvertexoperatoralgebrasChongyingDongandZongzhuLinAbstractForavertexoperatoralgebraVandavertexoperatorsubalgebraV′whichisinvariantunderanautomorphismgofVoffiniteorder,weintroduceag-twistedinductionfunctorfromthecategoryofg-twistedV′-modulestothecategoryofg-twistedV-modules.ThisfunctorsatisfiestheFrobeniusreciprocityandtransitivity.TheresultsareillustratedwithV′beingtheg-invariantsinsimpleVorV′beingg-rational.1IntroductionAlotofprogressontherepresentationtheoryforvertexoperatoralgebrashasbeenmadeinthelastfewyears.Forexample,therepresentationtheoryfortheconcretevertexoperatoralgebras,whichincludethemoonshinevertexoperatoralgebraV♮([FLM],[D3]),thevertexoperatoralgebrasbasedonevenpositivedefinitelattices[D1],thevertexoperatoralgebrasassociatedwiththeintegrablerepresentationsofaffineLiealgebrasandVirasoroalgebras([DMZ],[DL],[FZ],[W]),havebeenstudiedextensively.TherearealsoabstractapproachessuchasZhu’sonetoonecorrespondencebetweenthesetofinequivalentirreduciblemodulesforagivenvertexoperatoralgebraandthesetofinequivalentirreduciblemodulesforanassociativealgebraassociatedwiththevertexoperatoralgebra[Z],andthetensorproductsofmodules([HL]and[L]);Seealso[FHL]fortheresultsconcerningintertwiningoperatorsandcontragredientmodules.ManyoftheseresultsareanaloguesofthecorrespondingresultsintheclassicalLiealgebratheory.Thepurposeofthispaperistogiveaconstructionofinducedtwistedmodulesforvertexoperatoralgebrasandpresentsomeinitialresults.ThemainideainconstructingtheinducedmodulecomesfromtheinductiontheoryfortherepresentationsofLiegroups,algebraicgroups,quantumgroups,Hopfalgebras([V],[J],[APW],[Lin1-Lin2]).Inorderfortheinducedmoduletohavethefunctorialproperty,onehastoenlargethe1991MathematicsSubjectClassificationPrimary17B35ThefirstauthorwassupportedbyNSFgrantDMS-9303374andaresearchgrantfromtheCom-mitteeonResearch,UCSantaCruz.ThesecondauthorwassupportedbyNSFgrantDMS–9401389.1categoryofg-twistedmodulestoensuretheexistenceoftheinducedmodules.ThisresemblesHarish-Chandra’stheoryintherepresentationtheoryofLiegroups.Weprovethatinmostinterestingcases,theinducedmodulesfromasimplemoduleforavertexoperatorsubalgebraareindeedg-twistedmodules.Thestructuresofthesemodulesincertainspecialcasesarediscussed.Oneofthemainmotivationsforintroducingtheinducedmodulesistostudythe“orbifoldconformalfieldtheory.”Roughlyspeaking,anorbifoldtheoryisaconformalfieldtheorywhichisobtainedfromagivenconformalfieldtheorymodulotheactionofafinitesymmetrygroup(see[DVVV]).LetVbeavertexoperatoralgebraandGbeafinitesubgroupoftheautomorphismsofV.DenotebyVGthesubspaceofVconsistingofthefixedpointsundertheactionofG.ThenVGisavertexoperatorsubalgebraofV.Algebraically,theorbifoldtheoryistostudytherepresentationtheoryofVG.Themainnewfeatureoftheorbifoldtheoryistheintroductionoftwistedmodules.Ag-twistedmoduleforg∈GisautomaticallyanordinarymoduleforVGundertherestriction.Itisprovedin[DM2]thatifVisholomorphicandGisnilpotentthenanyirreducibleg-twistedV-moduleiscompletelyreducibleasaVG-module.ItisconjecturedthatthisistrueforarbitraryGandthatanyirreducibleVG-moduleappearsasanirreduciblecomponentofsomeirreducibletwistedmodule(seee.g.[DVVV],[DPR]and[DM2]).Thetheoryofinducedtwistedmodulesforvertexoperatoralgebras,discussedinthispaper,isbeingdevelopedwiththisconjectureinmind.Thispaperisorganizedasfollows:InSection2,afterrecallingthenotionoftwistedmoduleforavertexoperatoralgebrafrom[D2]and[FFR],wedefinethetwistedenvelopingalgebraA(g).WegiveanecessaryandsufficientconditionunderwhichanA(g)-moduleisag-twistedmodule.AlineartopologyisdefinedonA(g)byatopologicalbasisat0consistingofallAnnA(g)(m)foranelementminatwistedmodule.ItturnsoutthistopologyistheweakesttopologyonA(g)sothattheA(g)-modulestructureonanytwistedmoduleMgivesacontinuousmapfromA(g)toEndC(M),whichisequippedwiththepoint-wiseconvergencetopology.ItisimportanttonotethatweneedtherepresentationstobecontinuousinorderfortheJacobiidentitytohold.WealsointroduceacertainA(g)-modulecategory¯Cgwhichcontainstheg-twistedV-modulecategoryCgasafullsubcategory.Infact,¯CgconsistsoftheobjectsofCgtogetherwiththeirdirectlimitsinthecategoryofA(g)-modules.Section3whichdevotesthedefinitionofinducedmoduleisthecenterofthepaper.LetgbeafiniteorderautomorphismofVandV′asubalgebraofVwhichisg-invariant.DenotetherestrictionofgtoV′byg′.ThenthereisanalgebraembeddingfromA(g′)intoA(g).Forag′-twistedV′-moduleWwedefineInd¯Cg¯C′g′(W)tobethesubspaceofHomA(g′)(A(g),W)consistingofelementswhicharekilledbysomeAnnA(g)(m)forsomeminag-twistedV-module.Thenweprovethatthisinductionfunctorenjoysallthepropertiesofaninductionfunctorshouldhave,suchastheFrobeniusreciprocityandthetransitivity.InSection4weinvestigatetheg-inducedmodulesforvertexoperatoralgebrawith2onlyfinitelymanyirreducibleg-twistedmodules.InthecasethatVisg-rational,thatis,anyg-twistedmoduleiscompletelyreducible,weshowthattheinducedmoduleofW,whichhasacompositionseries,fromag-invariantsubalgebratoVisinfactinCg.Inparticular,theinducedmodulefromanyirreduciblemoduleinthiscaseisanordinaryg-twistedmodule.InSection