0911.0216v1Finite-dimensional vertex algebra modul

arXiv:0911.0216v1[math.QA]1Nov2009Finite-dimensionalvertexalgebramodulesoverfixedpointdifferentialsubfieldsKenichiroTanabe∗DepartmentofMathematicsHokkaidoUniversityKita10,Nishi8,Kita-Ku,Sapporo,Hokkaido,[e-mail]fferentialfieldoverCwithderivationD,GafinitelinearautomorphismgroupoverKwhichpreservesD,andKGthefixedpointsubfieldofKundertheactionofG.Weshowthatev-eryfinite-dimensionalvertexalgebraKG-moduleiscontainedinsometwistedvertexalgebraK-module.1IntroductionIn[2],Borcherdsdefinedthenotionofvertexalgebrasandshowedthatev-erycommutativeassociativealgebraAwithanarbitraryderivationDhasastructureofvertexalgebra.EveryassociativealgebraA-modulenaturallybecomesavertexalgebraA-module.However,aspointedoutbyBorcherds[2],thereisnoguaranteethatallvertexalgebraA-modulescomefromasso-ciativealgebraA-modules.Infact,in[9]forthepolynomialringC[s]inone∗PartiallysupportedbyJSPSGrant-in-AidforScientificResearchNo.20740002.1variableswithderivationD,IobtainedanecessaryandsufficientconditiononDthatthereexistfinite-dimensionalvertexalgebraC[s]-moduleswhichdonotcomefromassociativealgebraC[s]-modules.Thus,ingeneralvertexalgebraA-modulesandassociativealgebraA-modulesarecertainlydifferent.LetKbeadifferentialfieldwithderivationD,Gafinitelinearautomor-phismgroupofKwhichpreservesD,andKGthefixedpointsubfieldofKundertheactionofG.Inthispaper,westudyvertexalgebraKG-modules.Here,letusrecallthefollowingconjectureonvertexoperatoralgebras:letVbeavertexoperatoralgebraandGafiniteautomorphismgroupofV.ItisconjecturedthatundersomeconditionsonV,everyirreduciblemoduleoverthefixedpointvertexoperatorsubalgebraVGiscontainedinsomeirreducibleg-twistedV-moduleforsomeg∈G(cf.[4]).Themotivationforstudyingver-texalgebraKG-modulesistoinvestigatethisconjectureforvertexalgebras.InTheorem1,Ishallshowthateveryfinite-dimensionalindecomposablevertexalgebraKG-modulebecomesag-twistedvertexalgebraK-moduleforsomeg∈G.Namely,theconjectureholdsforallfinite-dimensionalvertexalgebraKG-modulesinastrongersense.Thispaperisorganizedasfollows.InSection2werecallsomepropertiesofvertexalgebrasandtheirmodules.InSection3weshowthateveryfinite-dimensionalindecomposablevertexalgebraKG-modulebecomesag-twistedvertexalgebraK-moduleforsomeg∈G.InSection4wegivetheclassifica-tionofthefinite-dimensionalvertexalgebraC(s)-modules.InSection5forallquadraticextensionsKofC(s)andallfinite-dimensionalindecomposablevertexalgebraC(s)-modulesM,westudytwistedvertexalgebraK-modulestructureofM.2PreliminaryWeassumethatthereaderisfamiliarwiththebasicknowledgeonvertexalgebrasaspresentedin[2,3,7].Throughoutthispaper,ζpisaprimitivep-throotofunityforapositiveintegerpand(V,Y,1)isavertexalgebra.RecallthatVistheunderlyingvectorspace,Y(·,x)isthelinearmapfromVto(EndV)[[x,x−1]],and1isthevacuumvector.LetDbetheendomorphismofVdefinedbyDv=v−21forv∈V.LetEndenotethen×nidentitymatrix.First,werecallsomeresultsin[2]foravertexalgebraconstructedfromacommutativeassociativealgebrawithaderivation.2Proposition1.[2]Thefollowinghold:(1)LetAbeacommutativeassociativeC-algebrawithidentityelement1andDaderivationofA.Fora∈A,defineY(a,x)∈(EndA)[[x]]byY(a,x)b=∞Xi=01i!(Dia)bxiforb∈A.Then,(A,Y,1)isavertexalgebra.(2)Let(V,Y,1)beavertexalgebrasuchthatY(u,x)∈(EndV)[[x]]forallu∈V.DefineamultiplicationonVbyuv=u−1vforu,v∈V.Then,VisacommutativeassociativeC-algebrawithidentityelement1andDisaderivationofV.Throughouttherestofthissection,AisacommutativeassociativeC-algebrawithidentityelement1overCandDaderivationofA.Let(A,Y,1)bethevertexalgebraconstructedfromAandDinProposition1andlet(M,YM)beamoduleoververtexalgebraA.WecallMavertexalgebra(A,D)(orA)-moduletodistinguishbetweenmodulesoververtexalgebraAandmodulesoverassociativealgebraA.Proposition2.[2]Thefollowinghold:(1)LetMbeanassociativealgebraA-module.Fora∈A,defineYM(a,x)∈(EndM)[[x]]byY(a,x)u=∞Xi=01i!(Dia)uxiforu∈M.Then,(M,YM)isavertexalgebra(A,D)-module.(2)Let(M,YM)beavertexalgebra(A,D)-modulesuchthatY(a,x)∈(EndM)[[x]]foralla∈A.DefineanactionofAonMbyau=a−1ufora∈Aandu∈M.Then,MisanassociativealgebraA-module.Remark1.(1)LetusconsiderthecaseofD=0.Let(M,YM)beanarbitraryvertexalgebra(A,0)-module.Foralla∈A,since0=YM(0a,x)=dYM(a,x)/dx,weseethatYM(a,x)isconstant.Thus,MisanassociativealgebraA-modulebyProposition2.3(2)LetusconsiderthecasethatAisfinite-dimensional.Let(M,YM)beanarbitraryvertexalgebra(A,D)-module.Supposethatthereexista∈Aandu∈MsuchthatYM(a,x)uisnotanelementofM[[x]].SinceYM(Dia,x)=diYM(a,x)/dxiforalli≥0,weseethat{YM(Dia,x)u|i=0,1,...}islinearlyindependent.ThiscontradictsthatAisfinite-dimensional.Thus,MisanassociativealgebraA-modulebyProposition2.ForaC-linearautomorphismgofVoffiniteorderp,setVr={u∈V|gu=ζrpu},0≤r≤p−1.Werecallthedefinitionofg-twistedV-modules.Definition1.Ag-twistedV-moduleMisavectorspaceequippedwithalinearmapYM(·,x):V∋v7→YM(v,x)=Xi∈(1/p)Zvix−i−1∈(EndM)[[x1/p,x−1/p]]whichsatisfiesthefollowingfourconditions:(1)YM(u,x)=Pi∈r/p+Zuix−i−1foru∈Vr.(2)YM(u,x)w∈M((x1/p))foru∈Vandw∈M.(3)YM(1,x)=idM.(4)Foru∈Vr,v∈Vs,m∈r/T+Z,n∈s/T+Z,andl∈Z,∞Xi=0mi(ul+iv)m+n−i=∞Xi=0li(−1)iul+m−ivn+i+(−1)l+1vl+n−ium+i