0210274v1 Classification of irreducible modules for the vertex operator algebra V_L+ General case

forthevertexoperatoralgebraVL+

:Generalcase

ToshiyukiAbe1

and

ChongyingDong2

DepartmentofMathematics,

UniversityofCalifornia,SantaCruz,CA95064

Abstract

TheirreduciblemodulesforthefixedpointvertexoperatorsubalgebraV+

ofthevertexoperatoralgebraVtoanarbitrarypositivedefiniteLLassociatedevenlatticeLundertheautomorphismliftedfromthe−1isometryofLareclassified.

1Introduction

ThispaperisacontinuationofourstudyoftheZ2-orbifoldvertexoperatoralge-braVL+

whichisthefixedpointsofvertexoperatoralgebraVL(see[B],[FLM2])associatedtopositivedefiniteevenlatticeLundertheautomorphismθliftedfrom

the−1isometryofL.WeclassifytheirreduciblemodulesforVL+

.Itturnsoutthat

anyirreducibleVL+

-moduleiseitherisomorphictoasubmoduleofanirreducibleVL-moduleorasubmoduleofanirreducibletwistedVL-module.InthecasethattherankofLisone,thisresultwasobtainedpreviouslyin[DN2].

TheVL+

formsabasicclassofvertexoperatoralgebrasbesidesaffine,Virasoroandlatticevertexoperatoralgebras.Thestructureandrepresentationtheoryforaffine,Virasoroandlatticevertexoperatoralgebrasarewellunderstood(see[B],[FLM2],[D1],[DL],[DLM2],[FZ],[Li],[W])withthehelpofaffineKac-Moody

Liealgebras,Virasoroalgebrasandlattices.AlthoughVL+

isalsorelatedtothelattice,butitsstructureandrepresentationtheoryaremuchmorecomplicated.Forexample,ifLdoesnothaveanyvectorofsquaredlength2,suchastheLeechlattice,VL+hasnoweightonevectorsandishardlyrelatedtoaffineKac-Moodyalgebras.

InthiscasetheweighttwosubspaceofVL+

isacommutativenonassociativealgebra

similartotheGriessalgebra[Gr].Insomesense,VL+

isa“typical”vertexoperator

algebra.SothestudyofVL+

shouldhelppeopletoinvestigategeneralvertexoperatoralgebras.

+

ItisworthytopointoutthatVLisaZ2-orbifoldvertexoperatoralgebra.InthecasethatL=ΛistheLeechlattice,suchvertexoperatoralgebrawasusedin[FLM2]toconstructthemoonshinevertexoperatoralgebraV♮.Therehavebeeneffortson

+

studyingtherepresentationsofVLforspecialLwiththehelpofrepresentationtheoryfortheVirasoroalgebrawithcentralcharge1/2(see[D3],[DGH]).Wereferthereaderto[DLM1],[DM],[DVVV]and[K]forgeneralorbifoldtheory.

+

Asin[DN2],themainideaistodetermineZhu’salgebraA(VL)whoseinequiv-alentsimplemoduleshaveaonetoonecorrespondencewiththeinequivalentirre-+ducible(admissible)modulesforVL(see[Z]).Theresultsandmethodsfrom[DN2]inthecaseofrankoneplayimportantrolesinthispaper.Thereisafundamentaldifferencebetweenrankonecaseandhigherrankcases.Zhu’salgebrainrankonecaseiscommutativewithoneexceptionbutisnotforhigherranks.IngeneralLisnotaorthogonalsumofrankonelattices.Nevertheless,foranyelementinLwecanconsidertherankonesublatticegeneratedbytheelementandapplyresultsin[DN2].Notethattheθ-invariantsM(1)+oftheHeisenbergvertexoperatoralgebra

+

M(1)isasubalgebraofVL.SothestructureandrepresentationtheoryforM(1)+obtainedin[DN3]havebeenextensivelyusedinthispaper.Althoughtheinforma-+

tionthatwegetonA(VL)isgoodenoughtoclassifythesimplemodules,wecould

+

notprovethatA(VL)issemisimple.

+

TherearestilltwomajorproblemsaboutVL.Oneistherationalityandtheotheristodeterminethefusionrulesamongirreduciblemodules.Itiscertainlybelieved

+

thatVLisrational.InthecasethatLissomerankonelattice,therationalitywasestablishedin[A].ButrationalityforanyLremainsopen.Thisproblemisrelated

+

toC2-cofinitenessofVL,whichisobtainedrecentlyin[Y]intherankonecaseandin[ABD]ingeneral.Thedeterminationofthefusionruleswillbecarriedoutinaseparatepaper.

Thepaperisorganizedasfollows:InSection2,wereviewvarious(twisted)modulesforavertexoperatoralgebraVanddefineZhu’salgebraA(V).Wedefine

+

vertexoperatoralgebraVLanditssubalgebraM(1)+inSection3.Wealsolist

+

theknownirreduciblemodulesforVL.Section4isaboutZhu’salgebrasA(M(1)+)

+

andA(VL).Inparticularweobtainmanyidentitiesforasetofgeneratorsforin

+A(VL)byusingresultsfrom[DN1],[DN2]and[DN3].Usingtheknownirreducible++VL-moduleswecomputehowthegeneratorsofA(VL)actontheknownsimple

++A(VL)-modules.InSection4wegiveaspanningsetforA(VL)whichisusedinthelatersectionsfortheclassification.Sections5and6aredevotedtothe

++

classificationofsimpleA(VL)-modulesandirreducibleVL-modules.SinceM(1)+

++

isasubalgebraofVLthereisanalgebrahomomorphismfromA(M(1)+)toA(VL).Thishomomorphismembedsthetwod×dmatrixsubalgebrasofA(M(1)+)(see

+

[DN3])intoA(VL)wheredistherankofL.Therearetwocasesintheclassification.InSection5wedealwiththecasethatthesimplemodulesareannihilatedbythe

2

twomatrixsubalgebras.Section6handlestheothercase:thatis,oneofthematrixalgebrasdoesnotannihilatethesimplemodules.

ThroughoutthepaperZ≥0isthesetofnonnegativeintegers.

2Vertexoperatoralgebrasandmodules

Inthissectionwerecallthedefinitionsofvarious(twisted)modulesforavertexoperatoralgebra(cf.[B],[D2],[FFR],[FLM1],[FLM2],[DLM4],[Le],[Z]).WealsoreviewtheZhu’salgebraA(V)associatedtoavertexoperatoralgebraV.

󰀎

AvertexoperatoralgebraVisaZ-gradedvectorspaceV=n∈ZVnequipped

󰀂−n−1

withalinearmapY:V→(EndV)[[z,z−1]],a→Y(a,z)=forn∈Zanz

a∈VsuchthatdimVnisfiniteforallintegernandthatVn=0forsufficientlysmallintegern(see[FLM2]).Therearetwodistinguishedvectors,thevacuumvector1∈V0andtheVirasoroelementω∈V2.BydefinitionY(1,z)=idV,andthe

󰀂

componentoperators{L(n)}ofY(ω,z)=n∈ZL(n)z−n−2givesarepresentationoftheVirasoroalgebraonVwithcentralchargec.EachhomogeneousspaceVn(n≥0)isaneigenspaceforL(0)witheigenvaluen.

AnautomorphismgofavertexoperatoralgebraVisalinearisomorphismofVsatisfyingg(ω)=ωandgY(a,z)g−1=Y(g(a),z)foranya∈V.WedenotebyAut(V)thegroupofallautomorphismsofV.ForasubgroupGLetgbeanautomorphismofavertexoperatoralgebraVoforderT.ThenVisadirectsumofeigenspacesforg:

V=

T−1󰀏r=0

Vr,Vr={a∈V|g(a)=e−

2πir

(3)YM(1,z)=idM,

T

anz−n−1,

3

(4)(thetwistedJacobiidentity)

󰀅

z1−z2−1

z0δ

−1

=z2

󰀅

z1−z0

T

δ

󰀅

−z0

z1−z0

󰀆

YM(b,z2)YM(a,z1)

dz

holdsforanya∈V(see[DLM2]).

Y(a,z)foralla∈V.(2.1)

Definition2.2.Anadmissibleg-twistedV-moduleMisaweakg-twistedV-modulewhichhasa1Z≥0M(n)suchthat

T

a(m)M(n)⊂M(wt(a)+n−m−1)

foranyhomogeneousa∈Vandm,n∈Q.

(2.2)

Inthecasegistheidentity,anyadmissibleg-twistedV-moduleiscalledanadmissibleV-module.Anyg-twistedweakV-submoduleNofag-twistedadmissible

󰀎

V-moduleiscalledag-twistedadmissibleV-submoduleifN=n∈1

Inthecasegistheidentity,ag-twistedV-moduleiscalledaV-module.AV-moduleMiscalledirreducibleifMisirreducibleasaweakV-module.Bydefinition,foravertexoperatoralgebra(V,Y,1,ω),(V,Y)becomesaV-module.IfavertexoperatoralgebraisirreducibleasaV-module,thenViscalledsimple.

Itisprovedin[DLM4]thatifVisg-rationalthenthereareonlyfinitelymanyinequivalentirreducibleadmissibleg-twistedV-modulesandanyirreducibleadmis-sibleg-twistedV-moduleisordinary.

WenextdefineZhu’salgebraA(V)whichisanassociativealgebrafollowing[Z].Foranyhomogeneousvectorsa∈V,andb∈V,wedefine

󰀅

(1+z)wt(a)

a∗b=Resz

󰀆

Y(a,z)b,z2andextendtoV×Vbilinearly.DenotebyO(V)thelinearspanofa◦b(a,b∈V)andsetA(V)=V/O(V).Wewrite[a]fortheimagea+O(V)ofa∈V.Thefollowingtheoremisdueto[Z](alsosee[DLM4]).

Theorem2.5.(1)Thebilinearoperation∗inducesA(V)anassociativealgebrastructure.Thevector[1]istheidentityand[ω]isinthecenterofA(V).

󰀎∞

(2)LetM=n=0M(n)beanadmissibleV-modulewithM(0)=0.Thenthelinearmap

o:V→EndM(0),a→o(a)|M(0)inducesanalgebrahomomorphismfromA(V)toEndM(0).ThusM(0)isaleftA(V)-module.

(3)ThemapM→M(0)inducesabijectionfromthesetofequivalenceclassesofirreducibleadmissibleV-modulestothesetofequivalenceclassesofirreducibleA(V)-modules.

3

+

VertexoperatoralgebrasM(1)+andVL

+

InthissectionwerecalltheconstructionofthevertexoperatoralgebraVLassociatedwithapositivedefiniteevenlatticeLandthevertexoperatoralgebraM(1)+(cf.[FLM2]).Wealsostateseveralresultsonclassificationsofirreduciblemodulesfor+VLwhentherankofLisone(see[DN2])andM(1)+(see[DN1]and[DN3]).

LetLbearankdevenlatticewithapositivedefinitesymmetricZ-bilinearform(·,·).Weseth=C⊗ZLandextend(·,·)toaC-bilinearformonh.Letˆ=C[t,t−1]⊗h⊕CCbetheaffinizationofcommutativeLiealgebrahdefinedbyh

ˆ]=0[β1⊗tm,β2⊗tn]=m(β1,β2)δm,−nCand[C,h

5

ˆ+=C[t]⊗h⊕CCisacommutativesubalgebra.foranyβi∈h,m,n∈Z.Thenh

ˆ+-moduleCeλbytheactionsForanyλ∈h,wecandefineaonedimensionalh

ρ(h⊗tm)eλ=(λ,h)δm,0eλandρ(C)eλ=eλforh∈handm≥0.Nowwedenoteby

ˆ)⊗ˆ+Ceλ∼M(1,λ)=U(h=S(t−1C[t−1])U(h)

ˆ-moduleinducedfromhˆ+-moduleCeλ.SetM(1)=M(1,0).Thenthereexiststheh

alinearmapY:M(1)→(EndM(1,λ)[[z,z−1]]suchthat(M(1),Y,1,ω)hasasimplevertexoperatoralgebrastructureand(M(1,λ),Y)becomesanirreducibleM(1)-moduleforanyλ∈h(see[FLM2]).ThevacuumvectorandtheVirasoroelementaregivenby1=e0andω=1

definedby

Y(h(−1)1,z)=h(z)=Y(eα,z)=exp

󰀌

󰀊

n∈Z

h(−n)z−n−1,

z−n

󰀍eαzα,

∞󰀊α(−n)n=1

n

whereh(−n)(h∈h,n∈Z)istheactionofh⊗tnonVλ+L,eαistheleftactionof

ˆonC[L◦]andzαistheoperatoronC[L◦]definedbyzαeλ=z(α,λ)eλ.L

Thevertexoperatorassociatedtothevectorv=β1(−n1)···βr(−nr)eαforβi∈h,ni≥1andα∈Lisdefinedby

(n1−1)

Y(v,z)=◦β1(z)···∂(nr−1)βr(z)Y(eα,z)◦◦∂◦,

◦

where∂=1dz)nandthenormalordering◦◦·◦isanoperationwhichreorderstheoperatorssothatβ(n)(β∈h,n<0)andeαtobeplacedtotheleftofX(n),(X∈h,n≥0)andzα.

WenotethatM(1)iscontainedinVLasavertexoperatorsubalgebrawithsameVirasoroelement.Foranyλ∈L◦,M(1)⊗eλisisomorphictoM(1,λ)as

󰀎

M(1)-modules.ThusVλ+Lisisomorphictoα∈LM(1,λ+α)asM(1)-modules.Nowwedefinealinearisomorphismθ:VL+λi→VL−λifori∈L◦/Lby

θ(β1(−n1)β2(−n2)···βk(−nk)eα+λi)=(−1)kβ1(−n1)β2(−n2)···βk(−nk)e−α−λiforβi∈h,ni≥1andα∈Lif2λi∈L,andθ(β1(−n1)β2(−n2)···βk(−nk)eα+λi)

=(−1)kc2λiǫ(α,2λi)β1(−n1)β2(−n2)···βk(−nk)e−α−λi

if2λi∈Lwherec2λiisasquarerootofǫ(2λi,2λi).Thenθdefinesalinearisomor-phismfromVL◦toitselfsuchthat

θY(u,z)v=Y(θu,z)θv

foru∈VLandv∈VL◦.Inparticular,θisanautomorphismofVLwhichinducesanautomorphismofM(1).

Foranyθ-stablesubspaceUofVL◦,letU±bethe±1-eigenspaceofUforθ.Wehavethefollowingproposition(see[DM]and[DLM1]):

Proposition3.1.(1)M(1)±,M(1,λ)forλ∈h−{0}areirreducibleM(1)+-modules,andM(1,λ)∼=M(1,−λ).

+

(2)(VL+λi+VL−λi)±fori∈L◦/LareirreducibleVL-modules.Moreoverif2λi∈Lthen(VL+λi+VL−λi)±,VL+λi,andVL−λiareisomorphic.

7

Nextwerecallaconstructionofθ-twistedmodulesforM(1)andVLfollowing

1

[FLM2]and[D2].Denotebyh[−1]=h⊗t

+Z.ThenthesymmetricalgebraM(1)(θ)=S(t−2

1

2

,z−

1

T,±±

+Zandt∈T.WedenotebyM(1)(θ)andVthe±1-≥0L2

T

eigenspaceforθofM(1)(θ)andVLrespectively.

ˆ-moduleassociatedtoacentralFollowing[FLM2],letTχbetheirreducibleL/K

characterχsatisfyingχ(κ)=−1.Thenanyirreducibleθ-twistedVL-moduleis

T

isomorphictoVLχforsomecentralcharacterχwithχ(κ)=−1(see[D2]).By[DLi]weget

Proposition3.2.(1)M(1)(θ)±areirreducibleM(1)+-modules.

ˆ(2)LetχbeacentralcharacterofL/Ksuchthatχ(ι(κ))=−1,andTχthe

ˆ-modulewithcentralcharacterχ.ThenV+-modulesVTχ,±areirre-irreducibleL/KLLducible.

+

Itisprovedin[DN1]and[DN2]thatanyirreduciblemodulesforM(1)+andVLwithrankonelatticeLisisomorphictooneofirreduciblemodulesinPropositions3.1and3.2:

Theorem3.3.([DN1])Theset

{M(1)±,M(1)(θ)±,M(1,λ)(∼=M(1,−λ))|λ∈h−{0}}

givesallinequivalentirreducibleM(1)+-modules.

(3.2)

8

Theorem3.4.([DN2])LetL=Zαbearankoneevenlatticesuchthat(α,α)=2kwithpositiveintegerk.Theset

Ti,±±±

{VL,Vα/,V,Vrα/2k+L|i=1,2,1≤r≤k−1}L2+L

(3.3)

+

givesallinequivalentirreducibleVL-modules,whereTiisanirreducibleL/2L-moduleonwhicheαactsbythescalar(−1)i−1fori=1,2.

4

+

Zhu’salgebrasA(M(1)+)andA(VZα)

+

InthissectionwerecallthestructureofZhu’salgebrasA(M(1)+)andA(VL)fol-lowing[DN2]and[DN3].

FirstwewritedownidentitiesinZhu’salgebraA(M(1)+).Let{ha}beanorthonormalbasisofh.Setωa=ωha=1

h(−2)21.ThevectorωaandJagenerateavertexoperator2a

algebraM(1)+associatedtotheonedimensionalvectorspaceCha(see[DG]).Next

u¯u,Et,E¯tandΛabasfollowswesetSab(m,n)=ha(−m)hb(−n)1,anddefineEab,Eababab

(see[DN3]);

uEab=5Sab(1,2)+25Sab(1,3)+36Sab(1,4)+16Sab(1,5)(a=b),u¯ba=Sab(1,1)+14Sab(1,2)+41Sab(1,3)+44Sab(1,4)+16Sab(1,5)(a=b),E

uuuEaa=Eab∗Eba,

tEab=−16(3Sab(1,2)+14Sab(1,3)+19Sab(1,4)+8Sab(1,5))(a=b),¯t=−16(5Sab(1,2)+18Sab(1,3)+21Sab(1,4)+8Sab(1,5))(a=b),EbatttEaa=EabEba,

Λab=45Sab(1,2)+190Sab(1,3)+240Sab(1,4)+96Sab(1,5).

¯u]=[Eu],[E¯t]=[Et]and[Λab]=[Λba]inA(M(1)+)Itisprovedin[DN3]that[Eabababab

ut¯andE¯for[Eu]and[Et]respectively.foranya,b.ThusweoftenuseEbabababa

By[DN3,Proposition5.3.12]wehaveProposition4.1.Foranya,b,c,d,

uuu[Eab]∗[Ecd]=δbc[Ead],

tu

[Eab]∗[Et∗cd]=δbc[Ead],

uttu

[Eab]∗[Ecd]=[Ecd]∗[Eab]=0,

tuut

]∗[Λab]=0(a=b).]=[Ecd]∗[Λab]=[Ecd[Λab]∗[Ecd]=[Λab]∗[Ecd

9

Remark4.2.Thevectors[Sab(1,n)](1≤n≤5)canbeexpressedaslinearcombi-ut

nationsof[Eab],[Eab]and[Λab]asfollows(see[DN3,Remark5.1.2]);

uu

[Sab(1,1)]=[Eab]+[Eba]+[Λab]+u[Sab(1,2)]=−2[Eab]−[Λab]−u[Sab(1,3)]=3[Eab]+[Λab]+

1

24

t

[Eba],

(4.1)(4.2)(4.3)(4.4)

t

[Eba].

3

t

[Eba],t[Eba],t[Eba],

1535

16315

u

[Sab(1,4)]=−4[Eab]−[Λab]−

32

uu[Sab(1,5)]=5[Eab]+[Eba]+[Λab]+

256

(4.5)

ut

LetAuandAtbethelinearsubspaceofA(M(1)+)spannedbyEabandEab,respectivelyfor1≤a,b≤d.Thenwehave(see[DN3,Proposition5.3.14]):

Proposition4.3.(1)AuareAtaretwosidedidealsofA(M(1)+)andthequotientalgebraA(M(1)+)/(Au+At)iscommutative.

(2)ThenaturalactionsofA(M(1)+)onM(1)−(0)andM(1)(θ)−induceiso-morphismsofalgebrasfromAuandAttoEndM(1)−(0)andEndM(1)(θ)−respec-tively.Underthebasis{h1(−1),...,hd(−1)}({h1(−1/2),...,hd(−1/2)}resp.)of

ut

M(1)−(0)(M(1)(θ)−(0)resp.),each[Eab]([Eab]resp.)correspondstothematrixelementEabwhose(i,j)-entryis1andzeroelsewhere.

Remark4.4.TheidealsAuandAtareindependentofthechoiceofanorthonormal

󰀂d󰀂duutt

basis{ha}.Inparticular,theunitsI=a=1[Eaa]andI=a=1[Eaa]ofAuandAtareindependentofthechoiceofanorthonormalbasis.

Wenextrecallsomerelations(see[DN3,Lemma5.2.2andLemma5.3.2])whichwillbeusedlater.

Proposition4.5.Foranyindicesa,b,c,

uu

[ωa]∗[Ebc]=δab[Ebc],

uu[Ebc]∗[ωa]=δac[Ebc],

󰀆󰀅

1tt

δab[Ebc],[ωa]∗[Ebc]=

2

󰀆󰀅

1tt

δac[Ebc],[Ebc]∗[ωa]=

2

[ωa]∗[Λbc]=[Λbc]∗[ωa]=0.

(4.6)(4.7)(4.8)(4.9)(4.10)

SetHa=Hha=Ja+ωa−4ωa∗ωa(see[DN3]).Thenthefollowingidentities(see[DN3,Proposition6.13])hold:

10

Proposition4.6.Fordistincta,bandc,󰀈󰀉

70[Ha]+1188[ωa]2−585[ωa]+27∗[Ha]=0,

󰀆󰀅

1

∗[Ha]=0,([ωa]−1)∗[ωa]−

1612uut

[Hb]=2[Eaa]−2[Ebb]+[Ebb],−

94

4

(2[ωb]+13)∗[Hb]−

135

15uu

=4([Eaa]−[Ebb])+

15

([ωa]−1)∗[Ha]+

9

1

1

(4.11)(4.12)(4.13)

uu

([Ha]+[Hb])−([Eaa]+[Ebb])−

M(1)+

10Ja

0

uEab

M(1,λ)(λ∈h−{0})

eλ

(ha,λ)2/2

−6δac

0

δbcha(−1)1

0

0

0

03/128

M(1)(θ)−hc(−1/2)11/16+1/2δac9/128−9/8δacδbcha(−1/2)1

0

Λab

2

h(−1)21,Jγ=h(−1)41−

11

2h(−3)h(−1)1+

3

󰀅

[ωα]−

k

(4k−1)(4k−9)

󰀆󰀅

9

[ωα]−16

󰀅

[ωα]−

k

4(8k−3)

󰀆

[Eα],

(4.18)

+VZα

VZα+r

α(−1)1

2k

α

+

VZα+α

−α

2+e

e

α

−VZα+α

2

0Jα

0Eα

0

T1,+

VZα

r2/4k

−6

0

1

α(−1/2)t1

T2,+

VZα

k/4

k4/4−k2/4

0

α(−1/2)t2

1/16

3/1289/1282−2k+11/163/1289/128−2−2k+1

+

Proposition4.9.Foranyα∈Lwith|α|2=2,A(VZα)isasemisimplealgebrageneratedby[ωα],[Jα],[Eα]andthefollowingidentitieshold;

[Eα]∗[Eα]=4ǫ(α,α)[ωα],

[Hα]∗[Eα]+[Eα]∗[Hα]=−12[ωα]∗[ωα]−

󰀆󰀅

1

∗[ωα]−4

󰀅

(4.20)

1

󰀆

∗[Eα]=0.

(4.22)

16

Proof.Thesemisimplicityisdueto[DLM3].SinceEα∗Eα=4ǫ(α,α)ωα,(4.20)isclear.Identities(4.20)–(4.22)areimmediateasthebothsideshavethesameactionsonthesimplemodulesbyTable3.Weonlyneedtoexplainhowtoreadtheactions

−−−α

onthetoplevelVZα(0)ofVZαinthetable.VZα(0)hasabasis{α(−1),F}whichareeigenvectorsfor[ωa],[Jα]and[Hα].Thevector[Eα]mapsα(−1)to−2FαandFα

+α

to2α(−1).InordertoseethatA(VZα)isgeneratedby[E],[ωα],[Jα],weobservethattheactionsof[ωα],[Jα],[Eα]distinguishthesimplemodules.

−VZα

1ωα

JαHαEα

1−6−9−2Fαt1

ωαJαHαEα

1302α(−1)1

T1,−VZα

e

α

2

2

2

−cαe−0

0−c3α

4

2

α

4

t2

T2,−

VZα

9/16

−45/128−135/128−3/29/16−45/128−135/1283/2

Table3.Theactionsofωα,Jα,HαandEαontoplevelsinthecase|α|2=2.

+

FinallywediscussthetoplevelsoftheknownirreducibleVL-modules.Letλ∈L◦suchthatλhastheminimallengthinL+λ,i.e.,λsatisfiesthat|λ+α|2≥|λ|2foranyα∈L.Weassumethatthecosetrepresentativesλihavetheminimallengths.Weset

∆(λ)={α∈L||λ+α|2=|λ|2}.

andL2={α∈L||α|2=2}.Weshallalsousethenotation

󰀇|λ|=

+

forλ∈L◦.ThenthetoplevelsW(0)ofirreducibleVL-modulesWaregivenasfollows;

󰀏

++

VL(0)=C1,VL(0)=h(−1)⊕C(eα−e−α),(4.23)

Vλi+L(0)=Vλ±(0)i+L

T,+

α∈∆(λi)

󰀏󰀊

α∈L2

Ceλi+α

(2λi∈/L),(4.24)(4.25)

=

C(eλi+α±θeλi+α)

T,−

(2λi∈L),

α∈∆(λi)

VLχ(0)=Tχ,VLχ(0)=h(−1/2)⊗Tχ.

(4.26)

Hereh(−1)={h(−1)|h∈h}⊂M(1)andh(−1/2)={h(−1/2)|h∈h}⊂M(1)(θ).

󰀂

Remark4.10.Notethatif2λi∈Lthesumα∈∆(λi)C(eλi+α±θe+λi+α)isnota

¯λi)bedirectsumsinceforanyα∈∆(λi),−2λi−αalsobelongsto∆(λi).Let∆(

¯λi)∩{α,−2λi−α}|=1foranyα∈∆(λi).Thenasubsetof∆(λi)suchthat|∆(

󰀏

±

Vλi+L(0)=C(eλi+α±θeλi+α).(4.27)

¯λi)α∈∆(

5

+

AspanningsetofA(VL)

+++

Foranyα∈L,setVL[α]=M(1)+⊗Eα⊕M(1)−⊗FαandA(VL)(α)=(VL[α]+

++++O(VL))/O(VL).ThenA(VL)isasumofA(VL)(α)forα∈L.Weobtainspanning

++

setsofA(VL)(α)forα∈LandthusforA(VL)inthissection.Thespanningsets

+

willbeusedinthenexttwosectionstoclassifythesimpleA(VL)-modules.

+

LetXbeavertexoperatorsubalgebraofVLwhoseVirasorovectormaydiffer

+

fromtheVirasorovectorofVL.Clearly,theidentitymapinducesanalgebrahomo-+morphismfromA(X)toA(VL).Fromnowonwewilluse[u]foru∈Xtodenote

+

bothu+O(X)andu+O(VL)ifthecontextisclear.Forexample,wewilluse

+uutt

,EabinA(VL],[Eab]fortheimagesofEab[Eab).

+

Considerthemapφα:M(1,α)→VL[α]definedbyu⊗eα→1

+α

(u−θ(u))⊗F.ClearlyφisanM(1)-moduleisomorphism.α2

FirstweconsideraspanningsetofM(1,α)forα∈h.

Proposition5.1.M(1,α)isspannedbyu(−1)eαandu(−1)h(−n)eαforu∈M(1)+,h∈handn≥1.

Proof.LetUbethesubsetofM(1,α)spannedbyu(−1)eαandu(−1)h(−n)eαforu∈M(1)+,h∈handn≥1.Weprovethat

β1(−n1)···βr(−nr)eα∈U

14

(5.1)

forr≥1,ni≥1andβi∈hbyinductiononthelengthr.Thecaser=1isclear.Letp≥2andsupposethat(5.1)holdsifr󰀌r󰀅󰀆󰀍󰀊󰀋−ij−1◦◦

(γ1(−n1)···γr(−nr)1)(−1)u=◦γ1(i1)···γr(ir)◦u

nj−1j=1foranyevenintegerr≥1,ni≥1,γi∈handu∈M(1,α),whereij(1≤j≤r)run

󰀂󰀂

throughintegerssatisfyingij=−nj.Weseethatforanyij∈Z(1≤j≤r)

󰀂󰀂󰀄r󰀈−ij−1󰀉

suchthatij=−nj,thecoefficientj=1nj−1isnonzeroifandonlyifij=−njorsomeijarenonnegative.Thusifpiseventhenweseethatβ1(−n1)···βp(−np)eα=(β1(−n1)···βp(−np)1)(−1)eα+lowerlengthterms,andifpisoddthenβ1(−n1)···βp(−np)eα

=(β1(−n1)···βp−1(−np−1)1)(−1)βp(−np)eα+lowerlengthterms.

Byinductionhypothesis,β1(−n1)···βp(−np)eα∈U.

+

ByProposition5.1andusingthemapφα,weseethatVL[α]isspannedbythevectorsu(−1)Eαandu(−1)h(−n)Fαforu∈M(1)+,h∈handn≥1.Since

u(−1)v=u∗v+lowerweightvectors,++

foranyhomogeneousvectorsu∈M(1)+,v∈VL[α]oneprovesthatVL[α]is

+

spannedbythevectorsu∗Eαandu∗h(−n)Fαforu∈M(1),h∈handn≥1byusinginductiononweight.

+

Lemma5.2.TheA(VL)(α)isspannedbythevectors[u]∗[Eα]∗[v]foru,v∈M(1)+.

Proof.Leth∈handn>0.Itisenoughtoprovethath(−n)Fαliesinthespanof[u]∗[Eα]∗[v]foru,v∈M(1)+.Ifh∈Cα,h(−n)FαiscontainedinthespanningsetbyProposition3.15(1)of[A].Nowweassumethat(h,α)=0.Then

α(−1)h(−n)1∗Eα−Eα∗α(−1)h(−n)1=(α,α)h(−n)Fα.

Againh(−n)Fαiscontainedinthespanningset.

+

NextwewillreducethesizeofthespanningsetofVL[α]further.Fixanor-thonormalbasis{ha}ofhsothath1∈Cα.ThenbyPropositions4.8and4.9,weseethat[Eα]∗[ωb],[Eα]∗[Hb]foranybarelinearcombinationsofvectorsoftheform[u]∗[Eα]foru∈M(1)+.SinceM(1)+isgeneratedby[ωa],[Ha],[Λab]and

15

Au,At,byLemma5.2andthefactthat[Λab]and[Eα]commuteifa=1,b=1,wehave:

+A(VL)(α)=span{[u]∗[Eα]|u∈M(1)+}

d󰀊b=2

+[E]∗A+[E]∗A+

αuαt

C[E]∗[Λ1b]

α

(5.2)

+u

whereAu(Atresp.)isunderstoodtobethesubalgebraofA(VL)generatedby[Eab]

T,−−t

([Eab]resp.)for1≤a,b≤d.SinceVL(0)containsh(−1)asasubspaceandVLχcontainsh(−1/2)asatensorfactorforanyχ(see(4.23)and(4.26)),byProposition4.3,AuandAtareisomorphictothematrixalgebraMd(C).

Lemma5.3.Foranyb=1,

[Λ1b]∗[Eα]+[Eα]∗[Λ1b]∈U1b∗[Eα]+[Eα]∗U1b,

+uuuu

whereUabisthesubspaceofA(VL)linearlyspannedby[Eab],[Eba],[Eab]and[Eba].

Proof.ByRemark4.2weseethat[S1b(1,1)]≡[Λ1b]≡−[S1b(1,2)]modU1b.Bydirectcalculations,wehave|α|S1b(1,n)∗Eα

=|α|h1(−1)hb(−n)Eα+n|α|2hb(−n−1)Fα+(n+1)|α|2hb(−n)Fα.(5.3)

α

Theidentity|α|h1(−1)hb(−n)Eα=−nhb(−n−L(−1)hb(−n)Fαshowsthat󰀁1)F+2|α|[h1(−1)hb(−n)eα]=−n[hb(−n−1)Fα]−n+|α|

2

Similarly,

|α|[Eα]∗[S1b(1,n)]

󰀅

|α|22α2

=n(|α|−1)[hb(−n−1)F]+n(|α|−1)−

󰀆

[hb(−n)Fα].(5.4)

Since[S1b(1,1)]+[S1b(1,2)]∈U1b,weseethat[hb(−2)Fα]+[hb(−1)Fα]liesinU1b∗[Eα]+[Eα]∗U1b.Using(5.4)and(5.5)gives

|α|([Λ1b]∗[Eα]+[Eα]∗[Λ1b])≡|α|([S1b(1,1)]∗[Eα]+[Eα]∗[S1b(1,1)])

modU1b∗[Eα]+[Eα]∗U1b

=(|α|2−1)([hb(−2)Fα]+[hb(−1)Fα])≡0modU1b∗[Eα]+[Eα]∗U1b,

asdesired.

ByLemma5.3and(5.2)weimmediatelyhaveProposition5.4.Foranyα∈L,

+A(VL)(α)=span{[u]∗[Eα]|u∈M(1)+}+[Eα]∗Au+[Eα]∗At

=span{[Eα]∗[u]|u∈M(1)+}+[Eα]∗Au+[Eα]∗At.

+

Inparticular,anyvectorinA(VL)isalinearcombinationofvectorsoftheform[u]∗[Eα]and[Eα]∗aforα∈L,u∈M(1)+anda∈Au+At.

+

WeremarkthatthesecondspanningsetofA(VL)(α)isprovedsimilarly.

Weconcludethissectionwiththefollowinglemmawhichwillbeusedinthenexttwosections.

Lemma5.5.LetItbetheunitofthesimplealgebraAt.Thenforanyα∈L,It∗[Eα]−[Eα]∗It=0.

Proof.WehavealreadypointedoutthatItisindependentofthechoiceofor-thonormalbasisofh.Takeanorthonormalbasis{ha}sothath1∈Cα.Then

󰀂d

tttt

].Itisclearthat[Eaa]∗[Eα]−[Eα]∗[Eaa]=0foranya≥2.HenceI=a=1[Eaa

tt

wehavetoshowthat[E11]∗[Eα]−[Eα]∗[E11]=0.Set

¯t=−16(3Saa(1,2)+18Saa(1,3)+21Saa(1,4)+8Saa(1,5)).Eaa

ttt¯t∗E¯t+u+v,forsomeItiseasytocheckbydefinitionsthatE11=E12∗E21=E1122

+++

u∈M(1)Ch1andv∈M(1)Ch2whereM(1)WisthesubspaceofM(1)+corresponding

¯t]and[v]commutewith[Eα]weonlyneedtotothesubspaceWofh.Since[E22

t¯provethat[E11]and[u]commutewith[Eα].Notethattheidentitymapinducesan

++¯t]andalgebrahomomorphismfromA(VZ)toA(V).Itsufficestoprovethat[E11Lα+α

[u]commutewith[E]inA(VZα).

+

Theresultisclearif|α|2=2asA(VZα)iscommutative(seeProposition4.8).

+

If|α|2=2,A(VZα)isasemisimplealgebraofdimension11byProposition4.9

¯t]and[u]commutewith[Eα]ontheandTable3.Soitisenoughtoprovethat[E11

17

−αt

unique2dimensionalmoduleVZα(0)spannedbyα(−1)andF.Since[E11]=0on−−−t¯tVL(0)weimmediatelyseethat[E11]actsonVZα(0)⊂VL(0)aszero.Clearly,[E22]

−−

and[v]actonVZα(0)trivially.Asaresult,[u]=0onVZα(0).Adirectcalculation

¯t)h1(−1)=o(E¯t)(Fα)=0.Thisshowsthat[E¯t]and[u]arezeroshowsthato(E111111

−α

onVZα(0),andcommutewith[E]inparticular.Sotheproofiscomplete.

6ClassificationI

+

InthissectionandthenextsectionweclassifythesimplemodulesforA(VL)and

+

thusclassifytheirreducibleadmissibleVL-modules.Weprovethatanyirreducible

+

admissibleVL-moduleisordinaryandisisomorphictoonegiveninPropositions

+

3.1and3.2.InthissectionwedealwithasimpleA(VL)-moduleWsuchthatAuW=AtW=0.Theothercaseswillbestudiedinthenextsection.

+

NowletWbeanirreducibleA(VL)-modulesuchthatAuW=AtW=0.An

+

elementu∈A(VL)iscalledsemisimpleonWifuactsonWdiagonally.Notethat

+

foranyα∈L,A(VZα)issemisimple,andthat[ωα]and[Jα]aresemisimpleonW.Asaresult,[Hα]issemisimpleonWsince[ωa]and[Ja]commute.

Takemutuallyorthogonalelementsαa∈Lwith1≤a≤d,andconsidertheorthonormalbasis{ha}ofhsuchthatha∈Cαa.SinceAtW=AuW=0weseethat[Sab(1,1)]=[Λab]onW.Usingidentity

1

2

[(αa+αb)(−1)21]=[ωa]+|αa||αb|[Λab]+[ωb]

(6.1)

onW.Since[ωa],[ωb]and[(αa+αb)(−1)21]aresemisimpleandcommuteeachotheronW,[Λab]isalsosemisimpleonW.NotethatA(M(1)+)isgeneratedby[ωa],[Ja]for1≤a≤d,[Λab]for1≤a=b≤dandAu,At.Weobtainthefollowinglemma.

+

Lemma6.1.LetWbeanirreducibleA(VL)-modulesuchthatAuW=AtW=0.ThenanyelementinA(M(1)+)issemisimpleonW.

SinceA(M(1)+)/(Au+At)isacommutativealgebrabyProposition4.3,Lemma6.1impliesthatWisadirectsumofone-dimensionalirreducibleA(M(1)+)-modulesonwhichAuandAtactaszero.ByTheorem3.3,anirreducibleA(M(1)+)-submoduleofWisisomorphictooneofthefollowing:M(1)+(0),M(1)(θ)+(0)orM(1,λ)(0)forsomeλ∈h−{0}.

Nowweconsiderthecase[Hγ]W=0forsomeγ∈h.Thenwetakeanorthonor-malbasis{ha}ofhsothath1∈Cγ.Byusing(4.13),wehave

[Ha]=[H1]=0

18

(6.2)

submodulesisomorphictoM(1,λ)(0)ifλ=0andW0isasumofsimpleA(M(1)+)-+

submodulesisomorphictoM(1)+(0)=C1.SinceM(1,λ)∼=M(1,−λ)asM(1)-modules,wecanassumethatλrangesinh−{0}/∼,wheretheequivalenceλ∼µisdefinedbyλ=±µ.

WefirstassumethatW0=0,andletvbeanonzerovectorinW0.ThenCvisisomorphictoC1asA(M(1)+)-modules.Thisshowsthat[ωα]v=0foranyα∈L.

+

Henceby(4.19)and(4.22)wehave[Eα]v=0.SinceA(VL)isgeneratedby[u]and

++

[Eα](u∈M(1)+,α∈L),CvisanA(VL)-moduleisomorphictoVL(0)=C1.This

+

showsthatWisisomorphictoVL(0).

NextweconsiderthecaseW0=0.WesetP(W)={λ∈h−{0}|Wλ=0}.Lemma6.2.(1)Foranyλ,µ∈P(W),|λ|2=|µ|2.

(2)Foranyα∈Landλ∈P(W),(λ,α)∈Z,i.e.,P(W)⊂L◦.(3)Foranyλ∈P(W),λhastheminimallengthinL+λ.

onWforanya.WenotefromTable1thatanyirreducibleA(M(1)+)-moduleonwhich[Ha]acts0fora=1,...,disisomorphictoM(1)+(0)orM(1,λ)(0)forλ∈h−{0}.SoWisadirectsumofM(1)+(0)andM(1,λ)(0)asanA(M(1)+)-module.Infact,[Hγ]W=0foranyγ󰀃∈hwith(γ,γ)=0againbyTable1.󰀁󰀂

+

WewriteW=W0⊕W,whereWisasumofsimpleA(M(1))-λλλ=0

Proof.(1)followsfromthefactthattheVirasoroelement[ω]actsonWasaconstant

2

andactsonWλbyscalar|λ|

.2|α|2

+

Ontheotherhand,byTables2and3,[ωα]actsonanyirreducibleA(VZα)-module22(λ,α)

).Thusforanyλ∈P(W),foronwhich[Hα]=0asr

22|α|2

somer.Thisshowsthat(λ,α)isaninteger.

4

Wenowprove(3)andletα,rbeasintheproofof(2).Then(λ,α)2=r2≤|α|

8

=

(λ,α)2

Proof.WehavealreadyknownthatQ(λ,W)⊂∆(λ)∪∆(−λ).Toprove∆(λ)∪∆(−λ)⊂Q(λ,W)itsufficestoshowthat[Eα]v=0foranynonzeroα∈∆(λ).Writeλ=λ1+λ2suchthatλ1∈Cαand(λ2,α)=0.Thenthecondition|λ+α|2=|λ|2impliesthat

2(λ1,α)=−|α|2.α.ByTheorem3.4andTable2andTable3,CvisanHenceλ1=(λ1,α)

2

++

irreducibleA(VZα)-moduleisomorphictoVα+Zα(0).ThusTable2and2Table3againimplies[Eα]v=0,asdesired.

Since∆(λ)∩∆(−λ)={0}foranyλ∈L◦andEα=E−αwehave

󰀊W=C([Eα]v).

α∈∆(λ)

(6.4)

¯λ)isasubsetof∆(λ)if2λ∈L.RecallfromSection4that∆(

Lemma6.4.(1)If2λ∈/L,thenWλ+α=C([Eα]v)forα∈∆(λ),Wλ+α=0for

󰀎

otherαandW=α∈∆(λ)C([Eα]v).(2)If2λ∈L,then

Wλ+α=C([Eα]v)=C([Eα+2λ]v)

forα∈∆(λ),Wλ+α=0forotherαandW=

󰀎

¯λ)α∈∆(

C([Eα]v).

α).WetakeanProof.Letα∈∆(λ)andh∈h.Thenh=(h,α)

|α|2

orthogonalbasis{ha}sothath1∈Cα.ByLemma5.3wehave

[Λ1i][Eα]v=−[Eα][Λ1i]v=−(h1,λ)(hi,λ)[Eα]v

for2≤i≤dand

[Λij][Eα]v=[Eα][Λij]v=−(hi,λ)(hj,λ)[Eα]v

20

2

≤d.Notethat[ωi][Eα]v=

(hi,λ)Eα

]v=

󰀊d(h,hi)2[ωi][Eα]v+2󰀊

(h,hi)(h,hj)[Λij][Eα]vi=1

1≤i=

󰀊d(λ,hi)(h,hi)2i=1

2

2󰀌((h,h󰀊d1)h1,λ)−

(h,hi)(λ,hi)

i=2

󰀍[Eα]v

=1

2󰀅

2(

(h,α)

2(−(h,α)−(h,λ))2[Eα]v

=

1

for2≤i2[h(−1)1][foranyu∈M(1)+andα∈L.RecallthatbothAuandAtacttriviallyonWandVL+λ(0).ByProposition5.4weseethat

f([u]v)=o(u)f(v)=o(u)eλ

+++

foranyu∈VL.Letw=[x]vforsomex∈VLandu∈VL.Then

f([u]w)=f([u∗x]v)=o(u∗x)f(v)=o(u)o(x)f(v)=o(u)f([x]v)=o(u)f(w).

+Thatis,fisanA(VL)-moduleisomorphism.

Thecasethat2λ∈Lismorecomplicated.Inthiscase[E2λ]v=±c32λv.We󰀎λ+αfirstassumethat[E2λ]v=c3+θeλ+α)with¯λ)C(e2λv.NotethatVL+λ(0)=α∈∆(

λλ

o(E2λ)(eλ+θeλ)=c32λ(e+θe)byLemma6.2(3)and(4.25),wherewetakeλtobearepresentativeofλ+L.Asbeforewedefinealinearmapf:W→Vλ++L(0)by

¯λ).ItisclearthatfisanA(M(1)+)-f([Eα]v)=o(Eα)(eλ+e−λ)foranyα∈∆(

moduleisomorphism.Usingtheproofforthecasethat2λ∈/Litisenoughtoprovethatf([Eα]v)=o(Eα)f(v)foranyα∈L.Sinceo(Eα)f(v)=f([Eα]v)=0foranyα∈L−∆(λ)∩∆(−λ)andEα=E−αweonlyneedtoshowthatf([E2λ+α]v)=

¯λ).o(E2λ+α)f(v)foranyα∈∆(

−2λα2λ

Let0=α∈∆(λ).Then(λ,α)isanegativeinteger.Thuseα=e−=0nene

+

foralln≥−1andEα∗E2λ∈VL[α+2λ].Asaresultweseethat[Eα]∗[E2λ]=

++

[u]∗[Eα+2λ]onbothWandVL+λ(0)forsomeu∈M(1)byProposition5.4.Since

+

Co(E2λ+α)(eλ+θeλ)∼=C[E2λ+α]v∼=M(1,λ+α)(0)asA(M(1))-modules,[u]actsonCo(E2λ+α)(eλ+θeλ)andC[E2λ+α]vasasameconstantp.Then[Eα]v=[Eα][E2λ]v=p[E2λ+α]vando(Eα)(eλ+θeλ)=po(Eα+2λ)(eλ+θeλ).Since[Eα]visnonzero,pisnonzero.Thisimpliesthatf([E2λ+α]v)=o(E2λ+α)f(v).Soif[E2λ]v=

+

c32λv,WisisomorphictoVL+λ.

−

Similarly,if[E2λ]v=−c32λv,WisisomorphictoVL+λ.Theproofiscomplete.

Nextweconsiderthecase[Hγ]W=0forsomenonzeroγ∈h.ByLemma6.1,thereexistsaneigenvectorv∈Wfor[Hγ].Wetakeanorthonormalbasis{ha}sothath1∈Cγ.Then(6.2)implies[Ha]v=[Hb]v=0foranya,b.By(4.14)wehave[ωa]v=[ωb]vforanyaandb.Thus(4.15)implies[wa]v=1

v.Wefinallyhave[Λab]v=0by(4.16).ByTheorem3.3128

andTable1,weseethatCvisisomorphictoM(1)(θ)+(0)asanA(M(1)+)-module.ThisshowsthatHβv=9

and0respectively128

foranyβ∈hwith(β,β)=0.

Letα∈L.WeclaimthatW9/128is[Eα]-invariant.Itisenoughtoshowthat[Hα][Eα]w=0if[Eα]wisnonzeroforw∈W.ByPropositions4.8and4.9,0=[Eα]wisaneigenvectorfor[Hα]withnonzeroeigenvaluebynotingthat[ωα]∗[Eα]=[Eα]∗[ωα]and[ωα]actsonW9/128asconstant1/16.ByProposition5.4W9/128isa

22

submoduleandmustbeWitself.ThusWisadirectsumofcopiesofM(1)(θ)+(0)andeachelementofA(M(1)+)actsonWasaconstant.Thisimpliesthattheactionof[Eα]foranyα∈LcommuteswiththeactionofA(M(1)+).Forany0=α∈Lwedefine

󰀅

2|α|22α|α|−1

E−Bα=2

7ClassificationII

+

AgainwefixanirreducibleA(VL)-moduleW.Inthissectionweconsiderthecase

+

AuW=0orAtW=0andcompleteourclassificationofirreducibleA(VL)-modules.FirstweassumethatAuWisnonzero.SinceAuisasimplealgebra,WcontainsasimpleAu-modulesisomorphictoh(−1).Sowecanassumethath(−1)isan

+

A(M(1)+)-submoduleofW.WewillusetheactionofA(VL)onh(−1)togetthewholeW.

Lemma7.1.Foranynonzeroα∈Lwith|α|2=2,Eαh(−1)=0.

Proof.Wetakeanorthonormalbasis{ha}sothath1∈Cα.Then[ωa]hb(−1)1=δa,bhb(−1)1.By(4.19),[Eα]hb(−1)1=0ifb=1.Inthecaseb=1,Ch1(−1)isa

+α

simplemoduleforA(VZα)suchthat[E]=0byTable2.¯α=[Eα]α(−1)1.Since[ωa]α(−1)1=Wenextassumethat|α|2=2.SetF

¯αisnonzero.Using(4.21)andα(−1)1andEα∗Eα=4ǫ(α,α)ωa,wehavethatF

¯α=0.Itisclearthatα(−1)1thefactthat[Hα]α(−1)1=−9α(−1)1gives[Hα]F

¯αarelinearindependentandthatCα(−1)1+CF¯αisclosedundertheactionsandF

+α

of[ωα],[Hα]and[Eα].SinceA(VZα)isgeneratedby[ωα],[Hα]and[E]weseethat

¯αisanA(V+)-module.ByTable3wehave:Cα(−1)1+CFZα¯αisProposition7.2.Foranyα∈L2,thetwodimensionalspaceCα(−1)1+CF+−

anA(VZα)-moduleisomorphictoVZα(0).¯αisanirreducibleA(M(1)+)-moduleNowwefixα∈L2andprovethatCF

isomorphictoM(1,α)(0).Wecontinuetofixanorthonormalbasis{ha}ofhsuch

¯α.Bytheproofofthath1∈CαandconsidertheactionofAu,AtandΛabonF

t¯αtt¯α

Lemma5.5weseethat[Eaa]F=[Eα][Eaa]α(−1)1=0foranya.Thus[Eab]F=tt¯α¯α=0.[Eab][Ebb]F=0foranya,bandAtF

¯α=[Hα]F¯α=0.Ifa>1then[Ha]F¯α=Wehavealreadymentionedthat[H1]F

u¯αu¯α

[Eα][Ha]α(−1)1=0byTable1.Thusby(4.14),[Eaa]F=[Ebb]Fforanya,b.

u¯α¯α=0as[Eu]F¯α=[Eu][Eu]F¯α=[Eu][EaaThisshowsthatAuF]F=0.ababbbab

WenowdealwithΛab.Ifb=1,onehas[Eα]hb(−1)1=0by(4.22).Henceut

[Eα][Eba]ha(−1)1=[Eα][Eba]ha(−1)1=0.Lemma5.3andTable1thenshowthat

¯α=−[Eα][Λ1b]ha(−1)1=0.Itisclearthat[Λab]F¯α=0foranya=1,b=1.[Λ1b]F

¯αisanirreducibleM(1)+-moduleisomorphictoConsequently,C([Eα])h(−1)=CF

M(1,α).

+

SinceA(VL)isgeneratedbyA(M(1)+)andEβforβ∈L,byLemma7.1,weseethat󰀍󰀌

󰀏󰀊

¯α.CFW=h(−1)

α∈L2

24

−

Lemma7.3.Letf:W→VL(0)bealinearmapdefinedbyf(h(−1)1)=h(−1)1

¯α)=−2Fα.ThenfisanA(V+)-moduleisomorphism.andf(FL

¯α=F¯−αandFα=F−α.WeonlyhavetoproveProof.Clearly,fiswell-definedasF

+

thatfisanA(VL)-modulehomomorphism.WehavealreadyprovedthatfisanA(M(1)+)-modulehomomorphism.Itsufficestoshowthat

o(Eβ)f(u)=f([Eβ]u)

(7.1)

+

foranyu∈Wandβ∈LbecauseA(VL)isgeneratedbyA(M(1)+)and[Eβ]forβ∈L.

Notethat[Eβ]h(−1)=0=o(Eβ)M(1)−(0)ifβ∈/L2.So(7.1)holdsforu∈h(−1)andβ∈/L2.Ifβ∈L2andu=h(−1)1suchthat(h,β)=0,then[Eβ]h(−1)1=0inWando(Eβ)h(−1)1=0inM(1)−(0).Again(7.1)holdsinthiscase.Ifβ∈L2andu=β(−1)1,(7.1)followsfromProposition7.2.Therefore(7.1)holdsforu∈h(−1)andβ∈L.

¯αforsomeα∈L2.Notethat[Eα]h(−1)=CF¯α.ByPropositionNowletu=F

5.4if(β,α)<0thenwehave

󰀊

βα

[E]∗[E]=[vi]∗[Eα+β]∗[wi]

i

¯α)=o(Eβ)Fα.forsomevi,wi∈M(1)+.Ifα+β∈/L2thenwehave0=f([Eβ]∗F

Hence(7.1)holdinthiscase.Ifα+β∈L2,C[Eα+β]h(−1)isanA(M(1)+)-moduleisomorphictoM(1,α+β)(0).Soeach[vi]actsasaconstantonC[Eα+β]h(−1).Since(7.1)holdsforanyu∈h(−1),

󰀊

β¯α

f([E]F)=f([vi][Eα+β][wi]α(−1)1)

==󰀊

iii

o(vi)o(Eα+β)o(wi)f(α(−1)1)o([vi]∗[Eα+β]∗[wi])α(−1)1

=o([Eβ])o([Eα])α(−1)1=−2o([Eβ])Fα

¯α).=o([Eβ])f(F

¯αandβ∈Lsuchthatα+β∈L2.Thisshows(7.1)foru=F

If(α,β)=0,wehaveEβ∗Eα=ǫ(α,β)(Eα+β+Eα−β).Since|α±β|2=

¯α.Thus(7.1)holdsinthis|α|2+|β|2≥4,weseethato(Eβ)Fα=0=[Eβ]F

case.

Thuswegetthefollowingproposition:

25

󰀊

+

Proposition7.4.LetWbeanirreducibleA(VL)-modulesuchthatAuW=0.

−+

ThenW∼(0)asA(VL)-module.=VL

FinallysupposethatAtW=0.ThenweseethatWcontainsanirreducibleA(M(1)+)-moduleh(−1/2)isomorphictoM(1)(θ)−(0).SetW0={u∈W|Atu=0}.Ifu∈W0,thenbyLemma5.5,At[Eα]u=At([It][Eα]u)=At([Eα][It]u)=0foranyα∈L.Thatis,W0is[Eα]-invariant.SinceAtisatwo-sidedidealof

+

A(M(1)+),Proposition5.4impliesthatW0isanA(VL)-submoduleofW.Thus

+

W0=0becauseWisanirreducibleA(VL)-modulesuchthatAtW=0.Therefore,wehaveW=AtW.InfactWisadirectsumoftheuniquesimplemoduleh(−1/2).Letα∈Landwetakeanorthonormalbasis{ha}ofhsothath1∈Cα.WeseethatAuW=0and[Λab]W=0forany1≤a=b≤d.Recalltheelement

+

[Bα]∈A(VL)in(6.5).

tLemma7.5.Forany1≤a,b≤d,[Bα]and[Eab]commuteonW.Therefore,[Bα]commuteswiththeactionofA(M(1)+).

Proof.Itisenoughtoshowthelemmainthecasea=1orb=1.SinceAuW=[Λ1b]W=0,by(4.1)and(4.2),wehave

t[Eab]=−[Sab(1,1)]−2[Sab(1,2)],

t

[Eba]=3[Sab(1,1)]+2[Sab(1,2)]

onW.Thus(5.4)and(5.5)givesthefollowingidentities;

tα2α

|α|[E1b]∗[E]=−4(|α|−1)[hb(−3)F]

−(6|α|2−5)[hb(−2)Fα]−

󰀅

3

󰀆

2

t2α|α|[Eα]∗[E1b]=−4(|α|−1)[hb(−3)F]

|α|2−3[hb(−1)Fα],

−(4|α|2−5)[hb(−2)Fα]−

󰀅

1

󰀆

2

Theseidentitiesimplies

tααt

(2|α|2−1)[E1b]∗[E]+[E]∗[E1b]

|α|2−3[hb(−1)Fα].

tα2αt

=−[Eb1]∗[E]−(2|α|−1)[E]∗[Eb1].(7.2)

26

+tt

Notethat[E1b]([Eb1]resp.)isaneigenvectorinA(VL)fortheleftmultiplicationof[ωb]ofeigenvalue116resp.)by(4.8).Multiplying(7.2)by[ωb]ontheleftandusingthefactthat[ωb]∗[Eα]=[Eα]∗[ωb]weobtain

1

tα2αt

[Eb]∗[E]+(2|α|−1)[E]∗[E]1b1.(7.3)

16

Combining(7.2)and(7.3)gives

󰀈

󰀉[E1tb]∗[Eα

]=−

1

++

5.2of[Y]andTheorem5.3of[ABD],VLisC2-cofiniteandthusA(VL)isfinite

+

dimensional.TheorbifoldtheoryconjecturesthatVLisarationalvertexoperator

+

algebra,whichimpliesA(VL)isafinitedimensionalsemisimpleassociativealgebra.

+

ButwecannotprovethesemisimplicityofA(VL)inthispaper.

Remark7.9.InthecasethatLisunimodular,VLhasauniqueirreduciblemodule

T+

VLandauniqueθ-twistedmoduleVLχ.SoVLhasexactly4inequivalentirreducible

T±

modules,VL,(VLχ)±.Inparticular,ifListheLeechlattice,thisgivesadifferent

+

proofoftheclassificationofirreducibleVL-modulespreviouslyobtainedin[D3].

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