Toroidal extended affine Lie algebras and vertex algebras
aa r X i v : . [ m a t h . QA ] F e b TOROIDAL EXTENDED AFFINE LIE ALGEBRAS ANDVERTEX ALGEBRAS
FULIN CHEN, HAISHENG LI, AND SHAOBIN TAN
Abstract.
In this paper, we study nullity-2 toroidal extended affine Lie alge-bras in the context of vertex algebras and their φ -coordinated modules. Amongthe main results, we introduce a variant of toroidal extended affine Lie algebras,associate vertex algebras to the variant Lie algebras, and establish a canoni-cal connection between modules for toroidal extended affine Lie algebras and φ -coordinated modules for these vertex algebras. Furthermore, by employingsome results of Billig, we obtain an explicit realization of irreducible modulesfor the variant Lie algebras. Introduction
Extended affine Lie algebras (of positive nullity) are a family of infinite-dimensionalLie algebras, which are natural generalizations of affine Kac-Moody Lie algebras.Of great importance in the study of these Lie algebras are those called toroidalextended affine Lie algebras. Let g be a finite-dimensional simple Lie algebra (over C ). Denote by R N the Laurent polynomial ring C [ t ± , t ± , . . . , t ± N ] (with N a posi-tive integer). By definition, the toroidal Lie algebra, denoted by t ( g ) in this paper,is a universal central extension of the multi-loop Lie algebra R N ⊗ g , and further-more the full toroidal Lie algebra is the semi-product extension of the toroidal Liealgebra by the (full) derivation Lie algebra D of the torus R N . While it is “per-fect” (for various reasons), the full toroidal Lie algebra is not an extended affineLie algebra in the precise sense of [BGK]. Replacing D with its subalgebra D div ,consisting of what were called divergence-zero derivations, one gets a Lie algebra,which in the precise sense is an extended affine Lie algebra, commonly called thetoroidal extended affine Lie algebra.In the study of this family of Lie algebras, one of the main focuses has beenthe classification of irreducible modules and their (explicit) vertex operator real-izations. The study on vertex operator realization of toroidal Lie algebras wasinitiated by Rao and Moody (see [EM], [MRY]), and since then extended affineLie algebras and their relations with vertex algebras have been extensively studied Mathematics Subject Classification.
Key words and phrases. extended affine Lie algebra, vertex algebra, φ -coordinated module. Partially supported by China NSF grant (No.11971397) and the Fundamental Research Fundsfor the Central Universities (Nos.20720190069, 20720200067). Partially supported by China NSF grants (No.11971397). in literature. The first precise connection with vertex algebras was obtained byBerman, Billig and Szmigielski (see [BB], [BBS]). Since then Billig has intensivelystudied representations of toroidal extended affine Lie algebras in the context ofvertex algebras and modules. In particular, Billig proved that full toroidal Liealgebras are vertex Lie algebras in the sense of [DLM1] and then associated vertexalgebras to these Lie algebras. He also gave a realization of irreducible modules forfull toroidal Lie algebras in terms of modules for certain concrete vertex operator al-gebras. Furthermore, Billig (see [B2]) studied modules for toroidal extended affineLie algebras by employing the natural embedding. A new phenomenon discoveredtherein is that toroidal extended affine Lie algebras are not vertex Lie algebras inthe precise sense (at least in the obvious way). Nevertheless, by making use of theaffinization of vertex algebras, Billig successfully obtained an explicit realizationof irreducible modules for toroidal extended affine Lie algebras in terms of mod-ules for certain vertex operator algebras. In the nullity-2 case, the construction ofBillig was later generalized in [CLT1], which leads to a classification of irreducibleintegrable modules for toroidal extended affine Lie algebras (see [CLT2]).In the present paper, on the basis of Billig’s work we explore natural connectionsbetween toroidal extended affine Lie algebras and vertex algebras with differentperspectives. We here concentrate ourselves to the nullity-2 case (with N = 1).An important gadget in this study is the theory of φ -coordinated modules forvertex algebras, which was developed in [Li2]. As a key ingredient, we introducea variation of the toroidal extended affine Lie algebras, which are shown to bevertex Lie algebras. Then we associate vertex algebras to the newly introduced Liealgebras naturally. Among the main results, we establish a canonical connectionbetween restricted modules for the nullity-2 toroidal extended affine Lie algebrasand φ -coordinated modules for the vertex algebras associated to the variants of thetoroidal extended affine Lie algebras. By using Billig’s results we obtain an explicitrealization of irreducible modules for the variant Lie algebras. Furthermore, byusing a result of Zhu (and Huang) and a result of Lepowsky we obtain an explicitrealization of irreducible modules for the toroidal extended affine Lie algebras,which recovers a result of [B2].To better explain the motivation and idea we continue this introduction withsome technical details. We start with a basic fact for vertex algebra modules. Let V be a vertex algebra and let ( W, Y W ) be a V -module. For u, v ∈ V , we have[ Y W ( u, z ) , Y W ( v, z )] = X j ≥ Y W ( u j v, z ) 1 j ! (cid:18) ∂∂z (cid:19) j z − δ (cid:18) z z (cid:19) (1.1)(the Borcherds commutator formula ). For the full toroidal Lie algebras (like othervertex Lie algebras such as untwisted affine Lie algebras and the Virasoro algebra),the commutator of two (suitably formulated) generating functions can be expressed
OROIDAL EALA AND VA 3 in the form: k X r =0 A r ( z ) 1 r ! (cid:18) ∂∂z (cid:19) r z − δ (cid:18) z z (cid:19) . It follows (from a conceptual result in [Li1]) that vertex algebras (and modules)can be associated to the full toroidal Lie algebras.For the toroidal extended affine Lie algebras, as it was noticed by Billig there isa discrepancy, which causes the nonexistence of a direct correspondence betweentoroidal extended affine Lie algebras and vertex algebras or their modules. In thispaper, as the first step we suitably modify the generating functions, so that thecommutator of two generating functions has the form: k X r =0 B r ( z ) 1 r ! (cid:18) z ∂∂z (cid:19) r δ (cid:18) z z (cid:19) . This still does not match the commutator formula for vertex algebras and modules.However, such commutator relations are in a full agreement with the commutatorformula in the theory of what were called φ -coordinated modules in [Li2] for vertexalgebras as we describe next.Note that a module ( W, Y W ) for a vertex algebra V can be defined by using theweak commutativity: For u, v ∈ V , there exists a nonnegative integer k such that( z − z ) k [ Y W ( u, z ) , Y W ( v, z )] = 0 , (1.2)a consequence of Borcherds’ commutator formula (1.1), together with the followingweak associativity z k Y W ( Y ( u, z ) v, z ) = (cid:0) ( x − x ) k Y W ( u, x ) Y ( v, x ) (cid:1) | x = x + z . (1.3)In the theory of φ -coordinated modules for vertex algebras, φ is what it wascalled an associate of the 1-dimensional additive formal group (law) F ( x, y ) = x + y ,which by definition is a formal series φ ( x, z ) ∈ C (( x ))[[ z ]], satisfying the condition φ ( x,
0) = x, φ ( φ ( x, z ) , z ) = φ ( x, z + z ) . It was proved (see [Li2]) that for every p ( x ) ∈ C (( x )), φ ( x, z ) = e zp ( x ) ddx ( x ) is anassociate of F ( x, y ) and every associate is of this form. As two particular cases,we have φ ( x, z ) = e z ddx ( x ) = x + z (the formal group itself) with p ( x ) = 1, and φ ( x, z ) = e zx ddx ( x ) = xe z with p ( x ) = x .For a vertex algebra V , a φ -coordinated V -module ( W, Y W ) can be defined byusing the weak commutativity (for u, v ∈ V , there exists a nonnegative integer k such that (1.2) holds) together with the following weak φ -associativity( φ ( x, z ) − x ) k Y W ( Y ( u, z ) v, x ) = (cid:0) ( x − x ) k Y W ( u, x ) Y ( v, x ) (cid:1) | x = φ ( x,z ) . (1.4) FULIN CHEN, HAISHENG LI, AND SHAOBIN TAN
Let φ ( x, z ) = xe z . If ( W, Y φW ) is a φ -coordinated module for a vertex algebra V ,then for u, v ∈ V , we have[ Y φW ( u, z ) , Y φW ( v, z )] = X j ≥ Y φW ( u j v, z ) 1 j ! (cid:18) z ∂∂z (cid:19) j δ (cid:18) z z (cid:19) . (1.5)This indicates that modules for the toroidal extended affine Lie algebra, denotedby b t ( g , µ ) in this paper, may be φ -coordinated modules for some vertex algebra.It is this fact that motivated us to introduce the variant Lie algebra b t ( g , µ ) o .Assume that V is a vertex operator algebra in the sense of [FLM] and [FHL].By using a result of Zhu (and Huang) and a result of Lepowsky we show (cf. [Li3])that there is a canonical category isomorphism between the module category andthe φ -coordinated category. Then using this isomorphism we obtain a realizationof irreducible b t ( g , µ )-modules of nonzero levels in terms of modules for certainconcrete vertex algebras, recovering a result of Billig.This paper is organized as follows: In Section 2, we review the full toroidal Liealgebra and the toroidal extended affine Lie algebra. In Section 3, we introducea variation b t ( g , µ ) o of the toroidal extended affine Lie algebra b t ( g , µ ). In Section4, we study vertex algebras V b t ( g ,µ ) o ( ℓ ), V int b t ( g ,µ ) o ( ℓ ), and their φ -coordinated modules.In Section 5, we give a realization of irreducible b t ( g , µ ) o -modules of nonzero levelsusing irreducible modules for certain concrete vertex algebras and determine allbounded irreducible b t ( g , µ ) o -modules. In Section 6, using certain results in ver-tex operator algebra theory we give a realization of irreducible b t ( g , µ )-modules ofnonzero levels in terms of modules for certain concrete vertex algebras.The following is a list of notations we frequently use in this paper: N : The set of nonnegative integers Z + : The set of positive integers Z × : The set of nonzero integers g : A finite-dimensional simple Lie algebra h : A Cartan subalgebra of g ∆ : The root system of ( g , h ) R : The Laurent polynomial ring C [ t ± , t ± ] K : A quotient space of K¨ahler differentials of C [ t ± , t ± ] D : The derivation Lie algebra of C [ t ± , t ± ] t ( g ) : The toroidal Lie algebra R ⊗ g + KT ( g , µ ) : The full (rank 2) toroidal Lie algebra OROIDAL EALA AND VA 5 e t ( g , µ ) : The (rank 2) toroidal EALA (Lie algebra) b t ( g , µ ) : A subalgebra of e t ( g , µ ) D div : The Lie algebra of divergence-zero derivations of C [ t ± , t ± ] D div ′ : A Lie subalgebra of D e t ( g , µ ) o : A subalgebra of the full toroidal Lie algebra T ( g , µ ) b t ( g , µ ) o : A Lie subalgebra of e t ( g , µ ) o . Toroidal extended affine Lie algebras
In this section, we briefly review the nullity-2 full toroidal Lie algebras andtoroidal extended affine Lie algebras.2.1.
Full toroidal Lie algebras.
We here follow [B1] and [BB] to present thenullity-2 full toroidal Lie algebras. Set R = C [ t , t − , t , t − ] , (2.1)the ring of Laurent polynomials in (commuting) variables t and t . LetΩ R = R dt ⊕ R dt be the space of 1 forms on R , which is a free R -module. Setk = t − d t , k = t − d t , (2.2)which also form a basis of Ω R over R . For f ∈ R , define d ( f ) = ∂f∂t dt + ∂f∂t dt = t ∂f∂t k + t ∂f∂t k ∈ Ω R . Set d ( R ) = { d ( f ) | f ∈ R} and form the vector space K = Ω R /d ( R ) . (2.3)The following relations hold in K for m , m ∈ Z : m t m t m k + m t m t m k = 0 . (2.4)For m , m ∈ Z , set(2.5) k m ,m = m t m t m k if m = 0 , − m t m k if m = 0 , m = 0 , m = m = 0 . Then the set B K := { k , k } ∪ { k m ,m | m , m ∈ Z with ( m , m ) = (0 , } (2.6)is a basis of K . We can also present B K slightly differently as B K = { k } ∪ { t m k , k m ,m | m ∈ Z , m ∈ Z × } . (2.7) FULIN CHEN, HAISHENG LI, AND SHAOBIN TAN
We have the following straightforward fact:
Lemma 2.1.
The following relations hold in K : at m t n k + bt m t n k = ( an − bm )k m,n + δ m, δ n, ( a k + b k )(2.8) for a, b ∈ C , m, n ∈ Z . In particular, we have t m k = − m k m, + δ m, k for m ∈ Z . On the other hand, set D = Der( R ) , (2.9)the derivation Lie algebra of R . Especially, setd = t ∂∂t , d = t ∂∂t . (2.10)Then D is a free R -module with basis { d , d } . The space K is a D -module with ψ ( f d g ) = ψ ( f )d g + f d ψ ( g ) for ψ ∈ D , f, g ∈ R . (2.11)Let g be any Lie algebra equipped with a non-degenerate symmetric invariantbilinear form h· , ·i . Form a central extension of the double loop algebra R ⊗ gt ( g ) = ( R ⊗ g ) ⊕ K , called the toroidal Lie algebra , where K is central and[ t m t m ⊗ u, t n t n ⊗ v ]= t m + n t m + n ⊗ [ u, v ] + h u, v i ( m t m + n t m + n k + m t m + n t m + n k )(2.12)for u, v ∈ g , m , n , m , n ∈ Z . When g is a finite-dimensional simple Lie algebra,the toroidal Lie algebra t ( g ) is a universal central extension (see [MRY]), which iscommonly called the nullity- toroidal Lie algebra .Lie algebra D acts on the Lie algebra R ⊗ g as a derivation Lie algebra by ψ ( a ⊗ u ) = ψ ( a ) ⊗ u for ψ ∈ D , a ∈ R , u ∈ g . Furthermore, D acts on the toroidal Lie algebra t ( g ) as a derivation Lie algebra.In particular, (considering the semi-product Lie algebra R ⊗ g ⋊ D ) we have[ t m t m d i , t n t n ⊗ u ] = n i ( t m + m t n + n ⊗ u )(2.13)for u ∈ g , m , n , m , n ∈ Z , i ∈ { , } , and[ t m t m d i , t n t n k j ] = n i t m + n t m + n k j + δ i,j X r =0 , m r t m + n t m + n k r (2.14)in K ⋊ D for m , n , m , n ∈ Z , i, j ∈ { , } .The following notion can be found in [B2]: OROIDAL EALA AND VA 7
Definition 2.2.
Let µ be a complex number, which is fixed throughout this paper.The nullity-2 full toroidal Lie algebra is the Lie algebra T ( g , µ ) := t ( g ) ⊕ D = ( R ⊗ g ) ⊕ K ⊕ D , (2.15)where in addition to the relations (2.12), (2.13) and (2.14), the following relationsare postulated:[ t m t m d i , t n t n d j ] = n i t m + n t m + n d j − m j t m + n t m + n d i − µm j n i ( m t m + n t m + n k + m t m + n t m + n k )(2.16)for m , n , m , n ∈ Z , i, j ∈ { , } . Remark 2.3.
Note that with (2.12)-(2.16) it can be readily seen that for any Liesubalgebra D of D , the subspace R ⊗ g + K + D of T ( g , µ ) is a Lie subalgebra. Definition 2.4.
We fix the particular Z -grading on T ( g , µ ) given by the adjointaction of − d : T ( g , µ ) = M m ∈ Z T ( g , µ ) ( m ) , (2.17)where T ( g , µ ) ( m ) = { x ∈ T ( g , µ ) | [d , x ] = − mx } . Set e g = C [ t , t − ] ⊗ g + C k + C d , (2.18) e g = C [ t , t − ] ⊗ g + C k + C d , (2.19)both of which are subalgebras of T ( g , µ ), canonically isomorphic to the affine Liealgebra e g = C [ t, t − ] ⊗ g + C k + C d. As a convention, for X ∈ e g (resp. e g ), wewrite t m X (resp. t m X ) for the corresponding elements of T ( g , µ ).For convenience, we formulate the following two technical lemmas, which followstraightforwardly from (2.14) and (2.16) together with Lemma 2.1: Lemma 2.5.
The following relations hold in T ( g , µ ) : [ at m t m d + bt m t m d , k n ,n ](2.20) = ( a ( m + n ) + b ( m + n ))k m + n ,m + n + δ m + n , δ m + n , ( b k − a k ) for a, b ∈ C , m , m , n , n ∈ Z . Lemma 2.6.
The following relations hold in T ( g , µ ) : [ a t m t m d + a t m t m d , b t n t n d + b t n t n d ](2.21) = ( b ( a n + a n ) − a ( b m + b m )) t m + n t m + n d + ( b ( a n + a n ) − a ( b m + b m )) t m + n t m + n d − µ ( a n + a n )( b m + b m ) · ( m n − m n )k m + n ,m + n − µ ( a n + a n )( b m + b m ) · δ m + n , δ m + n , ( m k + m k ) for a , a , b , b ∈ C , m , m , n , n ∈ Z . FULIN CHEN, HAISHENG LI, AND SHAOBIN TAN
Toroidal extended affine Lie algebra e t ( g , µ ) . Here, we follow [BGK, B2]to present the toroidal extended affine Lie algebra. Set D div = { f d + f d | f , f ∈ R , d ( f ) + d ( f ) = 0 } ⊂ D , (2.22)the Lie algebra of divergence-zero derivations, or namely skew derivations on R .For m , m ∈ Z , set(2.23) ˜d m ,m = m t m t m d − m t m t m d , which is an element of D div . A simple fact is that the set B := { d , d } ∪ { ˜d m ,m | ( m , m ) ∈ ( Z × Z ) \{ (0 , }} (2.24)is a C -basis of D div . With the relation˜d m, = mt m d for m ∈ Z , (2.25)we can also present B as (cf. (2.7)) B = { d } ∪ { t m d , ˜d m ,m | m ∈ Z , m ∈ Z × } . (2.26)Record the following relations in D div :[d i , ˜d m ,m ] = m i ˜d m ,m , (2.27) [˜d m ,m , ˜d n ,n ] = ( m n − m n )˜d m + n ,m + n (2.28)for i ∈ { , } , m , m , n , n ∈ Z . Remark 2.7.
It follows from the Lie bracket relations (2.27) and (2.28) that D div is indeed a subalgebra of D . Furthermore, we have D (1)div := [ D div , D div ] = Span { ˜d m,n | m, n ∈ Z } (2.29)and D div contains the following subalgebras: D (1)div , D (1)div ⋊ C d , D (1)div ⋊ C d , D (1)div ⋊ ( C d + C d ) = D div . (2.30) Definition 2.8.
Set e t ( g , µ ) = ( R ⊗ g ) + K + D div , (2.31) b t ( g , µ ) = ( R ⊗ g ) + K + D (1)div ⋊ C d , (2.32)both of which are Z -graded subalgebras of T ( g , µ ) (recall Remark 2.3). Remark 2.9.
Assume that g is a finite-dimensional simple Lie algebra. A specialfeature of e t ( g , µ ) is that it admits a suitable non-degenerate symmetric invariantbilinear form, whereas T ( g , µ ) does not. This makes e t ( g , µ ) an extended affine Liealgebra in the sense of [BGK], which is commonly known as the toroidal extendedaffine Lie algebra of nullity . OROIDAL EALA AND VA 9
Using Lemmas 2.1, 2.5 and 2.6 we straightforwardly have:[ t m t m ⊗ u, t n t n ⊗ v ] = t m + n t m + n ⊗ [ u, v ](2.33) + h u, v i (( m n − m n )k m + n ,m + n + δ m + n , δ m + n , ( m k + m k )) , [˜d i,m , t j t n ⊗ u ] = ( ni − mj ) t i + j t m + n ⊗ u, [˜d i,m , t j k ] = mj k i + j,m , (2.34) [ t i d , t j t n ⊗ u ] = nt i + j t n ⊗ u, [ t i d , k j,n ] = n k i + j,n , (2.35) [˜d i,m , k j,n ] = ( in − mj )k i + j,m + n + δ m + n, δ i + j, ( i k + m k ) , (2.36) [ t i d , t j k ] = iδ i + j, k , [ t i d , t j d ] = 0 , (2.37) [˜d i,m , ˜d j,n ] = ( in − mj )˜d i + j,m + n + µ ( in − mj ) k i + j,m + n , (2.38) [ t i d , ˜d j,n ] = n ˜d i + j,n + µn i k i + j,n (2.39)for u, v ∈ g , i, j, m, n ∈ Z .Introduce a vector space KD with a designated basis { K n , D n | n ∈ Z × } . Thenform a vector space A g = e g ⊕ KD = C [ t ± ] ⊗ g ⊕ C k ⊕ C d ⊕ X n ∈ Z × ( C K n ⊕ C D n ) . (2.40)Form generating functions in b t ( g , µ )[[ z, z − ]]: u [ z ] = X n ∈ Z ( t n u ) z − n , K r [ z ] = X n ∈ Z k n,r z − n , D m [ z ] = X n ∈ Z ˜d n,m z − n , (2.41)where u ∈ e g , r ∈ Z × , m ∈ Z . Note that the coefficients of a [ z ] for a ∈ A g linearlyspan b t ( g , µ ). In addition, setK [ z ] = X n ∈ Z k n, z − n + k (log z ) . (2.42)(Recall that k , = 0.) Note thatd [ z ] = X n ∈ Z ( t n d ) z − n , k [ z ] = X n ∈ Z ( t n k ) z − n . Then z ddz K [ z ] = k [ z ] , z ddz d [ z ] = − D [ z ] . (2.43)Now, we write the relations (2.33)-(2.39) in terms of the generating functions. Proposition 2.10.
For u, v ∈ g , m, n ∈ Z , we have (1) [( t m ⊗ u )[ z ] , ( t n ⊗ v )[ w ]]= ( t m + n ⊗ [ u, v ])[ w ] δ (cid:16) wz (cid:17) + h u, v i m (cid:18) w ∂∂w K m + n [ w ] (cid:19) δ (cid:16) wz (cid:17) + h u, v i ( m + n )K m + n [ w ] w ∂∂w δ (cid:16) wz (cid:17) + δ m + n, h u, v i w ∂∂w δ (cid:16) wz (cid:17) k , (2) [K m [ z ] , ( t n ⊗ u )[ w ]] = 0 , (3) [D m [ z ] , ( t n ⊗ u )[ w ]]= m (cid:18) w ∂∂w ( t m + n ⊗ u )[ w ] (cid:19) δ (cid:16) wz (cid:17) + ( m + n )( t m + n ⊗ u )[ w ] (cid:18) w ∂∂w (cid:19) δ (cid:16) wz (cid:17) , (4) [d [ z ] , ( t n ⊗ u )[ w ]] = n ( t n ⊗ u )[ w ] δ (cid:16) wz (cid:17) , (5) [K m [ z ] , K n [ w ]] = 0 , (6) [D m [ z ] , K n [ w ]] = m (cid:18) w ∂∂w K m + n [ w ] (cid:19) δ (cid:16) wz (cid:17) + ( m + n )K m + n [ w ] w ∂∂w δ (cid:16) wz (cid:17) + δ m + n, w ∂∂w δ (cid:16) wz (cid:17) k , (8) [d [ z ] , K n [ w ]] = n K n [ w ] δ (cid:16) wz (cid:17) + δ n, δ (cid:16) wz (cid:17) k , (9) [d [ z ] , k [ w ]] = w ∂∂w δ (cid:16) wz (cid:17) k , [d [ z ] , d [ w ]] = 0 , (10) [D m [ z ] , D n [ w ]]= m (cid:18) w ∂∂w D m + n [ w ] (cid:19) δ (cid:16) wz (cid:17) + ( m + n )D m + n [ w ] w ∂∂w δ (cid:16) wz (cid:17) + µ X r =0 (cid:18) r (cid:19) (cid:18)(cid:18) mw ∂∂w (cid:19) r K m + n [ w ] (cid:19) (cid:18) ( m + n ) w ∂∂w (cid:19) − r δ (cid:16) wz (cid:17) , (11) [d [ z ] , D n [ w ]] = n D n [ w ] δ (cid:16) wz (cid:17) + µn K n [ w ] (cid:18) w ∂∂w (cid:19) δ (cid:16) wz (cid:17) . Proof.
We here prove relation (10) while the others easier can be proved similarly.Using (2.38) and the identity in − mj = i ( m + n ) + m ( − i − j ), we get[D m [ z ] , D n [ w ]] = X i,j ∈ Z [˜d i,m z − i , ˜d j,n w − j ]= X i,j ∈ Z (cid:16) ( m + n )˜d i + j,m + n w − i − j ( iw i z − i ) + m ˜d i + j,m + n ( − i − j ) w − i − j ( w i z − i ) (cid:17) + µ X i,j ∈ Z X r =0 (cid:18) r (cid:19) m r ( − i − j ) r ( m + n ) − r i − r k i + j,m + n w − i − j ( w i z − i )= m (cid:18) w ∂∂w D m + n [ w ] (cid:19) δ (cid:16) wz (cid:17) + ( m + n )D m + n [ w ] (cid:18) w ∂∂w δ (cid:16) wz (cid:17)(cid:19) OROIDAL EALA AND VA 11 + µ X r =0 (cid:18) r (cid:19) (cid:18)(cid:18) mw ∂∂w (cid:19) r K m + n [ w ] (cid:19) (cid:18) ( m + n ) w ∂∂w (cid:19) − r δ (cid:16) wz (cid:17) , proving (10). (cid:3) Lie algebra b t ( g , µ ) o and vertex algebras V b t ( g ,µ ) o ( ℓ )In this section, as the key ingredients we introduce a particular subalgebra b t ( g , µ ) o of the full toroidal Lie algebra T ( g , µ ) and associate vertex algebras V b t ( g ,µ ) o ( ℓ )to this very Lie algebra b t ( g , µ ) o .3.1. Lie algebras b t ( g , µ ) o and e t ( g , µ ) o . First, consider a variant of D div . Insteadof using the free basis { d , d } of the R -module D , we now use free basis { t − d , d } .Set D div ′ = (cid:26) f ∂∂t + f d | f , f ∈ R , ∂∂t ( f ) + d ( f ) = 0 (cid:27) . (3.1)It is straightforward to show that D div ′ is also a Lie subalgebra of D . Note that t − d , t − d , d ∈ D div ′ , d / ∈ D div ′ . For m, n ∈ Z , set ¯d n,m = ( n + 1) t n t m d − mt n t m d , (3.2)which lies in D div ′ . Notice that ¯d − , = 0. It can be readily seen that the set B ′ := { t − d , t − d } ∪ { ¯d n,m | ( n, m ) ∈ ( Z × Z ) \{ ( − , }} (3.3)is a C -basis of D div ′ . By Lemma 2.6, the following relations hold in T ( g , µ ):[¯d m ,m , ¯d n ,n ] = (( m + 1) n − m ( n + 1))¯d m + n ,m + n (3.4) + µ ( m n − n ( m + 1))( m ( n + 1) − m n )( m n − m n )k m + n ,m + n + µm δ m + n , δ m + n , ( m k + m k )for m , m , n , n ∈ Z . In particular, the following relations in D div ′ (withoutcentral extension) hold:[ t − d , ¯d m ,m ] = ( m + 1)¯d m − ,m , [ t − d , ¯d m ,m ] = m ¯d m − ,m , (3.5) [¯d m ,m , ¯d n ,n ] = (( m + 1) n − m ( n + 1))¯d m + n ,m + n (3.6)for m , m , n , n ∈ Z . Remark 3.1.
Note that from (3.5) and (3.6), we see that D div ′ is indeed a subal-gebra of D . Furthermore, we have D (1)div ′ := [ D div ′ , D div ′ ] = Span { ¯d m,n | m, n ∈ Z } (3.7) and D div ′ contains the following subalgebras: D (1)div ′ , D (1)div ′ ⋊ C t − d , D (1)div ′ ⋊ C t − d , D (1)div ′ ⋊ ( C t − d + C t − d ) = D div ′ . Now, using Lie subalgebras D div ′ and D (1)div ′ ⋊ C t − d of D , we get two Z -gradedsubalgebras of the full toroidal Lie algebra T ( g , µ ) (recall Remark 2.3). Definition 3.2.
Define e t ( g , µ ) o = t ( g ) + D div ′ = R ⊗ g + K + C ( t − d ) + C ( t − d ) + X ( m,n ) ∈ Z × Z C ¯d m,n , b t ( g , µ ) o = t ( g ) + D (1)div ′ ⋊ C t − d = R ⊗ g + K + C ( t − d ) + X ( m,n ) ∈ Z × Z C ¯d m,n . For n, m ∈ Z , setd n,m = ¯d n,m + µm ( n + 1 / t n t m k = ¯d n,m + µ ( n + 1 / m k n,m ∈ b t ( g , µ ) o (3.8)(cf. [B1]). In particular, for n, m ∈ Z we haved n, = ¯d n, = ( n + 1) t n d , (3.9) d ,m = ¯d ,m + 12 µmt m k = t m d − mt m d + 12 µmt m k . (3.10)Especially, we have d − , = 0 , d , = d . (3.11)Form generating functions in b t ( g , µ ) o [[ z, z − ]][log z ]: u ( z ) = X n ∈ Z ( t n u ) z − n − , D m ( z ) = X n ∈ Z d n,m z − n − , (3.12) K m ( z ) = K m [ z ] = X n ∈ Z k n,m z − n + δ m, (log z )k (3.13)for u ∈ e g , m ∈ Z . (Recall that the square bracket notation such as u [ z ] is usedfor generating functions of b t ( g , µ ).) It is clear that the coefficients of all generatingfunctions linearly span b t ( g , µ ) o . Noticing thatk ( z ) = X n ∈ Z ( t n k ) z − n − , d ( z ) = X n ∈ Z ( t n d ) z − n − (with k , d ∈ e g ), we have ddz K ( z ) = k ( z ) , ddz d ( z ) = − D ( z ) . (3.14)In addition to (2.33), (2.35), and (2.37), the following relations hold in b t ( g , µ ) o :[d i,m , t j t n ⊗ u ] = (( i + 1) n − mj ) t i + j t m + n ⊗ u, (3.15) [d i,m , k j,n ](3.16) OROIDAL EALA AND VA 13 = (( i + 1)( m + n ) − m ( i + j ))k i + j,m + n + δ m + n, δ i + j, (( i + 1)k + m k ) , [d i,m , t j k ] = mj ( j − i + j,m , (3.17) [d i,m , d j,n ] = (( i + 1) n − ( j + 1) m )d i + j,m + n + 2 µm δ i + j, δ m + n, k (3.18) + µ X r =0 (cid:18) r (cid:19) n − r ( i + 1) (3 − r ) ( − m ) r ( j + 1) ( r ) k i + j,m + n , [ t i d , d j,n ] = n d i + j,n + µn i ( i − i + j,n (3.19)for u ∈ g , i, j, m, n ∈ Z , where for a ∈ C , r ∈ N , a ( r ) := a ( a − · · · ( a − r + 1) = r ! (cid:18) ar (cid:19) . (3.20)All the relations except (3.18) follow easily from Lemmas 2.1, 2.5 and 2.6. Inthe following, we prove (3.18). Using (3.4) we have[¯d i,m , ¯d j,n ] = [( i + 1) t i t m d − mt i t m d , ( j + 1) t j t n d − nt j t n d ]= (( i + 1) n − ( j + 1) m )¯d i + j,m + n + µ (( i + 1) n − jm )( in − ( j + 1) m ) (cid:0) ( in − jm )k i + j,m + n + δ i + j, δ m + n, ( i k + m k ) (cid:1) = (( i + 1) n − ( j + 1) m )¯d i + j,m + n + µ (( i + 1) n − jm )( in − ( j + 1) m )( in − jm )k i + j,m + n + µm δ i + j, δ m + n, ( i k + m k )= (( i + 1) n − ( j + 1) m ) (cid:18) d i + j,m + n − µ ( m + n ) ( i + j + 12 )k i + j,m + n (cid:19) + µ (( i + 1) n − jm )( in − ( j + 1) m )( in − jm )k i + j,m + n + µm δ i + j, δ m + n, ( i k + m k )= (( i + 1) n − ( j + 1) m )d i + j,m + n + µC i,jm,n k i + j,m + n + µm δ i + j, δ m + n, ( i k + m k ) , where C i,jm,n = n ( i + 1)( i − i − j −
12 ) − n m (cid:0) i + 1) i ( j + 1) − ( j + 12 )( j − (cid:1) + m n (cid:0) i + 1) j ( j + 1) − ( i + 12 )( i − (cid:1) − m ( j + 1)( j − j − i −
12 ) . On the other hand, using (3.16) we get[¯d i,m , µn ( j + 12 )k j,n ] − [¯d j,n , µm ( i + 12 )k i,m ]= µ (cid:18) (( i + 1) n − ( j − m ) n ( j + 12 ) − (( j + 1) m − ( i − n ) m ( i + 12 ) (cid:19) k i + j,m + n + µδ i + j, δ m + n, (cid:18) n ( j + 12 )(( i + 1)k + m k ) − m ( i + 12 )(( j + 1)k + n k ) (cid:19) = µ (cid:18) (( i + 1) n − ( j − m ) n ( j + 12 ) − (( j + 1) m − ( i − n ) m ( i + 12 ) (cid:19) k i + j,m + n + µm δ i + j, δ m + n, ( − i k + m k ) . Combining the two relations above we obtain (3.18).Rewriting these relations in terms of generating functions, we have:
Proposition 3.3.
For u, v ∈ g , m, n ∈ Z , we have (1) [( t m ⊗ u )( z ) , ( t n ⊗ v )( w )]= ( t m + n ⊗ [ u, v ])( w ) z − δ (cid:16) wz (cid:17) + h u, v i m (cid:18) ∂∂w K m + n ( w ) (cid:19) z − δ (cid:16) wz (cid:17) + ( m + n ) h u, v i K m + n ( w ) ∂∂w z − δ (cid:16) wz (cid:17) + δ m + n, h u, v i ∂∂w z − δ (cid:16) wz (cid:17) k , (2) [K m ( z ) , ( t n ⊗ u )( w )] = 0 , (3) [D m ( z ) , ( t n ⊗ u )( w )] = m (cid:18) ∂∂w ( t m + n ⊗ u )( w ) (cid:19) z − δ (cid:16) wz (cid:17) + ( m + n )( t m + n ⊗ u )( w ) ∂∂w z − δ (cid:16) wz (cid:17) , (4) [d ( z ) , ( t n ⊗ u )( w )] = n ( t n ⊗ u )( w ) z − δ (cid:16) wz (cid:17) , (5) [K m ( z ) , K n ( w )] = 0 , (6) [D m ( z ) , K n ( w )] = m (cid:18) ∂∂w K m + n ( w ) (cid:19) z − δ (cid:16) wz (cid:17) + ( m + n )K m + n ( w ) ∂∂w z − δ (cid:16) wz (cid:17) + δ m + n, ∂∂w z − δ (cid:16) wz (cid:17) k , (8) [d ( z ) , K n ( w )] = n K n ( w ) z − δ (cid:16) wz (cid:17) + δ n, z − δ (cid:16) wz (cid:17) k , (9) [d ( z ) , k ( w )] = ∂∂w z − δ (cid:16) wz (cid:17) k , [d ( z ) , d ( w )] = 0 , (10) [D m ( z ) , D n ( w )]= m (cid:18) ∂∂w D m + n ( w ) (cid:19) z − δ (cid:16) wz (cid:17) + ( m + n )D m + n ( w ) ∂∂w z − δ (cid:16) wz (cid:17) + µ X i =0 (cid:18) i (cid:19) (cid:18) m ∂∂w (cid:19) i K m + n ( w ) ! (cid:18) ( m + n ) ∂∂w (cid:19) − i z − δ (cid:16) wz (cid:17) , (11) [d ( z ) , D n ( w )] = n D n ( w ) z − δ (cid:16) wz (cid:17) + µn K n ( w ) (cid:18) ∂∂w (cid:19) z − δ (cid:16) wz (cid:17) . OROIDAL EALA AND VA 15
Proof.
Relations (1)-(9) and (11) can be proved straightforwardly by using thesimple identity an − bm = a ( m + n ) + m ( a + b ) for a, b ∈ Z and the fact ddw K ( w ) = X l ∈ Z − l k l, w − l − + k w − , (cid:18) ddw (cid:19) K ( w ) = X l ∈ Z ( − l ) (3) k l, w − l − + 2k w − . As for relation (10), using the technical result in Lemma 3.4 we have X r =0 (cid:18) r (cid:19) n − r ( i + 1) (3 − r ) ( − m ) r ( j + 1) ( r ) = X s =0 (cid:18) s (cid:19) ( m + n ) − s ( i + 1) (3 − s ) m s ( − i − j ) ( s ) . Then using this relation and (3.18), we obtain[D m ( z ) , D n ( w )] = X i,j ∈ Z [d i,m , d j,n ] z − i − w − j − ! = X i,j ∈ Z (cid:0) ( i + 1) n − ( j + 1) m )d i + j,m + n z − i − w − j − + µ X i,j ∈ Z X r =0 (cid:18) r (cid:19) n − r ( i + 1) (3 − r ) ( − m ) r ( j + 1) ( r ) k i + j,m + n z − i − w − j − + 2 µm δ m + n, X i,j ∈ Z δ i + j, k z − i − w − j − = m X i,j ∈ Z (cid:0) ( − i − j − i + j,m + n w − i − j − (cid:1) z − i − w i +1 + ( m + n ) X i,j ∈ Z d i + j,m + n w − i − j − (cid:0) ( i + 1) z − i − w i (cid:1) + µ X i,j ∈ Z X s =0 (cid:18) s (cid:19) ( m + n ) − s ( i + 1) (3 − s ) m s ( − i − j ) ( s ) (k i + j,m + n w − i − j − s )( z − i − w i − s )+ 2 µm δ m + n, X i ∈ Z (k w − )( z − i − w i +1 )= m (cid:18) ∂∂w D m + n ( w ) (cid:19) z − δ (cid:16) wz (cid:17) + ( m + n )D m + n ( w ) (cid:18) ∂∂w z − δ (cid:16) wz (cid:17)(cid:19) + µ X r =0 (cid:18) r (cid:19) (cid:18)(cid:18) m ∂∂w (cid:19) r K m + n ( w ) (cid:19) (cid:18) ( m + n ) ∂∂w (cid:19) − r z − δ (cid:16) wz (cid:17)! , proving relation (10). (cid:3) The following is the technical result we used in the proof above:
Lemma 3.4.
Let p ∈ N , a, b, α, β ∈ C . Then p X r =0 (cid:18) pr (cid:19) α p − r a ( p − r ) ( − β ) r b ( r ) (3.21) = p X t =0 (cid:18) pt (cid:19) ( α + β ) p − t a ( p − t ) β t ( − a − b − p ) ( t ) . Proof.
Note that we have the following version of the Newton identity( a + b ) ( q ) = q X i =0 (cid:18) qi (cid:19) a ( i ) b ( q − i ) . (3.22)Using this we get p X r =0 (cid:18) pr (cid:19) α p − r a ( p − r ) ( − β ) r b ( r ) = p X r =0 p − r X s =0 (cid:18) p − rs (cid:19)(cid:18) pr (cid:19) ( α + β ) p − r − s a ( p − r ) ( − β ) r + s b ( r ) = p X r =0 p − r X s =0 (cid:18) pr + s (cid:19)(cid:18) r + sr (cid:19) ( α + β ) p − r − s a ( p − r ) ( − β ) r + s b ( r ) = p X t =0 t X r =0 (cid:18) pt (cid:19)(cid:18) tr (cid:19) ( α + β ) p − t a ( p − r ) ( − β ) t b ( r ) = p X t =0 t X r =0 (cid:18) pt (cid:19)(cid:18) tr (cid:19) ( α + β ) p − t a ( p − t ) ( a − p + t ) ( t − r ) ( − β ) t b ( r ) = p X t =0 (cid:18) pt (cid:19) ( α + β ) p − t a ( p − t ) ( − β ) t ( a + b − p + t ) ( t ) = p X t =0 (cid:18) pt (cid:19) ( α + β ) p − t a ( p − t ) β t ( − a − b − p ) ( t ) , noticing that a ( p − r ) = a ( p − t ) ( a − p + t ) ( t − r ) with 0 ≤ r ≤ t ≤ p . (cid:3) OROIDAL EALA AND VA 17
Vertex Lie algebras and vertex algebras.
We here recall the notion ofvertex Lie algebra from [DLM2] (see also [B1, K2, P]) and the vertex algebrasassociated to a vertex Lie algebra.Let ∂ be an indeterminate. For a vector space U , denote by C [ ∂ ] U the free C [ ∂ ]-module over U , i.e., C [ ∂ ] U = C [ ∂ ] ⊗ U . A vertex Lie algebra is a Lie algebra L equipped with vector spaces A and C , and a linear bijection ρ : A ⊗ C [ t, t − ] ⊕ C → L a ⊗ t n + c a ( n ) + c (for a ∈ A , n ∈ Z , c ∈ C ) , (3.23)satisfying the condition that C is central in L and for a, b ∈ A , there exist finitelymany elements f i ( a, b ) ∈ C [ ∂ ] A + C for i = 0 , , . . . , r such that[ a ( z ) , b ( w )] = r X i =0 f i ( a, b )( w ) 1 i ! (cid:18) ∂∂w (cid:19) i z − δ (cid:16) wz (cid:17) , (3.24)where c ( z ) = c ∈ L for c ∈ C , and for a ∈ A , i ∈ N ,( ∂ i a )( z ) = (cid:18) ddz (cid:19) i X n ∈ Z a ( n ) z − n − ∈ L [[ z, z − ]] . (3.25)Furthermore, L is called a Z -graded vertex Lie algebra if L = ⊕ n ∈ Z L ( n ) is a Z -graded Lie algebra with A = ⊕ n ∈ Z A ( n ) a Z -graded vector space such thatdeg C = 0 and deg a ( n ) = m − n − a ∈ A ( m ) , m, n ∈ Z . Let ( L , A , C , ρ ) be a vertex Lie algebra. Set L − = Span { a ( n ) | a ∈ A , n < } , L + = Span { a ( n ) | a ∈ A , n ≥ } , which are subalgebras of L . Then we have a triangular decomposition of L : L = L + ⊕ C ⊕ L − . (3.26)Let γ ∈ C ∗ . View C as an ( L + + C )-module with L + acting trivially and with c acting as scalar γ ( c ) for c ∈ C . Then form an induced L -module V L ( γ ) := U ( L ) ⊗ U ( L + + C ) C . (3.27)Set = 1 ⊗ ∈ V L ( γ ). View A as a subspace of V L ( γ ) through the linear map a a ( − .The Lie algebra L admits a derivation d determined by d ( c ) = 0 , d ( a ( n )) = − na ( n −
1) for c ∈ C , a ∈ A , n ∈ Z . (3.28)View d as a derivation of U ( L ). As d preserves the subalgebra L + + C , d acts on V L ( γ ) with d = 0 and V L ( γ ) becomes an L ⋊ C d -module.A Z -graded vertex algebra is a vertex algebra V with a Z -grading V = ⊕ n ∈ Z V ( n ) such that ∈ V (0) and u m V ( n ) ⊂ V ( n + k − m − for u ∈ V ( k ) , m, n, k ∈ Z . (3.29) The following result can be found in [DLM2]:
Proposition 3.5.
Let ( L , A , C , ρ ) be a vertex Lie algebra and let γ be a linearfunctional on C . Then there exists a vertex algebra structure on V L ( γ ) , which isuniquely determined by the condition that is the vacuum vector and Y ( a, z ) = a ( z ) for a ∈ A . (3.30) Furthermore, if L is Z -graded, then V L ( γ ) is a Z -graded vertex algebra with deg ( a ( n ) · · · a s ( n s ) ) = deg a ( n ) + · · · + deg a s ( n s )(3.31) for a i ∈ A , n i ∈ Z . Let π : C [ ∂ ] A ⊕ C → V L ( γ ) be the linear map defined by π ( c ) = γ ( c ) , π ( ∂ i a ) = L ( − i a for c ∈ C , i ∈ N , a ∈ A , (3.32)where L ( −
1) denotes the canonical derivation of the vertex algebra V L ( γ ), definedby L ( − v = v − for v ∈ V L ( γ ). Then Y ( π ( X ) , z ) = X ( z ) for X ∈ C [ ∂ ] A ⊕ C . (3.33)Note that d on V L ( γ ) coincides with L ( − d , Y ( v, z )] = Y ( d ( v ) , z ) = ddz Y ( v, z ) for v ∈ V L ( γ ) . (3.34)From [LL], we immediately have: Lemma 3.6.
An ideal of the vertex algebra V L ( γ ) is the same as an L ⋊ C d -submodule of V L ( γ ) . For a vector space W , set E ( W ) = Hom( W, W (( z ))) . An L -module W is said to be restricted if for any a ∈ A , a ( z ) ∈ E ( W ), and oflevel γ ∈ C ∗ if c · w = γ ( c ) w for c ∈ C , w ∈ W . Then we have (see [DLM2]): Lemma 3.7.
Let W be a vector space and let γ ∈ C ∗ . Then a restricted L -modulestructure of level γ on W amounts to a V L ( γ ) -module structure Y W ( · , z ) on W suchthat Y W ( a, z ) = a ( z ) for a ∈ A . Vertex algebras V b t ( g ,µ ) o ( ℓ ) . We here show that Lie algebra b t ( g , µ ) o is a vertexLie algebra and associate vertex algebras V b t ( g ,µ ) o ( ℓ ) to b t ( g , µ ) o .Recall e g = C [ t , t − ] ⊗ g + C k + C d ⊂ b t ( g , µ ) o . Set b t ( g , µ ) o + = Span { t n u, k n +1 ,m , d n − ,m | u ∈ e g , n ∈ N , m ∈ Z } , (3.35) b t ( g , µ ) o − = Span { t − n − u, k − n,m , d − n − ,m | u ∈ e g , n ∈ N , m ∈ Z } . (3.36) OROIDAL EALA AND VA 19
It is clear that b t ( g , µ ) o ± are subalgebras of b t ( g , µ ) o and we have the following trian-gular decomposition b t ( g , µ ) o = b t ( g , µ ) o + ⊕ C k ⊕ b t ( g , µ ) o − . (3.37)Let ℓ be a complex number. View C as a ( b t ( g , µ ) o + + C k )-module with b t ( g , µ ) o + acting trivially and with k acting as scalar ℓ . Form an induced b t ( g , µ ) o -module V b t ( g ,µ ) o ( ℓ ) = U ( b t ( g , µ ) o ) ⊗ U ( b t ( g ,µ ) o + + C k ) C . (3.38)Set = 1 ⊗ ∈ V b t ( g ,µ ) o ( ℓ ). Notice that b t ( g , µ ) o + + C k is a graded subalgebra.Define deg C = 0, to make C a Z -graded ( b t ( g , µ ) o + + C k )-module. Then V b t ( g ,µ ) o ( ℓ )is a Z -graded b t ( g , µ ) o -module V b t ( g ,µ ) o ( ℓ ) = M m ∈ Z V b t ( g ,µ ) o ( ℓ ) ( m ) (3.39)with V b t ( g ,µ ) o ( ℓ ) ( m ) = 0 for m < A g = C [ t , t − ] ⊗ g ⊕ C k ⊕ C d ⊕ X n ∈ Z × ( C K n ⊕ C D n ) . Identify A g as a subspace of V b t ( g ,µ ) o ( ℓ ) through the linear map u ( t − u ) , K n k ,n , D n d − ,n for u ∈ e g , n ∈ Z × . (3.40)Define a linear map ρ : (cid:0) C [ t, t − ] ⊗ A g (cid:1) ⊕ C k → b t ( g , µ ) o by ρ ( t m ⊗ u ) = t m u, ρ ( t m ⊗ K n ) = k m +1 ,n , ρ ( t m ⊗ D n ) = d m − ,n , ρ (k ) = k for u ∈ e g , m ∈ Z , n ∈ Z × .With this setting we have: Proposition 3.8.
The quadruple ( b t ( g , µ ) o , A g , C k , ρ ) carries the structure of avertex Lie algebra. For any ℓ ∈ C , there is a vertex algebra structure on V b t ( g ,µ ) o ( ℓ ) ,which is uniquely determined by the condition that is the vacuum vector and Y ( a, z ) = a ( z ) for a ∈ A g . Furthermore, V b t ( g ,µ ) o ( ℓ ) is a Z -graded vertex algebra.Proof. The first assertion follows immediately from Proposition 3.3. Definedeg( t m ⊗ u ) = 1 , deg(k ) = deg(d ) = 1 , deg(K n ) = 0 , deg(D n ) = 2for u ∈ g , m ∈ Z , n ∈ Z × , to make A g a Z -graded vector space. Then b t ( g , µ ) o be-comes a Z -graded vertex Lie algebra. Let γ ℓ be the linear functional on C k definedby γ ℓ (k ) = ℓ . We see that V b t ( g ,µ ) o ( ℓ ) coincides with the b t ( g , µ ) o -module V b t ( g ,µ ) o ( γ ℓ ) defined in Section 3.2. Now, it follows from Proposition 3.5 that V b t ( g ,µ ) o ( ℓ ) is a Z -graded vertex algebra. (cid:3) Remark 3.9.
Recall that e t ( g , µ ) o = b t ( g , µ ) o ⋊C ( t − d ). It can be readily seen thatvertex algebra V b t ( g ,µ ) o ( ℓ ) is a e t ( g , µ ) o -module with − t − d acting as the canonicalderivation L ( − V b t ( g ,µ ) o ( ℓ ) is the same as a e t ( g , µ ) o -submodule (see Lemma 3.6).3.4. Vertex algebras V int b t ( g ,µ ) o ( ℓ ) . Now, we assume that g is a finite-dimensionalsimple Lie algebra with a Cartan subalgebra h . Denote by ∆ the root system of g with respect to h . Fix a simple root base Π of ∆ and denote by θ the highest longroot. Let h· , ·i be the Killing form which is normalized so that the squared lengthof the long roots equals 2. Note that for α ∈ ∆, we have h α, α i = 2, 1, or (for G ). Set ǫ α = 2 h α, α i ∈ { , , } . (3.41)In the following, we shall define a vertex algebra V int b t ( g ,µ ) o ( ℓ ) for every nonnega-tive integer ℓ and associate V int b t ( g ,µ ) o ( ℓ )-modules to restricted and integrable b t ( g , µ ) o -modules of level ℓ .For each α ∈ ∆, choose a Lie algebra embedding ρ α : sl → g such that ρ α ( e ) ∈ g α , ρ α ( f ) ∈ g − α , and set h α = ρ α ( h ) ∈ h ⊂ g , (3.42)where { e, h, f } is the standard basis of sl with[ e, f ] = h, [ h, e ] = 2 e, [ h, f ] = − f. Note that ρ α ( h ) is independent of the choice of ρ α and h ρ α ( e ) , ρ α ( f ) i = 12 h h α , h α i = 2 h α, α i = ǫ α (3.43)(also independent of the choice of ρ α ).Let m ∈ Z , α ∈ ∆. Notice that from (2.12) we have[ t p t m ρ α ( e ) , t q t − m ρ α ( f )] = t p + q h α + h ρ α ( e ) , ρ α ( f ) i ( mt p + q k + pδ p + q, k )for p, q ∈ Z . Set b sl ( m, α ) = C [ t ± ] t m g α + C [ t ± ] t − m g − α + C [ t ± ]( h α + mǫ α k ) + C k , (3.44)which is a subalgebra of t ( g ) and hence a subalgebra of both b t ( g , µ ) and b t ( g , µ ) o .It is straightforward to see that the linear map ρ m,α : b sl → b sl ( m, α ) , (3.45) OROIDAL EALA AND VA 21 defined by ρ m,α ( c ) = ǫ α k and ρ m,α ( t n e ) = t n t m ρ α ( e ) , ρ m,α ( t n f ) = t n t − m ρ α ( f ) , ρ m,α ( t n h ) = t n ( h α + mǫ α k )for n ∈ Z , is a Lie algebra isomorphism. Definition 3.10. A b t ( g , µ ) o -module W is said to be restricted if a ( z ) ∈ E ( W ) forevery a ∈ A g , is said to be of level ℓ ∈ C if k acts as scalar ℓ , and is said to be integrable if for any m , m ∈ Z , α ∈ ∆, t m t m g α acts locally nilpotently on W .Next, we present a characterization of integrable and restricted b t ( g , µ ) o -modulesas an analogue of a result for affine Kac-Moody algebras. Lemma 3.11.
Let W be a restricted b t ( g , µ ) o -module of level ℓ ∈ C . Then W isintegrable if and only if ℓ is a nonnegative integer and a ( z ) ǫ α ℓ +1 = 0 on W for any a ∈ t m g α with m ∈ Z , α ∈ ∆ . Proof.
Note that for any m ∈ Z , α ∈ ∆, via the homomorphism ρ m,α , W becomesa restricted module of level ǫ α ℓ for the affine Lie algebra b sl . We see that b t ( g , µ ) o -module W is integrable if and only if for any m ∈ Z , α ∈ ∆, W is an integrable b sl -module via ρ m,α . Recall from [DLM1] that a restricted b sl -module M of level ℓ ′ ∈ C is integrable if and only if ℓ ′ is a nonnegative integer and e ( z ) ℓ ′ +1 = 0 = f ( z ) ℓ ′ +1 = 0 on M. Then it follows immediately. (cid:3)
Definition 3.12.
Let ℓ be a nonnegative integer. Denote by J ( ℓ ) the b t ( g , µ ) o -submodule of V b t ( g ,µ ) o ( ℓ ), generated by the vectors( t − a ) ǫ α ℓ +1 for a ∈ t m g α with m ∈ Z , α ∈ ∆ . (3.46) Lemma 3.13.
Let ℓ be a nonnegative integer. Then the b t ( g , µ ) o -submodule J ( ℓ ) is a graded ideal of the vertex algebra V b t ( g ,µ ) o ( ℓ ) .Proof. As the generators of J ( ℓ ) are homogeneous, J ( ℓ ) is a graded submoduleof V b t ( g ,µ ) o ( ℓ ). To prove that J ( ℓ ) is a graded ideal of V b t ( g ,µ ) o ( ℓ ), by Remark 3.9 itsuffices to show that J ( ℓ ) is t − d -invariant. Let a ∈ t r g α with α ∈ ∆ , r ∈ Z . Notethat the quotient b t ( g , µ ) o -module V b t ( g ,µ ) o ( ℓ ) /J ( ℓ ) is integrable. Then it follows fromLemma 3.11 that a ( z ) ǫ α ℓ +1 ∈ J ( ℓ )[[ z, z − ]] . (3.47)Note that ( t − d ) = 0, [ t m a, t n a ] = 0 for m, n ∈ Z , and ( t p a ) = 0 for p ≥ t − d )( t − a ) ǫ α ℓ +1 = − ( ǫ α ℓ + 1)( t − a )( t − a ) ǫ α ℓ = − Res z z − a ( z ) ǫ α ℓ +1 ∈ J ( ℓ ) . It follows that ( t − d ) J ( ℓ ) ⊂ J ( ℓ ). Therefore, J ( ℓ ) is a graded ideal of V b t ( g ,µ ) o ( ℓ ). (cid:3) Now, we introduce another vertex algebra among the main objects of this paper.
Definition 3.14.
Let ℓ be a nonnegative integer. Set V int b t ( g ,µ ) o ( ℓ ) = V b t ( g ,µ ) o ( ℓ ) /J ( ℓ ) , (3.48)which is a Z -graded vertex algebra.We have: Proposition 3.15.
Let ℓ ∈ C and let W be a vector space. Then a level ℓ restricted b t ( g , µ ) o -module structure on W amounts to a V b t ( g ,µ ) o ( ℓ ) -module structure Y W ( · , z ) such that Y W ( a, z ) = a ( z ) for a ∈ A g . (3.49) Furthermore, restricted and integrable b t ( g , µ ) o -modules of level ℓ (which is neces-sarily a nonnegative integer) correspond exactly to V int b t ( g ,µ ) o ( ℓ ) -modules.Proof. The first assertion follows immediately from Lemma 3.7. For the secondassertion, let W be a restricted and integrable b t ( g , µ ) o -module of level ℓ , and let a ∈ t m g α with m ∈ Z , α ∈ ∆. In view of Lemma 3.11, we have Y W ( a, z ) ǫ α ℓ +1 = a ( z ) ǫ α ℓ +1 = 0 on W. By a result of [DL], we get Y W (( a − ) ǫ α ℓ +1 , z ) = 0. Then it follows that W isnaturally a V int b t ( g ,µ ) o ( ℓ )-module. On the other hand, let W be a V int b t ( g ,µ ) o ( ℓ )-module.Again by [DL], we have Y W ( a, z ) ǫ α ℓ +1 = Y W (( a − ) ǫ α ℓ +1 , z ) = 0 on W. Viewing W as a b t ( g , µ ) o -module we have a ( z ) ǫ α ℓ +1 = 0 on W. Then W is integrable by Lemma 3.11. (cid:3) Restricted b t ( g , µ ) -modules and φ -coordinated V b t ( g ,µ ) o ( ℓ ) -modules In this section, we assume that g is a finite-dimensional simple Lie algebra witha Cartan subalgebra h as in Section 3.4. As the main results, we give a canonicalconnection between restricted (resp. integrable and restricted) b t ( g , µ )-modules oflevel ℓ and φ -coordinated modules for V b t ( g ,µ ) o ( ℓ ) (resp. V int b t ( g ,µ ) o ( ℓ )). OROIDAL EALA AND VA 23 φ -coordinated modules for vertex algebras. We here recall from [Li2]the notion of φ -coordinated module for a vertex algebra and some basic results. Definition 4.1.
Let V be a vertex algebra. A φ -coordinated V -module is a vectorspace W equipped with a linear map Y W ( · , z ) : V → Hom(
W, W (( z ))) ⊂ (End W )[[ z, z − ]] v Y W ( v, z ) , satisfying the conditions that Y W ( , z ) = 1 W and that for u, v ∈ V , there exists k ∈ N such that( z − z ) k Y W ( u, z ) Y W ( v, z ) ∈ Hom(
W, W (( z , z ))) , (4.1) ( z e z − z ) k Y W ( Y ( u, z ) v, z ) = (cid:0) ( z − z ) k Y W ( u, z ) Y W ( v, z ) (cid:1) | z = z e z . (4.2) Remark 4.2.
The parameter φ in Definition 4.1 refers to the formal series φ ( x, z ) = xe z , which is a particular associate, as defined in [Li2], of the one-dimensional ad-ditive formal group (law) F ( x, y ) = x + y . Taking φ ( x, z ) = x + z (the formalgroup law itself) in Definition 4.1, we get an equivalent definition of a module for V (cf. [LTW]). Remark 4.3.
Let (
W, Y W ) be a φ -coordinated module for a vertex algebra V . Itwas proved in [Li2, Lemma 3.6] that for u, v ∈ V ,( z e z − z ) k Y W ( Y ( u, z ) v, z ) = (cid:0) ( z − z ) k Y W ( u, z ) Y W ( v, z ) (cid:1) | z = z e z holds for any k ∈ N such that( z − z ) k Y W ( u, z ) Y W ( v, z ) ∈ Hom(
W, W (( z , z ))) . Lemma 4.4.
Let V be a vertex algebra and let ( W, Y W ) be a φ -coordinated V -module. Then ker Y W is an ideal of V , where ker Y W = { v ∈ V | Y W ( v, z ) = 0 } .Proof. Let u, v ∈ V such that either Y W ( u, z ) = 0 or Y W ( v, z ) = 0. Then byRemark 4.3 with k = 0 we get Y W ( Y ( u, z ) v, z ) = Y W ( u, z ) Y W ( v, z ) | z = z e z = 0 , which implies that u n v ∈ ker Y W for all n ∈ Z . Thus ker Y W is an ideal of V . (cid:3) We have (see [Li2, Lemma 3.7], [Li4]):
Lemma 4.5.
Let V be a vertex algebra and let ( W, Y W ) be any φ -coordinated V -module. Then Y W ( L ( − v, z ) = z ddz Y W ( v, z ) for v ∈ V, (4.3) where L ( − is the linear operator on V defined by L ( − v = v − for v ∈ V . The following was proved in [Li2, Proposition 5.9] and [BLP, Theorem 3.19,Lemma 3.29]:
Proposition 4.6.
Let V be a vertex algebra and let ( W, Y W ) be a φ -coordinated V -module. Then for u, v ∈ V , [ Y W ( u, z ) , Y W ( v, w )] = X j ∈ N Y W ( u j v, w ) 1 j ! (cid:18) w ∂∂w (cid:19) j δ (cid:16) wz (cid:17) . (4.4) On the other hand, if W is faithful and if [ Y W ( u, z ) , Y W ( v, w )] = X j ∈ N Y W ( A j , w ) 1 j ! (cid:18) w ∂∂w (cid:19) j δ (cid:16) wz (cid:17) (4.5) with A j ∈ V , then A j = u j v for j ∈ N . Let (
W, Y W ) be a φ -coordinated V -module. For v ∈ V , write Y W ( v, z ) = X n ∈ Z v [ n ] z − n . (4.6)Then [ u [ m ] , v [ n ]] = X j ≥ m j ( u j v )[ m + n ](4.7)on W for u, v ∈ V, m, n ∈ Z . Definition 4.7.
Let V be a vertex algebra. A Z -graded φ -coordinated V -module is a φ -coordinated V -module W equipped with a Z -grading W = ⊕ n ∈ Z W ( n ) suchthat u [ m ] W ( n ) ⊂ W ( m + n ) for u ∈ V, m, n ∈ Z . (4.8)Let W be a vector space. Formal series a ( x ) , b ( x ) ∈ E ( W ) are said to be local ifthere exists a nonnegative integer k such that( z − w ) k [ a ( z ) , b ( w )] = 0 . A subset U of E ( W ) is said to be local if any a ( z ) , b ( z ) ∈ U are local. For alocal pair ( a ( x ) , b ( x )) as above, define a ( x ) φn b ( x ) ∈ E ( W ) for n ∈ Z in terms ofgenerating function Y φ E ( a ( x ) , z ) b ( x ) := X n ∈ Z (cid:0) a ( x ) φn b ( x ) (cid:1) z − n − ∈ E ( W )[[ z, z − ]]by Y φ E ( a ( x ) , z ) b ( x ) = ( xe z − x ) − k (cid:0) ( x − x ) k a ( x ) b ( x ) (cid:1) | x = xe z . A local subspace U of E ( W ) is said to be Y φ E -closed if a ( x ) φn b ( x ) ∈ U for all a ( x ) , b ( x ) ∈ U, n ∈ Z .The following result was obtained in ([Li2, Theorem 5.4], [BLP, Theorem 2.9]): OROIDAL EALA AND VA 25
Proposition 4.8.
Let U be a local subset of E ( W ) . Then there exists a Y φ E -closedlocal subspace of E ( W ) which contains W and U . Denote by h U i φ the smallestsuch local subspace. Then ( h U i φ , Y φ E , W ) is a vertex algebra with U as a generatingsubset and ( W, Y W ) is a faithful φ -coordinated module with Y W ( a ( x ) , z ) = a ( z ) for a ( x ) ∈ h U i φ . (4.9) Remark 4.9.
Let U be a local subset of E ( W ). For a ( z ) ∈ h U i φ , by (4.3) we have L ( − a ( z ) = Y W ( L ( − a ( x ) , z ) = z ddz Y W ( a ( x ) , z ) = z ddz a ( z ) . This in particular implies (cid:0) z ddz (cid:1) a ( z ) ∈ h U i φ .4.2. φ -coordinated modules for vertex Lie algebras. Let ( L , A , C , ρ ) be avertex Lie algebra and let γ be a linear functional on C . In this section, we intro-duce a notion of φ -coordinated L -module of level γ and prove that φ -coordinated L -modules exactly amount to φ -coordinated V L ( γ )-modules. Definition 4.10. A φ -coordinated L -module of level γ is a vector space W equippedwith a linear map ψ W ( · , z ) : A ⊕ C →
Hom (
W, W (( z ))); u ψ W ( u, z )such that ψ W ( c, z ) = γ ( c ) for c ∈ C and for a, b ∈ A ,[ ψ W ( a, z ) , ψ W ( b, w )] = r X i =0 ψ W ( f i ( a, b ) , w ) 1 i ! (cid:18) w ∂∂w (cid:19) i δ (cid:16) wz (cid:17) , (4.10)where f i ( a, b ) ∈ C [ ∂ ] A ⊕ C are the same as in (3.24) and where the linear map ψ W ( · , z ) is extended to C [ ∂ ] A ⊕ C linearly by ψ W ( ∂ i a, z ) = (cid:18) z ddz (cid:19) i ψ W ( a, z ) for i ∈ N , a ∈ A . (4.11)Recall that π : C [ ∂ ] A ⊕ C → V L ( γ ) is the linear map defined by π ( c ) = γ ( c ) , π ( ∂ i a ) = L ( − i a for c ∈ C , i ∈ N , a ∈ A (4.12)and that Y ( π ( X ) , z ) = X ( z ) for X ∈ C [ ∂ ] A ⊕ C . As the main result of this section, we have:
Proposition 4.11.
Let ( W, Y W ) be a φ -coordinated module for the vertex algebra V L ( γ ) . Then W is a φ -coordinated module of level γ for the vertex Lie algebra L with ψ W ( c, z ) = γ ( c ) for c ∈ C and ψ W ( a, z ) = Y W ( a, z ) for a ∈ A . (4.13) On the other hand, for any φ -coordinated L -module ( W, ψ W ) of level γ , there existsa φ -coordinated V L ( γ ) -module structure Y W ( · , z ) on W , which is uniquely deter-mined by Y W ( a, z ) = ψ W ( a, z ) for a ∈ A . (4.14) Proof.
Let a, b ∈ A and assume that [ a ( z ) , b ( w )] has the expression as in (4.1).Note that for any c ∈ C , c acts on V L ( γ ) as scalar γ ( c ). Then using (3.33) we get[ Y ( a, z ) , Y ( b, w )] = [ a ( z ) , b ( w )]= r X i =0 f i ( a, b )( w ) 1 i ! (cid:18) ∂∂w (cid:19) i z − δ (cid:16) wz (cid:17) = r X i =0 Y ( ¯ f i ( a, b ) , w ) 1 i ! (cid:18) ∂∂w (cid:19) i z − δ (cid:16) wz (cid:17) , where ¯ f i ( a, b ) = π ( f i ( a, b )) ∈ V L ( γ ). In view of Proposition 4.6, we have[ Y W ( a, z ) , Y W ( b, w )] = r X i =0 Y W ( ¯ f i ( a, b ) , w ) 1 i ! (cid:18) w ∂∂w (cid:19) i δ (cid:16) wz (cid:17) . Therefore, W is a φ -coordinated L -module of level γ with ψ W ( a, z ) = Y W ( a, z ) for a ∈ A . This proves the first assertion.For the second assertion, the uniqueness is clear as A generates V L ( γ ) as a vertexalgebra, so it remains to establish the existence. Set U = { ψ W ( a, x ) | a ∈ A} . From (4.10), U is a local subspace of E ( W ). Then by Proposition 4.8, U generatesa vertex algebra h U i φ and W is a faithful φ -coordinated module for h U i φ such that Y W ( ψ W ( a, x ) , z ) = ψ W ( a, z ) for a ∈ A . (4.15)For a, b ∈ A , by the second assertion of Proposition 4.6 we have[ Y φ E ( ψ W ( a, x ) , z ) , Y φ E ( ψ W ( b, x ) , z )]= r X i =0 Y φ E ( ψ W ( f i ( a, b ) , x ) , z ) 1 i ! (cid:18) ∂∂z (cid:19) i z − δ (cid:18) z z (cid:19) . From Remark 4.9, it follows that for a ∈ A , i ∈ N , (cid:0) x ∂∂x (cid:1) i ψ W ( a, x ) ∈ h U i φ and Y φ E (cid:18) x ∂∂x (cid:19) i ψ W ( a, x ) , z ! = Y φ E (cid:0) L ( − i ( ψ W ( a, x )) , z (cid:1) = (cid:18) ∂∂z (cid:19) i Y φ E ( ψ W ( a, x ) , z ) . Then it follows that h U i φ is an L -module with a ( z ) = Y φ E ( ψ W ( a, x ) , z ) , c = γ ( c ) for a ∈ A , c ∈ C . OROIDAL EALA AND VA 27
Since U generates the vertex algebra h U i φ , it follows that h U i φ is generated by1 W as an L -module. Moreover, the creation property of the vertex algebra h U i φ implies that L + W = 0. Then there exists an L -module homomorphism ϕ : V L ( γ ) → h U i φ (4.16)with ϕ ( ) = 1 W . For a ∈ A , we have ϕ ( Y ( a, z ) ) = ϕ ( a ( z ) ) = Y φ E ( ψ W ( a, x ) , z )1 W , which by setting z = 0 gives ϕ ( a ) = ψ W ( a, x ) . (4.17)Thus, ϕ ( Y ( a, z ) v ) = ϕ ( a ( z ) v ) = Y φ E ( ψ W ( a, x ) , z ) ϕ ( v ) = Y φ E ( ϕ ( a ) , z ) ϕ ( v )for a ∈ A , v ∈ V L ( γ ). As A generates V L ( γ ) as a vertex algebra, by [LL, Proposi-tion 5.7.9] ϕ is a vertex algebra homomorphism. Then W becomes a φ -coordinated V L ( γ )-module via ϕ . Moreover, by (4.15) and (4.17), we have Y W ( a, z ) = Y W ( ϕ ( a ) , z ) = Y W ( ψ W ( a, x ) , z ) = ψ W ( a, z ) for a ∈ A . This proves the existence and hence concludes the proof. (cid:3) b t ( g , µ ) -modules and φ -coordinated modules for V b t ( g ,µ ) o ( ℓ ) . Here, we givea canonical association of restricted b t ( g , µ )-modules of level ℓ with φ -coordinatedmodules for the vertex algebra V b t ( g ,µ ) o ( ℓ ).Just as for Lie algebra b t ( g , µ ) o , we formulate the following notions: Definition 4.12. A b t ( g , µ )-module W is said to be restricted if a [ z ] ∈ E ( W ) forevery a ∈ A g , is said to be of level ℓ ∈ C if k acts as scalar ℓ , and is said to be integrable if ( t m t m g α ) acts locally nilpotently on W for any m , m ∈ Z , α ∈ ∆.By the same arguments in the proof of Lemma 3.11, we have: Lemma 4.13.
Let W be a restricted b t ( g , µ ) -module of level ℓ ∈ C . Then W isintegrable if and only if ℓ is a nonnegative integer and a [ z ] ǫ α ℓ +1 = 0 on W for all a ∈ t m g α with m ∈ Z , α ∈ ∆ , where ǫ α = h α,α i . The following is an analogue of a result of Dong and Lepowsky (see [DL]):
Lemma 4.14.
Let V be a vertex algebra and let ( W, Y W ) be a φ -coordinated V -module. Assume that k is a nonnegative integer and a ∈ V such that a n a = 0 for n ≥ . If ( a − ) k +1 = 0 in V , then Y W ( a, z ) k +1 = 0 . (4.18) On the other hand, the converse is also true if ( W, Y W ) is faithful. Proof.
Note that for any u, v ∈ V , if u n v = 0 for n ≥
0, then by Proposition 4.6 wehave [ Y W ( u, z ) , Y W ( v, z )] = 0. Since a i a = 0 for all i ≥
0, we have [ a m , a n ] = 0on V for all m, n ∈ Z . Then for b = ( a − ) r with r ∈ N we have a i b = 0 for i ≥ Y W ( a, z ) , Y W ( b, z )] = 0. By Remark 4.3, we have Y W ( Y ( a, z ) b, z ) = Y W ( a, z ) Y W ( b, z ) | z = z e z . Setting z = 0 we get Y W ( a − b, z ) = Y W ( a, z ) Y W ( b, z ) . It then follows that Y W (( a − ) k +1 , z ) = Y W ( a, z ) k +1 . Therefore, if ( a − ) k +1 = 0, then Y W ( a, z ) k +1 = 0. Conversely, if Y W ( a, z ) ℓ +1 = 0and if W is faithful, then we have ( a − ) ℓ +1 = 0. (cid:3) Recall the vertex Lie algebra ( b t ( g , µ ) o , A g , C k , ρ ) from Proposition 3.8. ByPropositions 2.10 and 3.3, we immediately have: Lemma 4.15.
Let ℓ ∈ C and let W be a vector space. Then the structure of arestricted b t ( g , µ ) -module of level ℓ on W amounts to the structure ψ W ( · , z ) of a φ -coordinated b t ( g , µ ) o -module of level ℓ on W with a [ z ] = ψ W ( a, z ) for a ∈ A g . Now, we are in a position to present our first main result of this paper.
Theorem 4.16.
Let ℓ be any complex number. For any restricted b t ( g , µ ) -module W of level ℓ , there exists a φ -coordinated V b t ( g ,µ ) o ( ℓ ) -module structure Y W ( · , x ) on W , which is uniquely determined by Y W ( a, z ) = a [ z ] for a ∈ A g . (4.19) On the other hand, for any φ -coordinated V b t ( g ,µ ) o ( ℓ ) -module ( W, Y W ) , W is a re-stricted b t ( g , µ ) -module of level ℓ with a [ z ] = Y W ( a, z ) for a ∈ A g . (4.20) Furthermore, restricted and integrable b t ( g , µ ) -modules of level ℓ (which is necessar-ily a nonnegative integer) exactly correspond to φ -coordinated V int b t ( g ,µ ) o ( ℓ ) -modules.Proof. Recall the vertex Lie algebra ( b t ( g , µ ) o , A g , C k , ρ ) and the linear functional γ ℓ defined in the proof of Proposition 3.8. We have V b t ( g ,µ ) o ( γ ℓ ) = V b t ( g ,µ ) o ( ℓ ). Let W be a restricted b t ( g , µ )-module of level ℓ . From Lemma 4.15, W is a φ -coordinatedmodule of level ℓ for the vertex Lie algebra b t ( g , µ ) o with ψ W ( a, z ) = a [ z ] for a ∈ A g . OROIDAL EALA AND VA 29
Then by Proposition 4.11 there exists a φ -coordinated module structure Y W ( · , z )on W for the vertex algebra V b t ( g ,µ ) o ( ℓ ) (= V b t ( g ,µ ) o ( γ ℓ )) such that Y W ( a, z ) = ψ W ( a, z ) = a [ z ] for a ∈ A g . On the other hand, let (
W, Y W ) be a φ -coordinated V b t ( g ,µ ) o ( ℓ )-module. Then itfollows from the first assertion of Proposition 4.11 that W is a φ -coordinated b t ( g , µ ) o -module of level ℓ , and hence by Lemma 4.15 W is a restricted b t ( g , µ )-module of level ℓ with a [ z ] = ψ W ( a, z ) = Y W ( a, z ) for a ∈ A g . Next, we prove the furthermore assertion. Suppose W is an integrable andrestricted b t ( g , µ )-module of level ℓ . By the first part of this theorem, W is a φ -coordinated module for V b t ( g ,µ ) o ( ℓ ). Then W is naturally a faithful φ -coordinatedmodule for the quotient vertex algebra V b t ( g ,µ ) o ( ℓ ) / ker Y W . Let a ∈ t m g α with m ∈ Z , α ∈ ∆. Notice that [ a ( z ) , a ( w )] = 0 in b t ( g , µ ) o [[ z ± , w ± ]], which implies a n a = 0 in V b t ( g ,µ ) o ( ℓ ) for all n ≥ . (4.21)As W is integrable, by Lemma 4.13 we have a [ z ] ǫ α ℓ +1 = 0 on W , so that Y W ( a, z ) ǫ α ℓ +1 = a [ z ] ǫ α ℓ +1 = 0 on W. (4.22)Then by Lemma 4.14 we have( a − ) ǫ α ℓ +1 = 0 in V b t ( g ,µ ) o ( ℓ ) / ker Y W . It follows that J ( ℓ ) ⊂ ker Y W . Consequently, W is a φ -coordinated V int b t ( g ,µ ) o ( ℓ )-module.On the other hand, let W be a φ -coordinated V int b t ( g ,µ ) o ( ℓ )-module. From the firstpart, W is a restricted b t ( g , µ )-module of level ℓ with u [ z ] = Y W ( u, z ) for u ∈ A g . (4.23)Let a ∈ t m g α with m ∈ Z , α ∈ ∆. Denote the image of a in V int b t ( g ,µ ) o ( ℓ ) still by a . From the definition of V int b t ( g ,µ ) o ( ℓ ) we have ( a − ) ǫ α ℓ +1 = 0 in V int b t ( g ,µ ) o ( ℓ ). Thenby Lemma 4.14 we have Y W ( a, z ) ǫ α ℓ +1 = 0 on W . Thus a [ z ] ǫ α ℓ +1 = 0 on W .Therefore, W is an integrable b t ( g , µ )-module by Lemma 4.13. (cid:3) Irreducible V b t ( g ,µ ) o ( ℓ ) -modules In this section, we classify irreducible bounded N -graded modules (with finite-dimensional homogenous spaces) for Lie algebra b t ( g , µ ) o and vertex algebras V b t ( g ,µ ) o ( ℓ )and V int b t ( g ,µ ) o ( ℓ ). Realization of irreducible b t ( g , µ ) o -modules. Here, we give an explicit re-alization of a class of irreducible b t ( g , µ ) o -modules of nonzero levels. Recall thatboth e t ( g , µ ) o and b t ( g , µ ) o are Z -graded subalgebras of T ( g , µ ) with the gradinggiven by the adjoint action of − d . The degree-zero subalgebra of b t ( g , µ ) o is L (0) := b t ( g , µ ) o (0) = e g ⊕ X n ∈ Z × ( C d ,n + C k ,n ) ⊕ C k (5.1) = C [ t , t − ] ⊗ g + X n ∈ Z ( C d ,n + C t n k ) + C k (recall that d , = d , k , = 0 and n k ,n = t n k for n ∈ Z × ), where k is centraland [ t m k , t n k ] = 0 , [ t m k , t n ⊗ u ] = 0 , (5.2) [d ,m , d ,n ] = ( n − m )d ,m + n + 2 µm δ m + n, k , (5.3) [d ,m , t n k ] = nt m + n k + mδ m + n, k , (5.4) [d ,m , t n ⊗ u ] = n ( t m + n ⊗ u ) , (5.5) [ t m ⊗ u, t n ⊗ v ] = t m + n ⊗ [ u, v ] + mδ m + n, h u, v i k (5.6)for m, n ∈ Z , u, v ∈ g . Note that k (= t k ) is also central.Set P + = { λ ∈ h ∗ | λ ( α ∨ ) ∈ N for α ∈ Π } , the set of dominant integral weights of g . Let λ ∈ P + and let ℓ, α, β ∈ C with ℓ = 0.Denote by L g ( λ ) the (finite-dimensional) irreducible highest weight g -module withhighest weight λ . We then define an L (0) -module T ℓ,λ,α,β , where T ℓ,λ,α,β = C [ q, q − ] ⊗ L g ( λ )(5.7)as a vector space (with q an indeterminate) and where k acts trivially,( t m a )( q n ⊗ w ) = q m + n ⊗ aw, ( t m k )( q n ⊗ w ) = ℓq m + n ⊗ w, (5.8) d ,m ( q n ⊗ w ) = ( n + α + βm ) q m + n ⊗ w (5.9)for m, n ∈ Z , a ∈ g , w ∈ L g ( λ ). In particular, k acts as scalar ℓ . (It is knownthat this indeed gives an L (0) -module.) Also, it can be readily seen that T ℓ,λ,α,β isirreducible. We are particularly interested in T ℓ,λ,α,β with λ = 0 and α = β = 0. Definition 5.1.
For ℓ ∈ C × , define an irreducible L (0) -module T ℓ , where T ℓ = C [ q, q − ] as a vector space and where C [ t , t − ] ⊗ g = 0, k = 0, k = ℓ , and( t m k ) · q n = ℓq m + n , d ,m · q n = nq m + n for m, n ∈ Z . (5.10)Set b t ( g , µ ) o ( ± ) = P n> b t ( g , µ ) o ( ± n ) . Let λ, ℓ, α, β be given as before. We make T ℓ,λ,α,β a ( b t ( g , µ ) o ( − ) + L (0) )-module with b t ( g , µ ) o ( − ) acting trivially. Then form an OROIDAL EALA AND VA 31 induced b t ( g , µ ) o -module V b t ( g ,µ ) o ( T ℓ,λ,α,β ) := U ( b t ( g , µ ) o ) ⊗ U ( b t ( g ,µ ) o ( − ) + L (0) ) T ℓ,λ,α,β . (5.11)Note that V b t ( g ,µ ) o ( T ℓ,λ,α,β ) is naturally an N -graded b t ( g , µ ) o -module with deg T ℓ,λ,α,β =0. Denote by L b t ( g ,µ ) o ( T ℓ,λ,α,β ) the graded irreducible quotient of V b t ( g ,µ ) o ( T ℓ,λ,α,β ). Infact, L b t ( g ,µ ) o ( T ℓ,λ,α,β ) is an irreducible b t ( g , µ ) o -module.Recall that e t ( g , µ ) o and b t ( g , µ ) o share the same degree-zero subalgebra L (0) . Sim-ilarly, from the irreducible L (0) -module T ℓ,λ,α,β , we have an induced e t ( g , µ ) o -module V e t ( g ,µ ) o ( T ℓ,λ,α,β ) and its irreducible quotient L e t ( g ,µ ) o ( T ℓ,λ,α,β ). Note that as d = d , is semisimple on T ℓ,λ,α,β , it is semisimple on V b t ( g ,µ ) o ( T ℓ,λ,α,β ) and L b t ( g ,µ ) o ( T ℓ,λ,α,β ).In the following, we are going to use a result of Billig to give an explicit real-ization of these irreducible b t ( g , µ ) o -modules L b t ( g ,µ ) o ( T ℓ,λ,α,β ). First, follow [B1] toform a reductive Lie algebra f = g ⊕ C I (5.12)with 1-dimensional center C I . Consider the following (universal) central extensionof (Der C [ t ± ]) ⋉ ( C [ t ± ] ⊗ f ), with underlying space¯ f = (Der C [ t ± ]) ⋉ (cid:0) C [ t ± ] ⊗ f (cid:1) ⊕ C k ⊕ C k I ⊕ C k V I ⊕ C k V ir (5.13)and with the following Lie bracket relations:[ L ( m ) , L ( n )] = ( m − n ) L ( m + n ) + m − m δ m + n, k V ir , [( u + aI )( m ) , ( v + bI )( n )] = [ u, v ]( m + n ) + mδ m + n, ( h u, v i k + ab k I ) , [ L ( m ) , ( u + aI )( n )] = − n ( u + aI )( m + n ) − ( m + m ) aδ m + n, k V I for u, v ∈ g , m, n ∈ Z , a, b ∈ C , where L ( n ) = − t n +1 dd t , u ( n ) = t n ⊗ u for n ∈ Z , u ∈ f . This Lie algebra is often referred to as the twisted Virasoro-affine algebra . Remark 5.2.
Note that Lie algebra L (0) is a 1-dimensional central extension of(Der C [ t ± ]) ⋉ ( C [ t ± ] ⊗ f ). It then follows that L (0) is isomorphic to a quotient of(the universal central extension) ¯ f .Lie algebra ¯ f contains as subalgebras the following affine Lie algebra, Heisenbergalgebra, and the Virasoro algebra: b g = C [ t, t − ] ⊗ g + C k , Hei = C [ t, t − ] ⊗ C I + C k I , Vir = (Der C [ t, t − ]) + C k V ir . Furthermore, ¯ f contains as subalgebras the following Virasoro-affine algebra and twisted Virasoro-Heisenberg algebra : b g ⋊ Vir = (Der C [ t, t − ]) ⋉ (cid:0) C [ t, t − ] ⊗ g (cid:1) + C k + C k V ir , (5.14) HVir = (Der C [ t, t − ]) ⋉ (cid:0) C [ t, t − ] ⊗ C I (cid:1) + C k I + C k V I + C k V ir . (5.15)Set C = C k + C k I + C k V I + C k V ir , (5.16)the center of ¯ f , and set¯ f + = Span { u ( m ) , L ( n ) | u ∈ f , m ≥ , n ≥ − } , ¯ f − = Span { u ( m ) , L ( n ) | u ∈ f , m < , n < − } . From [B1], ¯ f is a vertex Lie algebra with triangular decomposition¯ f = ¯ f + ⊕ C ⊕ ¯ f − . (5.17)For any γ ∈ C ∗ , we have a vertex algebra V ¯ f ( γ ). In fact, V ¯ f ( γ ) is a vertex operatoralgebra with conformal vector ω f := L ( − , where V ¯ f ( γ ) ( n ) = 0 for n < , V ¯ f ( γ ) (0) = C , V ¯ f ( γ ) (1) = f . (5.18)Set H = C k + C d , an (independent) vector space with basis { k , d } , equippedwith the symmetric bilinear form h· , ·i given by h k , k i = 0 = h d , d i , h k , d i = 1 . (5.19)Viewing H as an abelian Lie algebra, we have a (general) affine algebra b H andthen a simple vertex operator algebra V b H (1 , L = Z k ⊂ H, (5.20)a free abelian group. Associated to the pair ( H, L ), we have a simple conformalvertex algebra (see [LX], [BBS], [BDT]) V ( H,L ) = V b H (1 , ⊗ C [ L ] , (5.21)which contains V b H (1 ,
0) as a vertex subalgebra, where C [ L ] = ⊕ m ∈ Z C e m k is thegroup algebra of L . Let ω H denote the standard conformal vector of V b H (1 , V ( H,L ) . We havewt k = 1 = wt d and wt e m k = 0 for m ∈ Z . (5.22)On the other hand, for every α ∈ C , we have an irreducible V ( H,L ) -module V ( H,L ) ( α ) := V b H (1 , ⊗ e α k C [ L ] , (5.23) OROIDAL EALA AND VA 33 where for m ∈ Z , Y ( e m k , z ) = E − ( − m k , z ) E + ( − m k , z ) e m k z m k . Noticing that z m k = 1 on V ( H,L ) ( α ) as h k , k i = 0, we have Y ( e m k , z ) e ( α + n ) k = E − ( − m k , z ) e ( α + m + n ) k for m, n ∈ Z . (5.24)The V ( H,L ) -module V ( H,L ) ( α ) is also Z -graded by the conformal weights with V ( H,L ) ( α ) ( n ) = 0 for n < V ( H,L ) ( α ) (0) = e α k C [ L ] . (5.25)(Notice that wt e ( α + m ) k = h ( α + m ) k , ( α + m ) k i = 0 for m ∈ Z .)Next, consider the tensor product V ¯ f ( γ ) ⊗ V ( H,L ) , which is also a conformal vertexalgebra with conformal vector ω = ω f ⊗ + ⊗ ω H . (5.26)Identify f and H as subspaces of V ¯ f ( γ ) ⊗ V ( H,L ) naturally. The following result isdue to Billig (see [B1]): Theorem 5.3.
Let ℓ ∈ C × . Define a linear functional γ ℓ on C by γ ℓ (k) = ℓ, γ ℓ (k I ) = 1 − µℓ, γ ℓ (k V I ) = 12 , γ ℓ (k V ir ) = 12 µℓ − . Then V ¯ f ( γ ℓ ) ⊗ V ( H,L ) is a T ( g , µ ) -module with k = ℓ and with X j ∈ Z ( t j t n k ) z − j − = ℓY ( e n k , z ) for n ∈ Z × , (5.27) X j ∈ Z ( t j k ) z − j − = ℓY ( k , z ) , (5.28) X j ∈ Z ( t j t m ⊗ a ) z − j − = Y ( a, z ) Y ( e m k , z ) for a ∈ g , m ∈ Z , (5.29) X j ∈ Z ( t j t m d ) z − j − =: Y ( d + mI, z ) Y ( e m k , z ) : , (5.30) X j ∈ Z ( − t j t m d + µ ( j + 1 / t j t m k ) z − j − =: Y ( ω, z ) Y ( e m k , z ) :(5.31) + Y ( I, z ) Y ( m k , z ) Y ( e m k , z ) + ( ℓµ − (cid:18) ddz Y ( m k , z ) (cid:19) Y ( e m k , z ) , where ω , defined in (5.26), is the conformal vector of V ¯ f ( γ ℓ ) ⊗ V ( H,L ) . In the following, we are going to use this result of Billig to obtain an explicitrealization of certain irreducible e t ( g , µ ) o -modules. For convenience, we formulatethe following general result: Lemma 5.4.
Let V be a vertex algebra and let ( L , A , C , ρ ) be a vertex Lie algebra.Suppose that ψ : C [ ∂ ] A ⊕ C → V is a linear map such that ψ ( C ) ⊂ C , ψ ( ∂ i a ) = L ( − i ψ ( a ) for i ∈ N , a ∈ A and such that V is an L -module with u ( z ) = Y ( ψ ( u ) , z ) for u ∈ A + C . Then every V -module W is an L -module with u ( z ) = Y ( ψ ( u ) , z ) for u ∈ A + C . Furthermore, W is an L ⋊ C d -module with d = − L ( − .Proof. Let a, b ∈ A . By definition, there exist f i ( a, b ) ∈ C [ ∂ ] A + C for i = 0 , , . . . , r such that (3.24) holds. As V is an L -module with u ( z ) = Y ( ψ ( u ) , z ) for u ∈ A + C ,the following commutator relation holds on V :[ Y ( ψ ( a ) , z ) , Y ( ψ ( b ) , w )] = r X i =0 Y ( ψ ( f i ( a, b )) , w ) 1 i ! (cid:18) ∂∂w (cid:19) i z − δ (cid:16) wz (cid:17) . (5.32)By [Li1, Lemma 2.3.5] (cf. [LL], Proposition 5.6.7), this relation holds on every V -module W . Thus, W is an L -module with u ( z ) = Y ( ψ ( u ) , z ) for u ∈ A + C . (cid:3) Corollary 5.5.
Let M be any V ¯ f ( γ ℓ ) ⊗ V ( H,L ) -module. Then M is a T ( g , µ ) -modulewith the action given as in Theorem 5.3. In particular, for any V ¯ f ( γ ℓ ) -module W ,any α ∈ C , W ⊗ V ( H,L ) ( α ) with the aforementioned action is a T ( g , µ ) -module.Proof. It follows from the argument of Billig in the proof of Theorem 5.3. We heregive a slightly different proof by using Theorem 5.3 instead of its proof. Let B bea basis of g . Then the following elements form a basis of T ( g , µ ): t r t m ⊗ u, K m,n , t m k , t r t m d , ( − t r t m d + µ ( r + 1 / t r t m k ) , k for u ∈ B, n ∈ Z × , m, r ∈ Z . Introduce a vector space U g := C [ t , t − ] ⊗ g + C k + X n ∈ Z × C K ( n ) + X m ∈ Z ( C D (0 ,m ) + C D (1 ,m ) )(5.33)with basis { k } ∪ { K ( n ) | n ∈ Z × } ∪ { t m u, D (0 ,m ) , D (1 ,m ) | m ∈ Z , u ∈ B } . Define a linear isomorphism ρ : C [ t, t − ] ⊗ U g ⊕ C k → T ( g , µ ) by ρ (k ) = k , ρ ( t r ⊗ K ( n ) ) = K r,n , ρ ( t m ⊗ k ) = t m k , ρ ( t r ⊗ t m u ) = t r t m ⊗ u,ρ ( t r ⊗ D (1 ,m ) ) = t r t m d , ρ ( t r ⊗ D (0 ,m ) ) = − t r t m d + µ ( r + 1 / t r t m k for n ∈ Z × , m, r ∈ Z , u ∈ g . From [B1], T ( g , µ ) is a vertex Lie algebra. Define alinear map ψ : U g ⊕ C k → V ¯ f ( γ ℓ ) ⊗ V ( H,L ) by ψ (k ) = ℓ , ψ ( t m ⊗ u ) = u ⊗ e m k , ψ (k ) = ⊗ ℓ k , ψ ( K ( n ) ) = 1 ⊗ n e n k ,ψ ( D (0 ,m ) ) = L ( − ⊗ e m k ) + I ⊗ ( m k ) − e m k + ( µℓ − ⊗ ( m k ) − e m k ) , OROIDAL EALA AND VA 35 ψ ( D (1 ,m ) ) = ⊗ d − e m k + I ⊗ me m k for u ∈ g , m ∈ Z , n ∈ Z × . Theorem 5.3 states that V ¯ f ( γ ℓ ) ⊗ V ( H,L ) is a T ( g , µ )-module with u ( z ) = Y ( ψ ( u ) , z ) for u ∈ U g + C k . Then it follows immediatelyfrom Lemma 5.4. (cid:3) Next, we discuss V ¯ f ( γ ℓ )-modules. We first recall the following result (see [B1]): Lemma 5.6.
The linear map η : b g ⋊ Vir → ¯ f = b g ⋊ HVir , (5.34) defined by η | b g = 1 , η (k V ir ) = k
V ir + 24k
V I − I , (5.35) η ( L ( m )) = L ( m ) + ( m + 1) I ( m ) for m ∈ Z , (5.36) is a Lie algebra embedding. For any pair ( ℓ, c ) of complex numbers, we have a vertex algebra V b g ⋊ Vir ( ℓ, c )associated to b g ⋊ Vir , where ℓ is the level of b g and c is the central charge of Vir .The embedding (5.34) naturally induces an embedding of vertex algebras η va : V b g ⋊ Vir ( ℓ, µℓ − ֒ → V ¯ f ( γ ℓ ) , where γ ℓ is the linear functional on C defined in Proposition 5.3. Furthermore, weget a vertex algebra embedding η va ⊗ V b g ⋊ Vir ( ℓ, µℓ − ⊗ V ( H,L ) → V ¯ f ( γ ℓ ) ⊗ V ( H,L ) . (5.37)As the first main result of this section, we have: Proposition 5.7.
Let ℓ ∈ C × . Then the conformal vertex algebra V := V b g ⋊ Vir ( ℓ, µℓ − ⊗ V ( H,L ) is a e t ( g , µ ) o -module of level ℓ , where the actions of the fields K n ( z ) , k ( z ) , ( t m ⊗ u )( z ) for n ∈ Z × , m ∈ Z , u ∈ g are given by (5.27) — (5.29) , and in addition d ( z ) = Y ( d , z ) , t − d = − ω V (= − L ( − , (5.38) D n ( z ) = nY ( ω V − e n k , z ) − ddz Y ( d − e n k , z ) + n ( ℓµ − Y (( n k ) − e n k , z )(5.39) for n ∈ Z × , where ω V denotes the conformal vector of V .Proof. With e t ( g , µ ) o a subalgebra of T ( g , µ ), V ¯ f ( γ ℓ ) ⊗ V ( H,L ) is naturally a e t ( g , µ ) o -module by Theorem 5.3. View V as a subspace of V ¯ f ( γ ℓ ) ⊗ V ( H,L ) via the embedding η va ⊗
1. Then it suffices to prove that V is a e t ( g , µ ) o -submodule of V ¯ f ( γ ℓ ) ⊗ V ( H,L ) .Note that under the embedding (5.37) we have (see Lemma 5.6) η V ( ω V ) = ω − L ( − I. Thus Y ( ω V , z ) = Y ( ω, z ) − ddz Y ( I, z ) . (5.40)In particular, we have ω V = ω , namely L V ( −
1) = L ( − n ( z ) , k ( z ) , ( t m ⊗ u )( z ), d ( z ), and the operator t − d preserve V .Thus it remains to show that the fields D n ( z ), n ∈ Z × also preserve V . Recall thatD n ( z ) = X j ∈ Z (cid:0) ( j + 1) t j t n d − nt j t n d + µn ( j + 1 / j,n (cid:1) z − j − . With (5.30) and (5.31), D n ( z ) acts on V ¯ f ( γ ℓ ) ⊗ V ( H,L ) as − ddz (: ( Y ( d , z ) + Y ( nI, z )) Y ( e n k , z ) :) + n : Y ( ω, z ) Y ( e n k , z ) :+ nY ( I, z ) Y ( n k , z ) Y ( e n k , z ) : + n ( ℓµ − (cid:18) ddz Y ( n k , z ) (cid:19) Y ( e n k , z )= n : ( Y ( ω, z ) − ddz Y ( I, z )) Y ( e n k , z ) : − ddz (: Y ( d , z ) Y ( e n k , z ) :)+ n ( ℓµ − (cid:18) ddz Y ( n k , z ) (cid:19) Y ( e n k , z )= n : Y ( ω V , z ) Y ( e n k , z ) : − ddz (cid:0) : Y ( d , z ) Y ( e n k , z ) : (cid:1) + n ( ℓµ − (cid:18) ddz Y ( n k , z ) (cid:19) Y ( e n k , z )= nY ( ω V − e n k , z ) − ddz Y ( d − e n k , z ) + n ( ℓµ − Y (( n k ) − e n k , z ) , where we are using the fact that ddz Y ( e n k , z ) = Y ( n k , z ) Y ( e n k , z ) for vertex algebra V ( H,L ) . With (5.40), this implies that the fields D n ( z ) preserve V (under the action(5.39)). This completes the proof. (cid:3) Equip b g ⋊ Vir with the Z -grading defined bydeg k = 0 = deg k Vir , deg( a ⊗ t n ) = − n, deg L ( n ) = − n for a ∈ g , n ∈ Z . Notice that the Lie algebra embedding η in Lemma 5.6 preserves the Z -gradings.The degree-zero subalgebra ( b g ⋊ Vir ) (0) equals g ⊕ ( C L (0) + C k + C k Vir ), a directproduct of g with the abelian Lie algebra C L (0) + C k + C k V ir . Set( b g ⋊ Vir ) ( − ) = X n< ( b g ⋊ Vir ) ( n ) = Span { L ( n ) , a ⊗ t n | n > , a ∈ g } . Let λ ∈ P + , β, ℓ, c ∈ C . Make the g -module L g ( λ ) a ( b g ⋊ Vir ) ( − ) + ( b g ⋊ Vir ) (0) -module by letting L (0) , k and k V ir acting as scalars β , ℓ and c , respectively, andletting ( b g ⋊ Vir ) ( − ) acting trivially. Then form an induced b g ⋊ Vir -module V b g ⋊ Vir ( ℓ, c, λ, β ) := U ( b g ⋊ Vir ) ⊗ U (( b g ⋊ Vir ) ( − ) +( b g ⋊ Vir ) (0) ) L g ( λ ) . (5.41) OROIDAL EALA AND VA 37
Notice that V b g ⋊ Vir ( ℓ, c, ,
0) = V b g ⋊ Vir ( ℓ, c ) (the associated vertex algebra). Denoteby L b g ⋊ Vir ( ℓ, c, λ, β ) the unique irreducible quotient of V b g ⋊ Vir ( ℓ, c, λ, β ).Recall that V = V b g ⋊ Vir ( ℓ, µℓ − ⊗ V ( H,L ) is a conformal vertex algebra, wherewt g = 1 , wt ω = 2 , wt k = 1 = wt d , wt e m k = 0 ( m ∈ Z ) . (5.42)As the main result of this section, we have (cf. [B1, Theorem 5.5]): Theorem 5.8.
Let λ ∈ P + , α, β, ℓ ∈ C with ℓ = 0 . Then L b g ⋊ Vir ( ℓ, µℓ − , λ, β ) ⊗ V ( H,L ) ( α ) is a e t ( g , µ ) o -module with the action given by (5.27) , (5.28) , (5.29) , (5.38) , and (5.39) . Furthermore, it is an irreducible module for e t ( g , µ ) o and b t ( g , µ ) o , which isisomorphic to L b t ( g ,µ ) o ( T ℓ,λ,α,β ) as a b t ( g , µ ) o -module, and isomorphic to L e t ( g ,µ ) o ( T ℓ,λ,α,β ) as a e t ( g , µ ) o -module.Proof. For convenience, set W = L b g ⋊ Vir ( ℓ, µℓ − , λ, β ) ⊗ V ( H,L ) ( α ) . Recall that e t ( g , µ ) o = b t ( g , µ ) o ⋊ C t − d and b t ( g , µ ) o is a vertex Lie algebra with A = A g and C = C k , where A g = C [ t , t − ] ⊗ g + C k + C d + X n ∈ Z × ( C K n + C D n ) . Define a linear map ψ : A g ⊕ C k → V (with V defined in Proposition 5.7) by ψ (k ) = ℓ , ψ (d ) = d , ψ (k ) = ⊗ ℓ k , (5.43) ψ (K n ) = ⊗ n e n k , ψ ( t m ⊗ a ) = a ⊗ e m k ,ψ (D n ) = nL ( − ⊗ e n k ) − L ( − ⊗ d − e n k )+ n ( ℓµ − ⊗ ( n k ) − e n k )for a ∈ g , m ∈ Z , n ∈ Z × . Proposition 5.7 states that V is a b t ( g , µ ) o -module with u ( z ) = Y ( ψ ( u ) , z ) for u ∈ A g + C k . In view of Lemma 5.4, W is a e t ( g , µ ) o -module(with t − d = − L ( − V is generated by g , k , d , e k , and ω . It follows that every V -submodule of W is a b t ( g , µ ) o -submodule of W . Thus W is an irreducible b t ( g , µ ) o -module and an irreducible e t ( g , µ ) o -module.Recall that e t ( g , µ ) o and b t ( g , µ ) o are Z -graded Lie algebras with L (0) = b t ( g , µ ) o (0) = e t ( g , µ ) o (0) and recall L (0) -module T ℓ,λ,α,β = C [ q, q − ] ⊗ L g ( λ ) as a vector space. Onthe other hand, for the V -module W , we have L g ( λ ) ⊂ L b g ⋊ Vir ( ℓ, µℓ − , λ, β ) , e α k C [ L ] ⊂ V ( H,L ) ( α ) . Recall that the V ( H,L ) -module V ( H,L ) ( α ) is Z -graded by the conformal weights with V ( H,L ) ( α ) ( n ) = 0 for n < V ( H,L ) ( α ) (0) = e α k C [ L ] . We see that W = ⊕ n ∈ N W ( n + β ) is C -graded by conformal weights with W ( β ) = L g ( λ ) ⊗ e α k C [ L ] . Note thatwt ψ (d ) = 1 = wt ψ (k ) , wt ψ ( t m ⊗ a ) = 1 , wt ψ (K n ) = 0 , ψ (D n ) = 2for u ∈ g , m ∈ Z , n ∈ Z × and thatwt v m = wt v − m − v ∈ V, m ∈ Z . It then follows from the action of e t ( g , µ ) o on W that the Z -grading of e t ( g , µ ) o agreeswith the conformal grading. Thus W with the conformal grading is a C -graded e t ( g , µ ) o -module. Consequently, we have e t ( g , µ ) o ( − ) (cid:0) L g ( λ ) ⊗ e α k C [ L ] (cid:1) = 0and L g ( λ ) ⊗ e α k C [ L ] is an irreducible L (0) -module.We now show that L g ( λ ) ⊗ e α k C [ L ] ≃ T ℓ,λ,α,β as an L (0) -module. Recall (5.27)—(5.29). Let u ∈ L g ( λ ) , r ∈ Z . We see that k acts as ℓ k = 0 on V ( H,L ) ( α ), k actsas scalar ℓ on W , d , = d acts as d and d ( u ⊗ e ( α + r ) k ) = u ⊗ h d , ( α + r ) k i e ( α + r ) k = ( α + r )( u ⊗ e ( α + r ) k ) . By (5.24) we have Y ( e m k , z )( u ⊗ e ( α + r ) k ) = u ⊗ E − ( − m k , z ) e ( α + m + r ) k ,Y ( a, z ) Y ( e m k , z )( u ⊗ e ( α + r ) k ) = Y ( a, z ) u ⊗ E − ( − m k , z ) e ( α + m + r ) k for m ∈ Z , a ∈ g . Then( t n k )( u ⊗ e ( α + r ) k ) = ℓ ( e n k ) − ( u ⊗ e ( α + r ) k ) = ℓ ( u ⊗ e ( α + n + r ) k ) , ( t m ⊗ a )( u ⊗ e ( α + r ) k ) = ψ ( t m ⊗ a ) ( u ⊗ e ( α + r ) k ) = au ⊗ e ( α + m + r ) k for n ∈ Z × .Note that d ,n = Res z z D n ( z ) (as D n ( z ) = P m ∈ Z d m,n z − m − ). Recall the actionof D n ( z ) from (5.39). We haveRes z z (cid:18) ddz Y ( n k , z ) (cid:19) Y ( e n k , z )( u ⊗ e ( α + r ) k )= Res z u ⊗ z (cid:18) ddz Y ( n k , z ) (cid:19) E − ( − n k , z ) e ( α + n + r ) k = Res z u ⊗ zE − ( − n k , z ) (cid:18) ddz Y ( n k , z ) (cid:19) e ( α + n + r ) k = 0 , noticing that ( n k ) j e ( α + n + r ) k = 0 for j ≥
0. We also have: Y ( d , z ) Y ( e n k , z ) : ( u ⊗ e ( α + r ) k ) OROIDAL EALA AND VA 39 = X j ≥ u ⊗ d − j z j − E − ( − n k , z ) e ( α + n + r ) k + X j ≥ u ⊗ Y ( e n k , z ) d j z − j − e ( α + r ) k = X j ≥ u ⊗ d − j z j − E − ( − n k , z ) e ( α + n + r ) k + u ⊗ ( α + r ) z − E − ( − n k , z ) e ( α + n + r ) k , from which we get − Res z z ddz : Y ( d , z ) Y ( e n k , z ) : ( u ⊗ e ( α + r ) k )= Res z : Y ( d , z ) Y ( e n k , z ) : ( u ⊗ e ( α + r ) k )= ( α + r )( u ⊗ e ( α + n + r ) k ) . On the other hand, we haveRes z z : Y ( ω, z ) Y ( e n k , z ) : ( u ⊗ e ( α + r ) k )= Res z z X j ≥ z j − L ( − j − Y ( e n k , z )( u ⊗ e ( α + r ) k )+ Res z z X j ≥ Y ( e n k , z ) z − j − L ( j )( u ⊗ e ( α + r ) k )= Res z z X j ≥ z j − L ( − j − (cid:0) u ⊗ E − ( − n k , z ) e ( α + n + r ) k (cid:1) + Res z z X j ≥ z − j − L ( j ) u ⊗ E − ( − n k , z ) e ( α + n + r ) k + Res z z X j ≥ z − j − Y ( e n k , z )( u ⊗ L ( j ) e ( α + r ) k )= L (0) u ⊗ e ( α + n + r ) k = β ( u ⊗ e ( α + n + r ) k ) , noticing that L ( j ) e ( α + r ) k = 0 for j ≥
0. Now, comparing these relations with (5.8)and (5.9), we conclude that C [ q, q − ] ⊗ L g ( λ ) ≃ L g ( λ ) ⊗ e α k C [ L ] as an L (0) -module.Then it follows that W ∼ = L e t ( g ,µ ) o ( T ℓ,λ,α,β ). (cid:3) As a consequence of Proposition 5.7 we have:
Corollary 5.9.
Let ℓ ∈ C × . Then there exists a vertex algebra epimorphism Θ : V b t ( g ,µ ) o ( ℓ ) → V b g ⋊ Vir ( ℓ, µℓ − ⊗ V ( H,L ) , which is uniquely determined by Θ( t m ⊗ u ) = u ⊗ e m k , Θ(k ) = ℓ k , Θ(d ) = d , Θ(K n ) = ℓn e n k , (5.44) Θ(D n ) = nL ( − e n k − L ( − d − e n k ) + n ( µℓ − n k − e n k )(5.45) for u ∈ g , m ∈ Z , n ∈ Z × . Proof.
As in the proof of Theorem 5.8, from Proposition 5.7 , V b g ⋊ Vir ( ℓ, µℓ − ⊗ V ( H,L ) is a b t ( g , µ ) o -module with u ( z ) = Y ( ψ ( u ) , z ) for u ∈ A g + C k (see (5.43)).Note that k ( ⊗ ) = ℓ ( ⊗ ) and b t ( g , µ ) o ( − ) ( ⊗ ) = 0 . It follows that there is a b t ( g , µ ) o -module epimorphismΘ : V b t ( g ,µ ) o ( ℓ ) → V b g ⋊ Vir ( ℓ, µℓ − ⊗ V ( H,L ) such that π ( ) = ⊗ . Furthermore, (5.44) and (5.45) hold. Since A g generates V b t ( g ,µ ) o ( ℓ ) as a vertex algebra, Θ is a vertex algebra homomorphism. (cid:3) On the other hand, we have:
Corollary 5.10.
Let ℓ ∈ C × . Then there exists a e t ( g , µ ) o -module epimorphism Ψ : V b t ( g ,µ ) o ( ℓ ) → L b t ( g ,µ ) o ( T ℓ ) , (5.46) which is uniquely determined by Ψ( ) = 1 ⊗ , and ker Ψ is an ideal of the vertexalgebra V b t ( g ,µ ) o ( ℓ ) . On the other hand, there exists a e t ( g , µ ) o -module isomorphism π : L b t ( g ,µ ) o ( T ℓ ) → L b g ⋊ Vir ( ℓ, µℓ − ⊗ V ( H,L ) , (5.47) which is uniquely determined by π (1 ⊗
1) = ⊗ . Furthermore, π is a vertex algebraisomorphism with L b t ( g ,µ ) o ( T ℓ ) equipped with the quotient vertex algebra structuresuch that Θ = π ◦ Ψ .Proof. From the second part of Theorem 5.8, there exists a e t ( g , µ ) o -module epimor-phism Ψ : V b t ( g ,µ ) o ( ℓ ) → L b t ( g ,µ ) o ( T ℓ ) such that Ψ( ) = 1 ⊗
1. As a e t ( g , µ ) o -submoduleof V b t ( g ,µ ) o ( ℓ ), ker Ψ is an ideal of the vertex algebra V b t ( g ,µ ) o ( ℓ ) from Remark 3.9.Then L b t ( g ,µ ) o ( T ℓ ) is naturally a vertex algebra with 1 ⊗ L b t ( g ,µ ) o ( T ℓ ) is a simple vertex algebra isomorphic to L b g ⋊ Vir ( ℓ, µℓ − ⊗ V ( H,L ) . (cid:3) Remark 5.11.
Assume ℓ = − h ∨ , where h ∨ denotes the dual Coxeter number of g . For the affine vertex operator algebra V b g ( ℓ, ω g and the central charge by c ℓ . Note that V b g ⋊ Vir ( ℓ, c ) contains V b g ( ℓ,
0) naturallyas a vertex subalgebra. Set ω ′ = ω − ω g ∈ V b g ⋊ Vir ( ℓ, c ) . From [FZ], the vertex subalgebra h ω ′ i generated by ω ′ is a vertex operator algebrawith ω ′ as its conformal vector of central charge c − c ℓ . It follows from the P-B-Wtheorem that V b g ⋊ Vir ( ℓ, c ) = V b g ( ℓ, ⊗ h ω ′ i ∼ = V b g ( ℓ, ⊗ V Vir ( c − c ℓ , , (5.48)an isomorphism of vertex operator algebras. Furthermore, from [FHL] we have L b g ⋊ Vir ( ℓ, c ) ∼ = L b g ( ℓ, ⊗ L Vir ( c − c ℓ , . (5.49) OROIDAL EALA AND VA 41
On the other hand, we have L b g ⋊ Vir ( ℓ, c, λ, β ) = L b g ( ℓ, λ ) ⊗ L Vir ( c − c ℓ , β ) , (5.50)where L b g ( ℓ, λ ) is the irreducible highest weight b g -module of level ℓ and L Vir ( c, β )is the irreducible highest weight Vir -module of central charge c .Recall that θ is the highest long root of g with h θ denoting the coroot. We have: Lemma 5.12.
Let λ ∈ P + , ℓ, α, β ∈ C with ℓ = 0 . Then the b t ( g , µ ) o -module L b t ( g ,µ ) o ( T ℓ,λ,α,β ) is integrable if and only if ℓ ∈ Z + and λ ( h θ ) ≤ ℓ . Furthermore, if ℓ ∈ Z + and λ ( h θ ) ≤ ℓ , then ( t m a ) ǫ γ ℓ +1 · (1 ⊗
1) = 0 in L b t ( g ,µ ) o ( T ℓ,λ,α,β ) for m ∈ Z , a ∈ g γ , γ ∈ ∆ .Proof. From [K2], the b g -module L b g ( ℓ, λ ) is integrable if and only if ℓ ∈ Z + and λ ( θ ∨ ) ≤ ℓ . Then the first assertion follows from Proposition 5.8 and (5.50). Thesecond assertion follows immediately from Lemma 3.11. (cid:3) Using Lemma 5.12, we immediately have:
Proposition 5.13.
Let ℓ be a positive integer. Then the vertex algebra epimor-phism Ψ : V b t ( g ,µ ) o ( ℓ ) → L b t ( g ,µ ) o ( T ℓ ) factors through V int b t ( g ,µ ) o ( ℓ ) : V b t ( g ,µ ) o ( ℓ ) ։ V int b t ( g ,µ ) o ( ℓ ) ։ L b t ( g ,µ ) o ( T ℓ ) . (5.51) In other words, Ψ reduces to a vertex algebra epimorphism Ψ int : V int b t ( g ,µ ) o ( ℓ ) → L b t ( g ,µ ) o ( T ℓ ) . (5.52)5.2. Classification of bounded N -graded modules for V b t ( g ,µ ) o ( ℓ ) and V int b t ( g ,µ ) o ( ℓ ) . Throughout this subsection, we assume that ℓ is a nonzero complex number.Recall that d ∈ b t ( g , µ ) o ⊂ e t ( g , µ ) o and d / ∈ e t ( g , µ ) o . Set b t ⋆ ( g , µ ) o = b t ( g , µ ) o ⋊ C d , (5.53)a subalgebra of T ( g , µ ). Set D = C d + C d ⊂ b t ⋆ ( g , µ ) o . (5.54)Following Billig (see [B1, B2]) we formulate the following notion: Definition 5.14. A b t ⋆ ( g , µ ) o -module W is called a D -weight module if D actssemisimply, and a D -weight b t ⋆ ( g , µ ) o -module W is called a bounded D -weight mod-ule if the real parts of the d -eigenvalues are bounded from above and if all D -weight subspaces of W are finite-dimensional. Remark 5.15.
Note that each D -weight b t ⋆ ( g , µ ) o -module with the grading givenby the eigenvalues of − d is naturally a C -graded b t ( g , µ ) o -module. On the otherhand, let W = ⊕ ν ∈ C W ( ν ) be a C -graded b t ( g , µ ) o -module. Then W becomes a b t ⋆ ( g , µ ) o -module with the action of d given by d | W ( ν ) = − ν for ν ∈ C . As-sume that W is an irreducible Z -graded b t ( g , µ ) o -module. As we work on C , it isstraightforward to show that d is semisimple on W , so that W becomes a D -weight b t ⋆ ( g , µ ) o -module. We then define a bounded N -graded b t ( g , µ ) o -module tobe an N -graded b t ( g , µ ) o -module which viewed as a b t ⋆ ( g , µ ) o -module is a bounded D -weight module.The following is the main result of this subsection: Theorem 5.16.
Let ℓ ∈ C × . Then for any λ ∈ P + , α, β ∈ C , L b t ( g ,µ ) o ( T ℓ,λ,α,β ) isan irreducible bounded b t ⋆ ( g , µ ) o -module of level ℓ . Furthermore, every irreduciblebounded b t ⋆ ( g , µ ) o -module of level ℓ is of this form. Before we present the proof of this theorem, we give a consequence. CombiningTheorem 5.8 and Lemma 5.12 with Theorem 5.16 we immediately have:
Corollary 5.17.
Let ℓ be a positive integer. Then all irreducible bounded N -graded V int b t ( g ,µ ) o ( ℓ ) -modules up to equivalence are exactly the irreducible b t ( g , µ ) o -modules L b t ( g ,µ ) o ( T ℓ,λ,α,β ) for α, β ∈ C , λ ∈ P + with λ ( h θ ) ≤ ℓ . Now we proceed to prove Theorem 5.16. The first part follows immediately fromthe explicit realization in Proposition 5.8. Now, assume that W is an irreduciblebounded b t ⋆ ( g , µ ) o -module of level ℓ . Then there is an eigenvalue β of − d on W such that all the eigenvalues of − d on W lie in β + N . For n ∈ Z , set W ( n ) = { u ∈ W | d ( u ) = − ( n + β ) u } . Then W = ⊕ m ∈ N W ( m ) is an irreducible bounded N -graded b t ( g , µ ) o -module where W (0) = 0 and it is an irreducible L (0) -module. For m ∈ Z , β ∈ C , set W ( m,β ) = { w ∈ W ( m ) | d w = βw } . Let α ∈ C such that W (0 ,α ) = 0. Since W irreducible, the eigenvalues of d on W belong to the single Z -coset α + Z of C .First, as an analogue of a result of Billig (see [B1, Lemma 2.1]) we have: Lemma 5.18.
The central element k acts trivially on W .Proof. As W is irreducible, k acts as a scalar, say c ∈ C . Assume c = 0. Takea nonzero vector v ∈ W (0 ,α ) . Note that d − , − n d − ,n v ∈ W (2 ,α ) for n ∈ Z + anddim W (2 ,α ) < ∞ . We claim that d − , − n d − ,n v for n = 1 , , . . . are linearly inde-pendent, which is a contradiction. Assume P n ≥ a n d − , − n d − ,n v = 0 with a n ∈ C .Notice that for r, s ∈ Z , from the Lie bracket relations (3.16) we havek ,r v ( ∈ W ( − ) = 0 , [d − ,r , d − ,s ] = 0 and [k ,r , d − ,s ] = rδ r + s, k . OROIDAL EALA AND VA 43
Using these facts, for m ≥
1, we get0 = k ,m k , − m X n ≥ a n d − , − n d − ,n v ! = a m m c v, which implies a m = 0. This proves the claim. Therefore, we must have c = 0,concluding the proof. (cid:3) Lemma 5.19.
There does not exist a positive integer N such that either W (0 ,α + n ) =0 for n > N or W (0 ,α + n ) = 0 for n < − N .Proof. We here prove that there does not exist N ∈ Z + such that W (0 ,α + n ) = 0 for n > N , while the proof for the other case is similar. Assume by contradiction thatthere exists N ∈ Z + such that W (0 ,α + N ) = 0 and W (0 ,α + n ) = 0 for n > N . Take anonzero vector v ∈ W (0 ,α + N ) . Note that k − ,n k , − n v ∈ W (1 ,α + N ) for n ∈ Z + , wheredim W (1 ,α + N ) < ∞ . We now show that these vectors are linearly independent, toget a contradiction. Assume that P n ≥ a n k − ,n k , − n v = 0 with a n ∈ C . Noticethat d , − m k , − n v ∈ W ( − = 0. On the other hand, from the Lie bracket relations(3.16), using Lemma 5.18 we have[d ,r , k − ,s ] = 2( r + s )k ,r + s + 2 ℓδ r + s, for r, s ∈ Z . Then we getd , − m X n ≥ a n k − ,n k , − n v ! = X n ≥ n − m ) a n k ,n − m k , − n + 2 ℓa m k , − m v (5.55)for m ∈ Z + . Notice that for m, n ∈ Z + with m = n , we havek ,n v ∈ W (0 ,α + N + n ) = 0 , d ,m v = 0 , k ,n − m k ,m − n v (= k ,m − n k ,n − m v ) = 0 . Using this and (3.16) we getd ,m k ,n − m k , − n v = ( n k ,n + k ,n − m d ,m )k , − n v = n k , − n k ,n v + k ,n − m (( m − n )k ,m − n + k , − n d ,m ) v = 0for m, n ∈ Z + with m = n . Then by (5.55) we obtain0 = d ,m d , − m X n ≥ a n k − ,n k , − n v ! = d ,m (2 ℓa m k , − m v ) = a m ℓ v, which implies a m = 0. This proves the linear independence, as desired. (cid:3) Fix a basis { v , . . . , v r } of W (0 ,α ) . For 1 ≤ i ≤ r , m ∈ Z , set v i ( m ) = ℓ ( t m k ) v i ∈ W (0 ,α + m ) . With Lemmas 5.18 and 5.19, it follows from (the proof of) [JM, Theorem3.1] that { v i ( m ) | i = 1 , . . . , r, m ∈ Z } is a basis of W (0) and1 ℓ ( t m k ) v i ( n ) = v i ( m + n ) for 1 ≤ i ≤ r, m, n ∈ Z . From this we see that W (0) is an irreducible jet module for L (0) / C k ∼ = Der C [ t ± ] ⋉ ( C [ t ± ] ⊗ f ) in the sense of [B3]. Then by [B3, Theorem 4] the L (0) -module W (0) is isomorphic to T ℓ,λ,α,β for some λ ∈ P + , α, β ∈ C . Consequently, W as an(irreducible) b t ( g , µ ) o -module is isomorphic to L b t ( g ,µ ) o ( T ℓ,λ,α,β ). This completes theproof of Theorem 5.16.6. Realization of a class of irreducible b t ( g , µ ) -modules In this section, we give a realization of a class of irreducible b t ( g , µ )-modulessimilar to the realization of irreducible b t ( g , µ ) o -modules in Section 5, which recoversa result of [B2]. To achieve this goal, by using a result of Zhu (and Huang) and aresult of Lepowsky, we show that for a general conformal vertex algebra V , thereis a canonical φ -coordinated V -module structure on any V -module W .We begin by defining a notion of a conformal vertex algebra. Roughly speaking,conformal vertex algebras are slight generalizations of vertex operator algebras inthe sense of [FLM] and [FHL]. A conformal vertex algebra is a vertex algebra V equipped with a vector ω , called the conformal vector, such that[ L ( m ) , L ( n )] = ( m − n ) L ( m + n ) + 112 ( m − m ) δ m + n, c for m, n ∈ Z , where Y ( ω, z ) = P n ∈ Z L ( n ) z − n − and c ∈ C , called the centralcharge, and such that V = ⊕ n ∈ Z V ( n ) , where V ( n ) = { v ∈ V | L (0) v = nv } ,Y ( L ( − v, z ) = ddz Y ( v, z ) for v ∈ V, and for every v ∈ V , there exists a positive integer N such that L ( n ) · · · L ( n r ) v = 0for any positive integers n , . . . , n r with r > N . Proposition 6.1.
Suppose that V is a conformal vertex algebra with conformalvector ω of central charge c . Let ( W, Y W ) be any V -module. Then there exists a φ -coordinated V -module structure Y φW ( · , z ) on W such that Y φW ( a, z ) = Y W ( z L (0) a, z ) for a ∈ P ( V ) , (6.1) Y φW ( ω, z ) = z Y W ( ω, z ) − c, (6.2) where P ( V ) = { v ∈ V | L ( n ) v = 0 for all n ≥ } , the space of primary vectors.Proof. Set e ω = ω − c , and for v ∈ V define Y [ v, z ] = Y ( e zL (0) v, e z − . It was proved by Zhu (see [Z], Theorem 4.2.1; cf. [Le1]) that (
V, Y [ · , z ] , , e ω ) is aconformal vertex algebra. Furthermore, it was proved by Zhu and by Huang (see OROIDAL EALA AND VA 45 [Z], Theorem 4.2.2, [H]) that the conformal vertex algebra (
V, Y [ · , z ] , , e ω ) is iso-morphic to ( V, Y, , ω ), where an isomorphism T from ( V, Y, , ω ) to ( V, Y [ · , z ] , , e ω )was constructed explicitly. From [Z], we have T ( ) = , T ( ω ) = e ω = ω − c , and T ( a ) = a for a ∈ P ( V ) . Let (
W, Y W ) be a V -module. For v ∈ V , set X W ( v, z ) = Y W ( z L (0) v, z ) . It was essentially proved by Lepowsky (see [Le2, Le3]; [Li3]) that (
W, X W ) is a φ -coordinated module for the vertex algebra ( V, Y [ · , z ] , ). With the aforementionedvertex algebra isomorphism T , it follows that there is a φ -coordinated V -modulestructure Y φW ( · , z ) on W , where Y φW ( v, z ) = X W ( T ( v ) , z ) = Y W ( z L (0) T ( v ) , z ) for v ∈ V. (6.3)For a ∈ P ( V ), we have Y φW ( a, z ) = Y W ( z L (0) T ( a ) , z ) = Y W ( z L (0) a, z ) . On the other hand, we have Y φW ( ω, z ) = Y W ( z L (0) e ω, z ) = Y W (cid:18) z ω − c , z (cid:19) = z Y W ( ω, z ) − c. This completes the proof. (cid:3)
Let (
V, Y, , ω ) be a conformal vertex algebra of central charge c . Recall theconformal vertex algebra ( V, Y [ · , z ] , , e ω ) defined in the proof of Proposition 6.1.For a ∈ V , write Y [ a, z ] = X m ∈ Z a [ m ] z − m − . (6.4)When a is homogeneous, we have (see [Z]) a [ m ] = Res z Y ( a, z )(log(1 + z )) m (1 + z ) wt a − , (6.5)where log(1 + z ) = X n ≥ ( − n − n z n = z − z + 13 z − z + · · · , (1 + z ) wt a − = X i ≥ (cid:18) wt a − i (cid:19) z i , which are elements of C [[ z ]] ⊂ C (( z )), and (log(1 + z )) − denotes the inverse oflog(1 + z ) in the field C (( z )). We have(log(1 + z )) − = z − (cid:18) z − z + 124 z + O ( z ) (cid:19) , (6.6) (log(1 + z )) − = z − (cid:18) z + 112 z + O ( z ) (cid:19) , (6.7) (log(1 + z )) − (1 + z ) − = z − (cid:18) − z + 512 z − z + O ( z ) (cid:19) , (6.8)and (log(1 + z )) − (1 + z ) − = z − (cid:18) z − z + O ( z ) (cid:19) . (6.9) Remark 6.2.
Here, we list a few cases we need: Case 1 with wt a = 0: a [ −
1] = a − − a + 512 a − a + · · · , (6.10) a [ −
2] = a − + 112 a − a + · · · . (6.11)Case 2 with wt a = 1: a [ −
1] = a − + 12 a − a + 124 a + · · · . (6.12)Case 3 with wt a = 2: a [ −
1] = a − + 32 a + 512 a − a + · · · . (6.13)Note that in the construction of the irreducible module L b g ⋊ Vir ( ℓ, µℓ − , λ, β )for b g ⋊ Vir , with L g ( λ ) replaced by a general irreducible g -module U we still get anirreducible b g ⋊ Vir -module. Denote this module by L b g ⋊ Vir ( ℓ, µℓ − , U, β ). We seethat for any α ∈ C , the tensor product space L b g ⋊ Vir ( ℓ, µℓ − , U, β ) ⊗ V ( H,L ) ( α )is naturally a module for the conformal vertex algebra V b g ⋊ Vir ( ℓ, µℓ − ⊗ V ( H,L ) .We have (cf. [B2]): Theorem 6.3.
Let ℓ, α, β ∈ C with ℓ = 0 and let U be an irreducible g -module.Then the V b g ⋊ Vir ( ℓ, µℓ − ⊗ V ( H,L ) -module L b g ⋊ Vir ( ℓ, µℓ − , U, β ) ⊗ V ( H,L ) ( α ) is an irreducible b t ( g , µ ) -module with ( t m ⊗ u )[ z ] = zY W ( u, z ) Y W ( e m k , z ) , k [ z ] = ℓzY W ( k , z ) , d [ z ] = zY W ( d , z ) , K n [ z ] = ℓn Y W ( e n k , z ) , D n [ z ] = nz : Y W ( ω, z ) Y W ( e n k , z ) : + nz ddz Y W ( e n k , z ) − ncY W ( e n k , z ) − z ddz z : Y W ( d , z ) Y W ( e n k , z ) : + n ( ℓµ − (cid:18) z ddz Y W ( e n k , z ) (cid:19) Y W ( e n k , z ) for m ∈ Z , u ∈ g , n ∈ Z × . OROIDAL EALA AND VA 47
Proof.
Set V = V b g ⋊ Vir ( ℓ, µℓ − ⊗ V ( H,L ) , W = L b g ⋊ Vir ( ℓ, µℓ − , U, β ) ⊗ V ( H,L ) ( α ) . With W a V -module, by Proposition 6.1, ( W, Y φW ) is a φ -coordinated V -module.Recall from Corollary 5.9 the vertex algebra homomorphism Θ : V b t ( g ,µ ) o ( ℓ ) → V ,where Θ( t m ⊗ u ) = u ⊗ e m k , Θ(k ) = ℓ k , Θ(d ) = d , Θ(K n ) = ℓn e n k , Θ(D n ) = nL ( − e n k − L ( − d − e n k ) + n ( µℓ − n k − e n k )for u ∈ g , m ∈ Z , n ∈ Z × . Then W becomes a φ -coordinated V b t ( g ,µ ) o ( ℓ )-modulevia Θ. Furthermore, by Theorem 4.16 W is a b t ( g , µ )-module of level ℓ with v [ z ] = Y φW (Θ( v ) , z ) = Y W ( z L (0) T Θ( v ) , z ) for v ∈ A g ⊂ V b t ( g ,µ ) o ( ℓ ) , where T is an isomorphism from ( V, Y, , ω ) to ( V, Y [ · , z ] , , e ω ), described in theproof of Proposition 6.1. Note that u ⊗ e m k , k , d , e n k ∈ P ( V ) with wt ( u ⊗ e m k ) = wt k = wt d = 1 , wt e n k = 0for u ∈ g , m ∈ Z , n ∈ Z × as above, so we have T ( u ⊗ e m k ) = u ⊗ e m k , T ( k ) = k , T ( d ) = d , T ( e n k ) = e n k and ( t m ⊗ u )[ z ] = Y W ( z L (0) ( u ⊗ e m k ) , z ) = zY W ( u, z ) Y W ( e m k , z ) , k [ z ] = ℓzY W ( k , z ) , d [ z ] = zY W ( d , z ) , K n [ z ] = ℓn Y W ( e n k , z ) . As e n k ∈ P ( V ) and wt e n k = 0, by Remark 6.2 Case 3, we have ω [ − e n k = ω − e n k + 32 ω e n k + 512 ω e n k + · · · = L ( − e n k + 32 L ( − e n k , so T ( ω − e n k ) = T ( ω )[ − T ( e n k ) = ( ω − c )[ − e n k = ω [ − e n k − ce n k = L ( − e n k + 32 L ( − e n k − ce n k . Then Y φW ( ω − e n k , z ) = Y W ( z L (0) T ( ω − e n k ) , z )= Y W (cid:18) z L ( − e n k + 32 zL ( − e n k − ce n k , z (cid:19) = z Y W ( ω − e n k , z ) + 32 z ddz Y W ( e n k , z ) − cY W ( e n k , z ) . (6.14) As d , e n k ∈ P ( V ), wt d = 1, wt e n k = 0, and d i e n k = nδ i, e n k for i ≥
0, we have d [ − b = d − b + 12 d b − d b + 124 d b + · · · = d − b + 12 nb. Furthermore, we have Y φW ( d − e n k , z ) = Y W ( z L (0) T ( d − e n k ) , z ) = Y W ( z L (0) d [ − e n k , z )= zY W ( d − e n k , z ) + 12 nY W ( e n k , z ) , so Y φW ( L ( − d − e n k , z ) = z ddz Y φW ( d − e n k , z )(6.15) = z ddz zY W ( d − e n k , z ) + 12 nz ddz Y W ( e n k , z ) . Set a, b = e n k ∈ P ( V ), where wt a = wt b = 0 and a i b = 0 for i ≥
0. Then a [ − b = a − b + 112 a b − a b + · · · = a − b, and Y φW ( a − b, z ) = Y W ( z L (0) T ( a − b ) , z ) = Y W ( z L (0) a [ − b, z )(6.16) = zY W ( a − b, z ) = (cid:18) z ddz Y W ( a, z ) (cid:19) Y W ( b, z ) . Using all the facts above, for n ∈ Z × we obtain D n [ z ]= nY φW ( ω − e n k , z ) − Y φW ( L ( − d − e n k , z ) + n ( ℓµ − Y φW (( n k ) − e n k , z )= nz Y W ( ω − e n k , z ) + 32 nz ddz Y W ( e n k , z ) − ncY W ( e n k , z ) − z ddz zY W ( d − e n k , z ) − nz ddz Y W ( e n k , z )+ n ( ℓµ − (cid:18) z ddz Y W ( e n k , z ) (cid:19) Y W ( e n k , z )= nz : Y W ( ω, z ) Y W ( e n k , z ) : + nz ddz Y W ( e n k , z ) − ncY W ( e n k , z ) − z ddz z : Y W ( d , z ) Y W ( e n k , z ) : + n ( ℓµ − (cid:18) z ddz Y W ( e n k , z ) (cid:19) Y W ( e n k , z ) . This completes the proof. (cid:3)
OROIDAL EALA AND VA 49
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Email address : [email protected] Department of Mathematical Sciences, Rutgers University, Camden, NJ 08102,USA
Email address : [email protected] School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China
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