Algebra of q-difference operators, affine vertex algebras, and their modules
aa r X i v : . [ m a t h . QA ] J a n Algebra of q -difference operators, affine vertexalgebras, and their modules Hongyan Guo School of Mathematics and Statistics, Central China Normal University,Wuhan 430079, China
Abstract
In this paper, we explore a canonical connection between the algebra of q -difference operators e V q , affine Lie algebra and affine vertex algebras associatedto certain subalgebra A of the Lie algebra gl ∞ . We also introduce and studya category O of e V q -modules. More precisely, we obtain a realization of e V q asa covariant algebra of the affine Lie algebra c A ∗ , where A ∗ is a 1-dimensionalcentral extension of A . We prove that restricted f V q - modules of level ℓ cor-respond to Z -equivariant φ -coordinated quasi-modules for the vertex algebra V e A ( ℓ , e A is a generalized affine Lie algebra of A . In the end, weshow that objects in the category O are restricted f V q -modules, and we classifysimple modules in the category O . The algebra of q -difference operators, also named q -analog Virasoro-like algebra, hasa lot of background. It can be viewed as a q -deformation of Virasoro-like algebra(cf. [8]). It is the (2-dimensional) universal central extension of the Lie algebraof inner derivations of the quantum torus C q [ x ± , y ± ] (cf. [1], [8], etc.). It canalso be viewed as a Lie algebra whose universal enveloping algebra is the toroidalalgebra which correspond to the quantum toroidal gl algebra U q,t ( ¨ gl ) at q = t (cf. [1], [2], etc.). Certain deformation of the universal enveloping algebra of thealgebra of q -difference operators has been introduced and studied in [18], and it hasa connection with the deformed Virasoro algebra and W N algebras. The structuretheory and representation theory of the algebra of q -difference operators has beenwidely studied ([2], [15], [16], [17], [19], to name a few.).More explicitly, the algebra of q -difference operators e V q is a Lie algebra spannedby the elements E k,l , c , c , where ( k, l ) ∈ Z , c and c are central elements, withLie bracket[ E k,l , E r,s ] = ( q rl − sk − q sk − rl ) E k + r,l + s + δ k, − r δ l, − s ( kc + lc ) , for ( k, l ) , ( r, s ) ∈ Z . We add E , = 0 in the definition. This paper is twofold.On the one hand, we study the algebra of q -difference operators from the vertexalgebras points of view. We build a connection between restricted modules of the Partially supported by NSFC (No.11901224) and NSF of Hubei Province (No.2019CFB160) q -difference operators and certain modules of generalized affine vertexalgebras. On the other hand, we study a category of e V q -modules which are locallyfinite with respect to some subalgebra of e V q . Modules in the category are proved tobe restricted e V q -modules and simple modules are classified.Constructing vertex algebras from Lie algebras and build a connection betweenthe representations of the two algebraic objects is an important subject in both Liealgebra and vertex algebra areas. For certain Lie algebras L , like Virasoro algebra,Heisenberg algebra and affine Lie algebras, etc., the corresponding vertex algebras(denote by V L ) are the vacuum modules of L , and restricted L -modules at fixedlevels are in one-to-one correspondence to V L -modules (cf. [3], [9], etc.). However,there are Lie algebras (by abuse of notation, we still denote by L ) whose restrictedmodules are not related to the vertex algebras V L , but vertex algebras constructedfrom other Lie algebras (cf. [5], [6], [14], etc.). And restricted L -modules are notcorrespond to vertex algebra modules, but rather quasi modules or (equivariant) φ -coordinated (quasi) modules which are introduced and studied in a series of papers([10], [11], [12], [13], etc.).Affine Kac-Moody algebra b g is the 1-dimensional universal central extensionof the loop algebra (also called current algebra) g ⊗ C [ t, t − ], where g is a finite-dimensional simple Lie algebra. Affine vertex algebras V b g ( ℓ,
0) are constructed fromaffine Lie algebras (with g possibly infinite-dimensional). The relations betweenthe representations of affine Lie algebra b g and the representations of affine vertexalgebras V b g ( ℓ,
0) are investigated by many authors (cf. [3], [9], [11], etc.). In trying torelate the algebra of q -difference operators with vertex algebras and their modules,we need to consider the following generalized affine Lie algebras.Let g be a (possibly infinite-dimensional) Lie algebra equipped with a symmetricinvariant bilinear form h , i . Suppose that ψ : g × g −→ C is a 2-cocycle on g .Consider the following generalized affine Lie algebra e g = g ⊗ C [ t, t − ] ⊕ C K ⊕ C K with Lie bracket[ a ⊗ t i , b ⊗ t j ] = [ a, b ] ⊗ t i + j + ψ ( a, b ) δ i + j +1 , K + i h a, b i δ i + j, K , where K , K are central elements, a, b ∈ g , i, j ∈ Z . It is easy to see that e g canalso be viewed as a 1-dimensional central extension of the (usual) affine Lie algebra b g = g ⊗ C [ t, t − ] ⊕ C K (see Remark 2.2). The corresponding vertex algebras V e g ( ℓ ,
0) are straightforward to construct. Restricted modules of the Lie algebra e g and modules of the vertex algebra V e g ( ℓ ,
0) are closely related. The relations ofthe generalized affine vertex algebra V e g ( ℓ ,
0) and affine vertex algebra V b g ( ℓ ,
0) arediscussed in this paper.Let Γ be a group of automorphisms of g , preserving the bilinear form and sat-isfying [ ga, b ] = 0 = h ga, b i for all but finitely many g ∈ Γ. Let χ : Γ −→ C × be agroup homomorphism. The covariant algebra b g [Γ] which is a quotient space of the2ffine Lie algebra b g has been construct and studied in [11] (see also [4]). b g [Γ] is ageneralization of twisted affine Lie algebras. If χ is injective, then restricted mod-ules of level ℓ for b g [Γ] correspond to quasi-modules for V b g ( ℓ,
0) viewed as a Γ-vertexalgebra (Theorem 4.9 of [11]).In the study of q -Virasoro algebra and trigonometric Lie algebras, certain Liealgebra A plays an important role (cf. [5], [6], [14]). More specifically, A is asubalgebra of the Lie algebra gl ∞ which is spanned by the basis elements (the el-ementary matrices) E m,n with m, n ∈ Z , m + n ∈ Z . Rewrite the basis elementsas G α,m = E m + α,m − α for α, m ∈ Z . Then A is spanned by G α,m , α, m ∈ Z . Let α, β, m, n ∈ Z . There is a natural symmetric invariant bilinear form on A : h G α,m , G β,n i = δ α + β, δ m − n, . Consider the 2-cocycle ψ on A defined by ψ ( G α,m , G β,n ) = αδ α + β, δ m − n, . Even though ψ is proved to be a trivial 2-cocycle (Proposition 4.3), in order to relatethe algebra of q -difference operators e V q with vertex algebras we need to consider this2-cocycle. We first show that e V q is isomorphic to a covariant algebra of the affine Liealgebra c A ∗ , where A ∗ is the 1-dimensional central extension of A via the 2-cocycle ψ . Then, for any ℓ , ℓ ∈ C , we establish an equivalence between the category ofrestricted e V q -modules of level ℓ (means that c , c act as scalars ℓ , ℓ respectively)and the category of Z -equivariant φ -coordinated quasi modules for the vertex algebra V e A ( ℓ , t be any positive integer. Consider the subalgebra e V q ( t ) = X k ∈ Z ,l ≥ t C E k,l . To study the restricted modules for the algebra of q -difference operators, we intro-duce a category O of e V q -modules. It is a subcategory of modules with the propertythat they are locally finite e V q ( t ) -modules for some t . We prove that objects in thecategory O are restricted e V q -modules.Finally, we classify simple modules in the category O . Let b = M l ≥ ,k ∈ Z C E k,l ⊕ C c ⊕ C c . Consider the induced moduleInd ℓ ( V ) = U ( e V q ) ⊗ U ( b ) V, where V is a b -module with c , c act as ℓ , ℓ . We first show that if V is a simple b -module, then under certain conditions Ind ℓ ( V ) is a simple e V q -module. Then we3rove that simple modules in the category O are either highest weight modules orinduced modules Ind ℓ V for some simple b -module V .This paper is organized as follows. In Section 2, we study generalized affine Lie al-gebras and the corresponding affine vertex algebras as well as their modules. In Sec-tion 3, we give a realization of the algebra of q -difference operators e V q in terms of thecovariant algebra of the affine Lie algebra c A ∗ . In Section 4, we show that restricted e V q -module category of level ℓ is equivalent to the Z -equivariant φ -coordinated quasimodule category for the generalized affine vertex algebra V e A ( ℓ , O of e V q -modules. We prove that all the modulesin the category O are restricted e V q -modules. We also give a classification of simplemodules in the category O .Throughout this paper, we denote by Z , N , Z + and C the sets of integers,nonnegative integers, positive integers and complex numbers respectively. In this section, we construct and study certain generalized affine Lie algebras e g andthe corresponding generalized affine vertex algebras V e g ( ℓ , V e g ( ℓ ,
0) and affine vertex algebra V b g ( ℓ , Definition 2.1.
Let g be a Lie algebra. A ψ : g × g −→ C on g is askew-symmetric bilinear form satisfying ψ ( x, [ y, z ]) + ψ ( y, [ z, x ]) + ψ ( z, [ x, y ]) = 0for any x, y, z ∈ g .Let g be a (possibly infinite-dimensional) Lie algebra over the complex numberfield C . Let h , i : g × g −→ C be a symmetric invariant bilinear form on g . Supposethat ψ : g × g −→ C is a 2-cocycle on g .With respect to (cid:16) g , h , i , ψ (cid:17) , there are the following three ways that we can getaffine-type Lie algebras:(1) The affine Lie algebra b g = g ⊗ C [ t, t − ] ⊕ C K with Lie bracket[ a ⊗ t i , b ⊗ t j ] = [ a, b ] ⊗ t i + j + i h a, b i δ i + j, K , (2.1)where K is a central element, a, b ∈ g , i, j ∈ Z .(2) Let g ∗ = g ⊕ C K be the 1-dimensional central extension of g by K via ψ .For a, b ∈ g , λ, µ ∈ C , the Lie bracket on g ∗ is given by[ a + λK , b + µK ] = [ a, b ] + ψ ( a, b ) K . (2.2)The bilinear form h , i on g can be naturally extended to a symmetric invariantbilinear form on g ∗ with h g ∗ , K i = h K , g ∗ i = 0. Then we have the affine Lie4lgebra b g ∗ = g ∗ ⊗ C [ t, t − ] ⊕ C K with Lie bracket[ a ∗ ⊗ t i , b ∗ ⊗ t j ] = [ a ∗ , b ∗ ] ⊗ t i + j + i h a ∗ , b ∗ i δ i + j, K , (2.3)where K is a central element, a ∗ , b ∗ ∈ g ∗ , i, j ∈ Z . Clearly, K ⊗ C [ t, t − ] isalso in the center of the affine Lie algebra b g ∗ .(3) The generalized affine Lie algebra e g = g ⊗ C [ t, t − ] ⊕ C K ⊕ C K with Liebracket[ a ⊗ t i , b ⊗ t j ] = [ a, b ] ⊗ t i + j + ψ ( a, b ) δ i + j +1 , K + i h a, b i δ i + j, K , (2.4)where K , K are central elements, a, b ∈ g , i, j ∈ Z . Remark 2.2. (1) The 2-cocycle ψ can be extended to a 2-cocycle ψ on b g with ψ ( a ⊗ t i , b ⊗ t j ) = ψ ( a, b ) δ i + j +1 , ,ψ ( b g , K ) = ψ ( K , b g ) = 0for a, b ∈ g , i, j ∈ Z . Then e g can be viewed as a 1-dimensional centralextension of the affine Lie algebra b g by K via ψ . More precisely, we canwrite e g = b g ⊕ C K with Lie bracket[ b a + λK , b b + µK ] = [ b a, b b ] + ψ ( b a, b b ) K , where b a, b b ∈ b g , λ, µ ∈ C . Note that b g may not be a subalgebra of e g .(2) It is easy to see that b g ∗ / ( K ⊗ C [ t, t − ]) ∼ = b g and e g / C K ∼ = b g as Lie algebras.In the following, we consider the (general) affine Lie algebra e g . For a ∈ g , forma generating function a ( x ) = X n ∈ Z a ( n ) x − n − , where a ( n ) stands for a ⊗ t n . The defining relation (2.4) of e g can be written as[ a ( x ) , b ( x )] = [ a, b ]( x ) x − δ (cid:16) x x (cid:17) + ψ ( a, b ) x − δ (cid:16) x x (cid:17) K + h a, b i ∂∂x x − δ (cid:16) x x (cid:17) K (2.5)for a, b ∈ g .Let ℓ , ℓ be complex numbers. View C as a module for the subalgebra g ⊗ C [ t ] + C K + C K of e g with g ⊗ C [ t ] acting trivially and with K , K acting as scalars ℓ , ℓ respectively. Then form an induced module V e g ( ℓ ,
0) = U ( e g ) ⊗ U ( g ⊗ C [ t ]+ C K + C K ) C , U ( · ) denotes the universal enveloping algebra of a Lie algebra.Set = 1 ⊗ ∈ V e g ( ℓ ,
0) and identify g as a subspace of V e g ( ℓ ,
0) through linearmap g −→ V e g ( ℓ , a a ( − . Then there is a unique vertex algebra structure on ( V e g ( ℓ , , Y, ) given by Y ( a, x ) = a ( x ) for a ∈ g . Furthermore, V e g ( ℓ ,
0) is a Z -graded vertex algebra: V e g ( ℓ ,
0) = a n ≥ V e g ( ℓ , ( n ) , (2.6)where V e g ( ℓ , ( n ) is spanned by the vectors a ( − m ) · · · a r ( − m r ) for r ≥ a i ∈ g , m i ≥ m + · · · + m r = n .For a vector space W , set E ( W ) = Hom( W, W (( x ))) ⊂ (End W )[[ x, x − ]] . Definition 2.3.
We say that a e g -module W is restricted if for any w ∈ W , a ∈ g , a ( n ) w = 0 for n sufficiently large, or equivalently, a ( x ) ∈ E ( W ). We say W is of level ℓ if the central element K i acts as scalar ℓ i , i = 1 , Theorem 2.4.
Let ℓ , ℓ be any complex numbers. Let W be any restricted e g -moduleof level ℓ . Then there exists a unique V e g ( ℓ , -module structure on W such thatfor a ∈ g , Y W ( a, x ) = a W ( x )(= P n ∈ Z a ( n ) x − n − ) . Conversely, any module W for the vertex algebra V e g ( ℓ , is naturally a restricted e g -module of level ℓ , with a W ( x ) = Y W ( a, x ) for a ∈ g . Furthermore, the V e g ( ℓ , -submodules of W coincidewith the e g -submodules of W . For affine Lie algebras b g and b g ∗ , we denote by V b g ( ℓ ,
0) and V b g ∗ ( ℓ ,
0) their cor-responding affine vertex algebras with K acts as ℓ (cf. [3], [9], etc.).Let J b g ( ℓ , J b g ∗ ( ℓ , J e g ( ℓ ,
0) be the maximal (two-sided) ideals of the ver-tex algebras V b g ( ℓ , V b g ∗ ( ℓ , V e g ( ℓ ,
0) respectively. Denote by L e g ( ℓ ,
0) = V e g ( ℓ , /J e g ( ℓ , L b g ∗ ( ℓ ,
0) = V b g ∗ ( ℓ , /J b g ∗ ( ℓ , L b g ( ℓ ,
0) = V b g ( ℓ , /J b g ( ℓ , V b g ( ℓ , V b g ∗ ( ℓ , V e g ( ℓ ,
0) as well as their simple quotients.
Remark 2.5.
There is a natural surjective vertex algebra homomorphism ϕ : V b g ∗ ( ℓ , −→ V b g ( ℓ ,
0) with K ( − m ) m ≥
1. Then Ker( ϕ ) = h K ( − m ) , m ≥ i . It induces a vertex algebra isomorphism V b g ∗ ( ℓ , / h K ( − m ) , m ≥ i ∼ = V b g ( ℓ , . (2.7)And L b g ∗ ( ℓ , ∼ = L b g ( ℓ ,
0) (2.8)as simple vertex algebras, for any ℓ ∈ C .6learly, if ℓ = 0, then V e g ( ℓ ,
0) = V b g ( ℓ ,
0) and L e g ( ℓ ,
0) = L b g ( ℓ , Definition 2.6.
A 2-cocycle ψ : g × g −→ C is said to be trivial if there exists alinear map µ : g −→ C such that for any x, y ∈ g , ψ ( x, y ) = µ ([ x, y ]) . Proposition 2.7.
Suppose that ψ : g × g −→ C is a trivial 2-cocycle. Then g ∗ ∼ = g ⊕ C K and e g ∼ = b g ⊕ C K as Lie algebras.Proof. Let µ : g −→ C be a linear map such that ψ ( x, y ) = µ ([ x, y ]) for any x, y ∈ g .It is easy to check that f : g ⊕ C K −→ g ∗ , ( a, λK ) a + ( µ ( a ) + λ ) K , a ∈ g , λ ∈ C , is a Lie algebra isomorphism.Note that the 2-cocycle ψ in Remark 2.2 then is also a trivial 2-cocycle. Let µ : b g −→ C be a linear map defined by µ ( a ⊗ t i + λK ) = µ ( a ) δ i +1 , for any a ∈ g , i ∈ Z , λ ∈ C . Then ψ ( b a, b b ) = µ ([ b a, b b ]) for any b a, b b ∈ b g . Similarly, f : b g ⊕ C K −→ e g , ( b a, λK ) b a + ( µ ( b a ) + λ ) K , b a ∈ b g , λ ∈ C , is a Lie algebraisomorphism.Then we can get the following assertion. Proposition 2.8.
Let ℓ be any nonzero complex number. If ψ is a trivial 2-cocycleon g . Then we have V e g ( ℓ , ∼ = V b g ( ℓ , as vertex algebras, for any ℓ ∈ C . Hence also L e g ( ℓ , ∼ = L b g ( ℓ , for any ℓ ∈ C .Proof. V b g ( ℓ ,
0) is naturally a b g -module. It can be viewed as a b g ⊕ C K -modulewith K acts as a scalar ℓ . Explicitly, ( b a, λK ) .w = b a.w + λℓ w for b a ∈ b g , λ ∈ C , w ∈ V b g ( ℓ , V b g ( ℓ ,
0) is then a e g -module with a ⊗ t i acts as( a ⊗ t i , − µ ( a ) δ i +1 , K ), K acts as ℓ , K acts as ℓ , for any a ∈ g , i ∈ Z .Let ϕ : V e g ( ℓ , −→ V b g ( ℓ ,
0) be a e g -module homomorphism with ϕ ( ) = .Then ϕ ( a ( − ) = a ( − . = a ( − − µ ( a ) ℓ for any a ∈ g . It is straightforward to check that ϕ ( a n b ) = ϕ ( a ) n ϕ ( b ) for any a, b ∈ g , n ∈ Z (note that n b = δ n +1 , b ). Hence ϕ is a vertex algebra homomorphism.Similarly, V e g ( ℓ ,
0) can be viewed as a b g -module with K acts as ℓ , a ⊗ t i actsas a ⊗ t i + µ ( a ) δ i +1 , ℓ for any a ∈ g , i ∈ Z . Let Φ : V b g ( ℓ , −→ V e g ( ℓ ,
0) be a b g -module homomorphism with Φ( ) = . Then ϕ is a vertex algebra isomorphismwith inverse Φ. 7t the end of this section, we give two examples. The second example plays animportant role in our study of the algebra of q -difference operators. Example 2.9. g = gl ∞ is the Lie algebra of doubly infinite complex matrices withonly finitely many nonzero entries under the usual commutator bracket. For any m, n ∈ Z , denote by E m,n the matrix whose only nonzero entry is the ( m, n )-entrywhich is 1. Then E m,n , m, n ∈ Z , form a bases of gl ∞ . Let m, n, p, q ∈ Z . The Liebracket on gl ∞ is given by[ E m,n , E p,q ] = δ n,p E m,q − δ q,m E p,n . gl ∞ is equipped with a natural symmetric invariant bilinear form: h E m,n , E p,q i = trace( E m,n E p,q ) = δ m,q δ n,p . There is a 2-cocycle ψ on the Lie algebra gl ∞ defined by ψ ( E m,n , E n,m ) = 1 = − ψ ( E n,m , E m,n ) if m ≤ n ≥ ,ψ ( E m,n , E p,q ) = 0 otherwise . The corresponding 1-dimensional central extension gl ∗∞ and its restricted moduleshas been related to certain quantum vertex algebras and their φ -coordinated modulesin [7]. Example 2.10.
Let g = A := span { E m,n | m + n ∈ Z } . Then A is a subalgebra of gl ∞ . For α, m ∈ Z , denote by G α,m = E m + α,m − α ∈ A . Then A = span { G α,m | α, m ∈ Z } with Lie bracket[ G α,m , G β,n ] = δ α + β,m − n G α + β,α + n − δ α + β,n − m G α + β,n − α (2.9)for α, β, m, n ∈ Z . The symmetric invariant bilinear form on gl ∞ in Example 2.9restricted to A is a symmetric invariant bilinear form on A with h G α,m , G β,n i = δ α + β, δ m − n, , for α, β, m, n ∈ Z . Let ψ : A × A −→ C be a bilinear map defined by ψ ( G α,m , G β,n ) = αδ α + β, δ m − n, . It is straightforward to check that ψ is a 2-cocycle. The Lie algebra A , vertexalgebras related to b A and their (equivariant) quasi modules are in connection withrestricted modules of certain Lie algebras in [5], [6], [14], etc.The importance and significance of vertex algebras V c A ∗ ( ℓ ,
0) and V e A ( ℓ ,
0) aswell as their certain module categories show up in the study of the algebra of q -difference operators as we will see in the following two sections.8 Affine Lie algebra c A ∗ and the algebra of q -differenceoperators In this section, we first review the algebra of q -difference operators e V q (cf. [2], [8],etc.). Then we show that e V q is isomorphic to the Z -covariant algebra of c A ∗ . At last,we show that restricted e V q -modules of level ℓ with ℓ = 0 are one-to-one correspondto equivariant quasi modules for the affine vertex algebra V c A ∗ ( ℓ , q ∈ C × be generic, i.e. q is not a root of unity. The algebra of q -differenceoperators is the universal central extension of the Lie algebra of inner derivations ofthe quantum torus C q [ x ± , y ± ] ([8]). We rewrite its definition (and add E , = 0)in the following way. Definition 3.1.
The algebra of q -difference operators e V q is a Lie algebra spannedby E k,l , c , c , where ( k, l ) ∈ Z , c and c are central elements, with Lie bracket[ E k,l , E r,s ] = ( q rl − sk − q sk − rl ) E k + r,l + s + δ k, − r δ l, − s ( kc + lc ) , (3.1)for ( k, l ) , ( r, s ) ∈ Z . We first give a characterization of e V q as certain covariant algebra of the affineLie algebra c A ∗ .For any r ∈ Z , define σ r ( G α,m ) = G α,m + r , σ r ( K ) = K . It is easy to see that σ r is an automorphism of A ∗ , for any r ∈ Z . Let Γ = { σ r | r ∈ Z } . Then Γ ∼ = Z , and Γ preserves h , i . Γ extends canonically to an automorphismgroup of c A ∗ and furthermore to an automorphism group of the Z -graded vertexalgebra V c A ∗ ( ℓ , χ q : Z −→ C × ; r q r for r ∈ Z . We have (cf. [4], [11]):
Proposition 3.2.
The algebra of q -difference operators e V q is isomorphic to the ( Z , χ q ) -covariant algebra c A ∗ [ Z ] of the affine Lie algebra c A ∗ with c = K ⊗ , c = K , and E α,m = G α, ⊗ t m for α, m ∈ Z . Proof.
From the definition, e V q has a basis { E α,m , c , c | α, m ∈ Z } with the givenbracket relations. On the other hand, from Proposition 4.4 of [11], c A ∗ [ Z ] as a vectorspace is the quotient space of c A ∗ , modulo the subspace linearly spanned by σ r ( a ) ⊗ t m − q − mr ( a ⊗ t m ) for a ∈ A ∗ , r, m ∈ Z , a ⊗ t m , b ⊗ t n ] Γ = X r ∈ Z q mr (cid:16) [ σ r ( a ) , b ] ⊗ t m + n + m h σ r ( a ) , b i δ m + n, K (cid:17) , (3.2)where a ⊗ t m denotes the image of a ⊗ t m in c A ∗ [ Z ] for a ∈ A ∗ , m ∈ Z , and K isidentified with its image. Note that we have ( q is generic) G α + β, − α ⊗ t m + n = q ( m + n ) α G α + β, ⊗ t m + n , (3.3) G α + β,α ⊗ t m + n = q − ( m + n ) α G α + β, ⊗ t m + n , (3.4) K ⊗ t m + n = δ m + n, K ⊗ . (3.5)We see that K , K ⊗ G α, ⊗ t m ( α, m ∈ Z ) form a bases of c A ∗ [ Z ]. Let α, β, m, n ∈ Z , then[ G α, ⊗ t m , G β, ⊗ t n ] Γ = q m ( α + β ) G α + β,α ⊗ t m + n − q − m ( α + β ) G α + β, − α ⊗ t m + n + αδ α + β, K ⊗ t m + n + mδ α + β, δ m + n, K = ( q mβ − nα − q nα − mβ ) G α + β, ⊗ t m + n + αδ α + β, δ m + n, K ⊗ mδ α + β, δ m + n, K . (3.6)It follows that e V q is isomorphic to c A ∗ [ Z ] with E α,m corresponding to G α, ⊗ t m for α, m ∈ Z , c corresponding to K ⊗ c corresponding to K .For k ∈ Z , form a generating function E k ( x ) = X l ∈ Z E k,l x − l − . Then the defining relation (3.1) can be written as[ E k ( x ) , E r ( x )]= q k E k + r ( q k x ) x − δ (cid:16) q k + r x x (cid:17) − q − k E k + r ( q − k x ) x − δ (cid:16) q − k − r x x (cid:17) + kδ k, − r x − x − δ (cid:16) x x (cid:17) c + δ k, − r ∂∂x x − δ (cid:16) x x (cid:17) c (3.7)for k, r ∈ Z . Definition 3.3. A e V q -module W is said to be restricted if for any w ∈ W , k ∈ Z , E k,l w = 0 for l sufficiently large, or equivalently, E k ( x ) ∈ E ( W ) for any k ∈ Z .We say a e V q -module W is of level ℓ if the central elements c , c act as scalars ℓ , ℓ ∈ C . 10 emark 3.4. Let V be a Z -graded vertex algebra. Denote by L (0) the degreeoperator on V . Let Γ be an automorphism group of the Z -graded vertex algebra V and let χ : Γ → C × be a group homomorphism. Then V becomes a Γ-vertex algebrawith R g = χ ( g ) − L (0) g for g ∈ Γ (see [10], [11] for the details).By Remark 3.4, V c A ∗ ( ℓ ,
0) becomes a Γ-vertex algebra with Γ = Z and R r = q − rL (0) σ r for r ∈ Z .Now we give a connection between affine vertex algebra V c A ∗ ( ℓ ,
0) and restricted e V q -modules. Theorem 3.5.
Assume that q is not a root of unity and let ℓ ∈ C . Then for anyrestricted e V q -module W of level ℓ with ℓ = 0 , there exists an equivariant quasi V c A ∗ ( ℓ , -module structure Y W ( · , x ) on W , which is uniquely determined by Y W ( K , x ) = 0 , Y W ( G α,m , x ) = q m E α ( q m x ) for α, m ∈ Z . (3.8) On the other hand, for any equivariant quasi V c A ∗ ( ℓ , -module ( W, Y W ) such that Y W ( K , x ) = 0 , W becomes a restricted e V q -module of level ℓ with ℓ = 0 and E α ( x ) = Y W ( G α, , x ) for α ∈ Z . (3.9) Proof.
Let W be a restricted e V q -module of level ℓ with c acts as 0. In view ofProposition 3.2, W is a restricted c A ∗ [ Z ]-module with K ⊗ K acts as ℓ , and with E α,m = G α, ⊗ t m for α, m ∈ Z . Note that q is not a root of unity, χ q : Z −→ C × is one-to-one. By Theorem 4.9of [11], there exists an quasi V c A ∗ ( ℓ , Y W ( · , x ) on W , which isuniquely determined by Y W ( K , x ) = K ( x )(= X n ∈ Z K ⊗ t n x − n − = K ⊗ x − ) = 0 ,Y W ( G α,m , x ) = G α,m ( x )(= X n ∈ Z G α,m ⊗ t n x − n − ) for α, m ∈ Z . For α, m ∈ Z , we have Y W ( G α,m , x ) = G α,m ( x ) = σ m ( G α, )( x ) = q m G α, ( q m x ) = q m E α ( q m x ) . Hence Y W ( σ r ( G α,m ) , x ) = Y W ( q r G α,m , q r x ) for any r ∈ Z . Consequently, there eixts an equivariant quasi V c A ∗ ( ℓ , Y W ( · , x )on W such that Y W ( K , x ) = 0 , Y W ( G α,m , x ) = q m E α ( q m x ) for α, m ∈ Z . The other direction follows from Proposition 3.2, and Theorem 4.9 of [11].
Remark 3.6.
Theorem 3.5 holds for all ℓ ∈ C if we require a less natural conditionthat Y W ( K , x ) = ℓ x − (so that K ⊗ ℓ ).11 Vertex algebra V e A ( ℓ , and e V q In order to naturally relate restricted e V q -modules of level ℓ for any complex num-bers ℓ , ℓ to vertex algebras and their corresponding modules, in this section,we consider φ -coordinated quasi modules for vertex algebras. More precisely, weshow that restricted e V q -module of level ℓ are in one-to-one correspondence to Z -equivariant φ -coordinated quasi modules for the vertex algebra V e A ( ℓ , Z -equivariant) φ -coordinated quasimodules, we refer to [12] and [13]. Set φ = φ ( x, z ) = xe z ∈ C (( x ))[[ z ]], which isfixed throughout this section.For k ∈ Z , we modify the generating function E k ( x ) by b E k ( x ) = xE k ( x ) = X l ∈ Z E k,l x − l . Then, for any k, r ∈ Z , we have[ b E k ( x ) , b E r ( x )] = b E k + r ( q k x ) δ (cid:16) q k + r x x (cid:17) − b E k + r ( q − k x ) δ (cid:16) q − k − r x x (cid:17) + kδ k, − r δ (cid:16) x x (cid:17) c + δ k, − r x ∂∂x δ (cid:16) x x (cid:17) c . (4.1)Furthermore, let e E k,m ( x ) = b E k ( q m x ) for k, m ∈ Z . Then[ e E k,m ( x ) , e E r,n ( x )] = e E k + r,n + k ( x ) δ (cid:16) q − m + n + k + r x (cid:17) − e E k + r,n − k ( x ) δ (cid:16) q − m + n − k − r x (cid:17) + kδ k, − r δ (cid:16) q n − m x x (cid:17) c + δ k, − r x ∂∂x δ (cid:16) q n − m x x (cid:17) c (4.2)for any k, m, r, n ∈ Z .For k, m ∈ Z , recall the formal operator G k,m ( x ) = X i ∈ Z ( G k,m ⊗ t i ) x − i − . Then in e A , for any k, m, r, n ∈ Z , we have[ G k,m ( x ) , G r,n ( x )]= δ − m + n + k + r, G k + r,n + k ( x ) x − δ (cid:16) x x (cid:17) − δ − m + n − k − r, G k + r,n − k ( x ) x − δ (cid:16) x x (cid:17) + kδ k, − r δ n − m, x − δ (cid:16) x x (cid:17) K + δ k, − r δ n − m, ∂∂x x − δ (cid:16) x x (cid:17) K . (4.3)12or any r ∈ Z , define a linear map τ r on e A by τ r ( G k,m ⊗ t i ) = G k,m + r ⊗ t i , τ r ( K j ) = K j , for k, m, i ∈ Z , j = 1 ,
2. Then Γ = { τ r | r ∈ Z } ∼ = Z is an automorphism groupof the Lie algebra e A . It then gives rise to an automorphism group of the vertexalgebra V e A ( ℓ , Theorem 4.1.
Let W be a restricted e V q -module of level ℓ . Then there exists a Z -equivariant φ -coordinated quasi V e A ( ℓ , -module structure Y W ( · , x ) on W , whichis uniquely determined by Y W ( G k,m , x ) = e E k,m ( x ) for k, m ∈ Z . Proof.
Since T = { G k,m , | k, m ∈ Z } generates V e A ( ℓ ,
0) as a vertex algebra, theuniqueness is clear. We now establish the existence. Set U W = { W , e E k,m ( x ) | ( k, m ) ∈ Z } ⊂ E ( W ) . Let ( k, m ) , ( r, n ) ∈ Z . From (4.2) we have( x − q − m + n + k + r x )( x − q − m + n − k − r x )( x − q n − m x ) [ e E k,m ( x ) , e E r,n ( x )] = 0 . (4.4)So U W is a quasi local subset of E ( W ). In view of Theorem 5.4 of [12] or Theorem4.10 of [13], U W generates a vertex algebra h U W i e under the vertex operation Y e E with W a φ -coordinated quasi module, where Y W ( a ( x ) , z ) = a ( z ) for a ( x ) ∈ h U W i e . By Lemma 4.13 of [13], from (4.2), we have e E k,m ( x ) en e E r,n ( x ) = 0 for n ≥ , e E k,m ( x ) e e E r,n ( x ) = δ k, − r δ n − m, ℓ W , e E k,m ( x ) e e E r,n ( x ) = δ − m + n + k + r, e E k + r,n + k ( x ) − δ − m + n − k − r, e E k + r,n − k ( x ) + kδ k, − r δ n − m, ℓ W . Then by Borcherds’ commutator formula we have[ Y e E ( e E k,m ( x ) , x ) , Y e E ( e E r,n ( x ) , x )]= X n ≥ Y e E ( e E k,m ( x ) en e E r,n ( x ) , x ) 1 n ! (cid:16) ∂∂x (cid:17) n x − δ (cid:16) x x (cid:17) = (cid:16) δ − m + n + k + r, Y e E ( e E k + r,n + k ( x ) , x ) − δ − m + n − k − r, Y e E ( e E k + r,n − k ( x ) , x ) (cid:17) x − δ (cid:16) x x (cid:17) + kδ k, − r δ n − m, x − δ (cid:16) x x (cid:17) ℓ W + δ k, − r δ n − m, ∂∂x x − δ (cid:16) x x (cid:17) ℓ W . (4.5)With (4.3), we see that h U W i e is a e A -module of level ℓ with G k,m ( z ) acting as Y e E ( e E k,m ( x ) , z ) for ( k, m ) ∈ Z . 13et ρ : V e A ( ℓ , −→ h U W i e be a e A -module homomorphism with ρ ( ) = W .Then for any ( k, m ) ∈ Z , n ∈ Z , v ∈ V e A ( ℓ , ρ (( G k,m ) n v ) = e E k,m ( x ) en ρ ( v ) = ρ ( G k,m ) n ρ ( v ) . It follows that ρ is a homomorphism of vertex algebras. Consequently, W becomesa φ -coordinated quasi V e A ( ℓ , Y W ( G k,m , x ) = e E k,m ( x ), ( k, m ) ∈ Z . Moreover, for any k, m, d ∈ Z , we have Y W ( σ r ( G k,m ) , x ) = Y W ( G k,m + r , x ) = e E k,m + r ( x ) = e E k,m ( q r x ) = Y W ( G k,m , χ q ( σ r ) x ) , and it is clear that { Y W ( v, x ) | v ∈ T } is χ q ( Z )-quasi local. Then it follows fromLemma 4.21 of [13] that ( W, Y W ) is a Z -equivariant φ -coordinated quasi V e A ( ℓ , Theorem 4.2.
Let W be a Z -equivariant φ -coordinated quasi V e A ( ℓ , -module.Then W is a restricted e V q -module of level ℓ with e E k,m ( x ) = Y W ( G k,m , x ) for k, m ∈ Z . Proof.
For k, m, r ∈ Z , we have Y W ( G k,m + r , x ) = Y W ( σ r ( G k,m ) , x ) = Y W ( G k,m , χ q ( σ r ) x ) = Y W ( G k,m , q r x ) . Let ( k, m ) , ( r, n ) ∈ Z . For any r ∈ Z , s ≥
0, there is( σ r ( G k,m )) s G l,n = ( G k,m + r ) s G l,n = ( G k,m + r ) s ( G l,n ) − = [ G k,m + r ⊗ t s , G l,n ⊗ t − ] = δ s, [ G k,m + r , G l,n ] − + δ s, ψ ( G k,m + r , G l,n ) ℓ + s h G k,m + r , G l,n i δ s − , ℓ . (4.6)Noticing that χ q is injective ( q is not a root of unity), by Theorem 4.19 of [13], we14ave [ Y W ( G k,m , x ) , Y W ( G l,n , x )]= Res x X r ∈ Z Y W ( Y ( σ r ( G k,m ) , x ) G l,n , x ) e x ( x ∂∂x ) δ (cid:16) χ q ( σ r ) x x (cid:17) = X r ∈ Z X s ≥ Y W ( σ r ( G k,m ) s G l,n , x ) 1 s ! (cid:16) x ∂∂x (cid:17) s δ (cid:16) q r x x (cid:17) = X r ∈ Z Y W ([ G k,m + r , G l,n ] , x ) δ (cid:16) q r x x (cid:17) + X r ∈ Z ψ ( G k,m + r , G l,n ) δ (cid:16) q r x x (cid:17) ℓ W + X r ∈ Z h G k,m + r , G l,n i x ∂∂x δ (cid:16) q r x x (cid:17) ℓ W = Y W ( G k + l,n + k , x ) δ (cid:16) q − m + n + k + l x x (cid:17) − Y W ( G k + l,n − k , x ) δ (cid:16) q − m + n − k − l x x (cid:17) + kδ k, − l δ (cid:16) q n − m x x (cid:17) ℓ W + δ k, − l x ∂∂x δ (cid:16) q n − m x x (cid:17) ℓ W . (4.7)With (4.2), we see that W is a e V q -module of level ℓ with e E k,m ( x ) = Y W ( G k,m , x ), for( k, m ) ∈ Z . And Y W ( G k,m , x ) ∈ E ( W ) for ( k, m ) ∈ Z . Therefore, W is a restricted e V q -module of level ℓ .We then wish to study the generalized affine vertex algebra V e A ( ℓ ,
0) and itssimple quotient L e A ( ℓ , Proposition 4.3.
The 2-cocycle ψ of A in Example 2.10 gives trivial central exten-sion of A . Similarly, the 2-cocycle ψ in Remark 2.2 gives trivial central extensionof the affine Lie algebra b A .Proof. Let µ : A −→ C be a linear map defined by µ ( G α,m ) = 12 δ α, m, for α, m ∈ Z . Then ψ ( G α,m , G β,n ) = µ ([ G α,m , G β,n ]) for any α, β, m, n ∈ Z .Let µ : b A −→ C be a linear map defined by µ ( G α,m ⊗ t i ) = 12 δ α, δ i +1 , m, and µ ( K ) = 0 , for α, m, i ∈ Z . Then ψ ( G α,m ⊗ t i , G β,n ⊗ t j ) = µ ([ G α,m ⊗ t i , G β,n ⊗ t j ]), for any α, β, m, n ∈ Z .Therefore, by Proposition 2.8, we have L e A ( ℓ , ∼ = L b A ( ℓ ,
0) for any ℓ , ℓ ∈ C .Vertex algebra L b A ( ℓ ,
0) and its modules has been studied in Section 4 of [14].15
Category O of e V q -modules In this section, we introduce and study a category O of e V q -modules. We prove thatobjects in the category O are restricted e V q -modules. We also classify simple modulesin the category O .Let b = M l ≥ ,k ∈ Z C E k,l ⊕ C c ⊕ C c . Let ℓ , ℓ ∈ C . Given a b -module V with c , c act as scalars ℓ , ℓ respectively.Consider the induced moduleInd ℓ ( V ) = U ( e V q ) ⊗ U ( b ) V. We first show that under certain conditions the induced module Ind ℓ ( V ) is asimple e V q -module. Theorem 5.1.
Let V be a simple b -module and assume that there exists t ∈ Z + such that(1) E k,t acts injectively on V for all k ∈ Z .(2) E k,l V = 0 for all k ∈ Z , l > t .Then for any ℓ , ℓ ∈ C , the induced module Ind ℓ ( V ) is a simple e V q -module.Proof. Let v be a nonzero element in V . Let k ′ , k ∈ Z , j ∈ Z + . It is straightforwardto show (by induction) that for any i ∈ N we have E k ′ ,t + j E i k , − j v ∈ C E k ′ + i k ,t + j − i j v ⊂ V for all j ≥ i j . Let n ≥ k , k , . . . , k n ∈ Z , j , j , . . . , j n ∈ Z + . Similarly, one can get that for any i , i , . . . , i n ∈ N we have E k ′ ,t + j E i n k n , − j n · · · E i k , − j E i k , − j v ∈ C E k ′ + P nl =1 i l k l ,t + j − ( P nl =1 i l j l ) v ⊂ V for all j ≥ P nl =1 i l j l . In particular, E k ′ ,t + j E i n k n , − j n · · · E i k , − j E i k , − j v = 0 for j > n X l =1 i l j l and E k ′ ,t + P nl =1 i l j l E i n k , − j n · · · E i k , − j E i k , − j v ∈ C E k ′ + P nl =1 i l k l ,t v ⊂ V. By PBW theorem, every nonzero element v of Ind ℓ ( V ) can be uniquely writtenin the form of a finite sum w = P m ≥ E k m ,j m · · · E k ,j E k ,j v m , where v m ∈ V . Withabove and the assumptions (1) and (2), for any w ∈ Ind ℓ ( V ), we can always arriveat a nonzero element in V . The simplicity of V then tells us that Ind ℓ ( V ) is asimple e V q -module. 16ecall that a module V over a Lie algebra L is locally finite if for any v ∈ V ,dim( P n ∈ Z + L n v ) < + ∞ .For any t ∈ Z + , denote by e V q ( t ) = X k ∈ Z ,l ≥ t C E k,l . It is a subalgebra of e V q . Let O be a category of e V q -modules which are locally finitewith respect to some e V q ( t ) for t ∈ Z + .The following Lemma tells us that all the modules in the category O are restrictedmodules. Lemma 5.2.
Let V be a e V q -module in the category O . Then for any nonzero vector v ∈ V , there exists t ∈ Z + such that e V q ( t ) v = 0 .Proof. Let 0 = v ∈ V . By assumption, there exists r ∈ Z + such that e V q ( r ) actslocally finite on V . Hence W = U ( e V q ( r ) ) v is a finite dimensional e V q ( r ) -module. Let I ⊂ e V q ( r ) be the kernel of the representation map, i.e. I = { x ∈ e V q ( r ) | x.v = 0 } .Then I is an ideal of e V q ( r ) of finite codimension. We claim that there exists k, l ∈ Z , l ≥ r , such that E k,l ∈ I . If not, then there exists a minimal m ∈ N such that I contains E n,m := ( a ( s )1 E k ,s + · · · + a ( s ) n E k n ,s ) + · · · + ( a ( s + m )1 E k m ,s + m + · · · + a ( s + m ) n m E k mnm ,s + m )for some s ≥ r , k ij ∈ Z , i = 1 , . . . , m , j = 1 , . . . , n , . . . , n m , and complex numbers a ( l ) j satisfying a ( s )1 = 0, a ( s ) n = 0 if m = 0 (and then n is taken to be the minimalone) or a ( s + m ) n m = 0 if m ∈ Z + . Then I contains [ E k ,s , E n,m ] which is an element ofthe form ( b ( s )2 E k + k , s + · · · + b ( s ) n E k + k n , s ) + · · · + ( b ( s + m )1 E k + k m , s + m + · · · + b ( s + m ) n m E k + k mnm , s + m ) . If m = 0, then it contradicts to our choice of n . If m ∈ Z + ,then I contains[ E P n − i =1 n − − i k i + k n , n − s , · · · , [ E k + k , s , [ E k ,s , E n,m ]] · · · ] , which is of the form ( c ( s +1)1 E r , (2 n − s + s +1 + · · · + c ( s +1) n E r n , (2 n − s + s +1 ) + · · · +( c ( s + m )1 E r m , (2 n − s + s + m + · · · + c ( s + m ) n m E r mnm , (2 n − s + s + m ) , this contradicts to ourchoice of m .Let E k,l ∈ I . Then l ≥ r , k ∈ Z , E k,l v = 0. We claim then e V q (2 r + l ) ⊂ I . For any E m,n ∈ e V q (2 r + l ) . Then n ≥ r + l . We have[ E m − k,n − l − r , E k,l ] = ( q k ( n − r ) − lm − q lm − k ( n − r ) ) E m,n − r . ase 1: If k ( n − r ) = lm . Take a nonzero integer s with ns = mr (for latter use).Then [ E m − k − s,n − l − r , E k,l ] = ( q sl − q − sl ) E m − s,n − r gives that E m − s,n − r ∈ I . And then[ E m − s,n − r , E s,r ] = ( q ns − mr − q mr − ns ) E m,n gives that E m,n ∈ I . Case 2: If k ( n − r ) = lm , then E m,n − r ∈ I . And then if m = 0,[ E m,n − r , E ,r ] = ( q − mr − q mr ) E m,n gives that E m,n ∈ I . For m = 0. If k = 0, then[ E − k,n − l , E k,l ] = ( q kn − q − kn ) E ,n gives that E ,n ∈ I . For m = 0 and k = 0, we are in Case 1. Hence the resultholds.As we need, we introduce the following notion. If a e V q -module V is generated bya vector 0 = v ∈ V with E k,l v = 0 for all k ∈ Z and l >
0, then V is called a highestweight module . Harish-Chandra modules in [15] are highest weight modules.Finally, we give a classification of all simple e V q -module in the category O . Theorem 5.3.
Let S be a simple e V q -module in the category O . Then S is a highestweight module, or there exists ℓ , ℓ ∈ C , t ∈ Z + and a simple b -module V such thatboth conditions (1) and (2) of Theorem 5.1 are satisfied and S ∼ = Ind ℓ V .Proof. By Lemma 5.2, for any nonzero v ∈ S , there exists some j ∈ N such that v is annihilated by all E k,l , k ∈ Z , l > j . Consider the following vector space N j = { v ∈ S | E k,l v = 0 for all k ∈ Z , l > j } . We have N j = 0 for some j by the above argument. If N = 0, then S is anirreducible highest weight module. Assume j ≥
1. Thus we can find a smallestpositive integer, say t , with V := N t = 0. It is easy to check that V is a b -module.Note that c , c act as scalars on V , say ℓ , ℓ respectively. It follows from t issmallest that assumption (1) of Theorem 5.1 is satisfied.Then there is a canonical surjective (since S is simple) morphism π : Ind ℓ V −→ S, ⊗ v v, ∀ v ∈ V. It remains to show that π is also injective. Let K = Ker( π ). If K = 0. Take0 = v ∈ K . Then v ∈ Ind ℓ V . By the same analysis as the proof of Theorem5.1, from v we can arrive at 0 = u ∈ V . Since K is a e V q -module, u ∈ K . Hence u ∈ V ∩ K . But V ∩ K = 0, contradiction! Hence K = 0. S ∼ = Ind ℓ V as e V q -modules. Then V is a simple b -module by the property of induced modules.18 eferences [1] S. Berman, Y. Gao, Y. S. Krylyuk, Quantum tori and the structure of ellipticquasi-simple Lie algebras. J. Funct. Anal. (135) (1996), no. 2, 339-389.[2] M. Bershtein, R. Gonin, Twisted representations of algebra of q -difference op-erators, twisted q - W algebras and conformal blocks. SIGMA Symmetry Inte-grability Geom. Methods Appl. (16) (2020), Paper No. 077, 55 pp.[3] I. B. Frenkel, Y. C. Zhu, Vertex operator algebras associated to representationsof affine and Virasoro algebras.
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