Algebra of Q-Difference Operators, Affine Vertex Algebras, And

Home , Affine Lie algebra

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finite with respect to some subalgebra of Vq. Modules ine the category are proved to be restricted Vq-modules and simple modulese are classified. Constructinge vertex algebras from Lie algebras and build a connection between the representations of the two algebraic objects is an important subject in both Lie algebra 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 VL) are the vacuum modules of L, and restricted L-modules at fixed levels are in one-to-one correspondence to VL-modules (cf. [3], [9], etc.). However, there are Lie algebras (by abuse of notation, we still denote by L) whose restricted modules are not related to the vertex algebras VL, but vertex algebras constructed from other Lie algebras (cf. [5], [6], [14], etc.). And restricted L-modules are not correspond 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 g is the 1-dimensional universal central extension −1 of the loop algebra (also calledb current algebra) g ⊗ C[t, t ], where g is a finite- dimensional simple Lie algebra. Affine vertex algebras Vg(ℓ, 0) are constructed from affine Lie algebras (with g possibly infinite-dimensional).b The relations between the representations of affine Lie algebra g and the representations of affine vertex algebras Vg(ℓ, 0) are investigated by manyb authors (cf. [3], [9], [11], etc.). In trying to relate theb 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 symmetric invariant bilinear form h , i. Suppose that ψ : g × g −→ C is a 2-cocycle on g. Consider the following generalized affine Lie algebra

−1 g = g ⊗ C[t, t ] ⊕ CK1 ⊕ CK2 e with Lie bracket

i j i+j [a ⊗ t , b ⊗ t ] = [a, b] ⊗ t + ψ(a, b)δi+j+1,0K1 + iha, biδi+j,0K2, where K1, K2 are central elements, a, b ∈ g, i, j ∈ Z. It is easy to see that g can also be viewed as a 1-dimensional central extension of the (usual) affine Lie algebrae −1 g = g ⊗ C[t, t ] ⊕ CK2 (see Remark 2.2). The corresponding vertex algebras Vg(ℓ12, 0) are straightforward to construct. Restricted modules of the Lie algebra be g and modules of the vertex algebra Vg(ℓ12, 0) are closely related. The relations of e ethe generalized affine vertex algebra Vg(ℓ12, 0) and affine vertex algebra Vg(ℓ2, 0) are discussed in this paper. e b Let Γ be a group of automorphisms of g, preserving the bilinear form and satisfying [ga, b]=0= hga, bi for all but finitely many g ∈ Γ. Let χ : Γ −→ C× be a group homomorphism. The covariant algebra g[Γ] which is a quotient space of the b 2 affine Lie algebra g has been construct and studied in [11] (see also [4]). g[Γ] is a generalization of twistedb affine Lie algebras. If χ is injective, then restrictedb modules of level ℓ for g[Γ] correspond to quasi-modules for Vg(ℓ, 0) viewed as a Γ-vertex b algebra (Theoremb 4.9 of [11]). In the study of q-Virasoro algebra and trigonometric Lie algebras, certain Lie algebra A plays an important role (cf. [5], [6], [14]). More specifically, A is a subalgebra of the Lie algebra gl∞ which is spanned by the basis elements (the el- ementary matrices) Em,n with m, n ∈ Z, m + n ∈ 2Z. Rewrite the basis elements as Gα,m = Em+α,m−α for α, m ∈ Z. Then A is spanned by Gα,m, α, m ∈ Z. Let α,β,m,n ∈ Z. There is a natural symmetric invariant bilinear form on A:

hGα,m,Gβ,ni = δα+β,0δm−n,0.

Consider the 2-cocycle ψ on A deﬁned by

ψ(Gα,m,Gβ,n)= αδα+β,0δm−n,0.

Even though ψ is proved to be a trivial 2-cocycle (Proposition 4.3), in order to relate the algebra of q-diﬀerence operators Vq with vertex algebras we need to consider this

2-cocycle. We ﬁrst show that Vq is isomorphice to a covariant algebra of the aﬃne Lie algebra A∗, where A∗ is the 1-dimensionale central extension of A via the 2-cocycle ψ. Then,c for any ℓ1,ℓ2 ∈ C, we establish an equivalence between the category of restricted Vq-modules of level ℓ12 (means that c1,c2 act as scalars ℓ1,ℓ2 respectively) and the categorye of Z-equivariant φ-coordinated quasi modules for the vertex algebra VA(ℓ12, 0). e Let t be any positive integer. Consider the subalgebra

(t) V = CE . q X k,l e k∈Z,l≥t To study the restricted modules for the algebra of q-diﬀerence operators, we introduce a category O of Vq-modules. It is a subcategory of modules with the property e (t) that they are locally ﬁnite Vq -modules for some t. We prove that objects in the category O are restricted Vqe-modules. Finally, we classify simplee modules in the category O. Let

b = CE ⊕ Cc ⊕ Cc . M k,l 1 2 l≥0,k∈Z

Consider the induced module

Indℓ12 (V )= U(Vq) ⊗U(b) V, e where V is a b-module with c1,c2 act as ℓ1,ℓ2. We first show that if V is a simple b-module, then under certain conditions Indℓ12 (V ) is a simple Vq-module. Then we e 3 prove that simple modules in the category O are either highest weight modules or induced modules Indℓ12 V for some simple b-module V . This paper is organized as follows. In Section 2, we study generalized affine Lie algebras 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 Vq in terms of the covariant algebra of the affine Lie algebra A∗. In Section 4, we showe that restricted Vq-module category of level ℓ12 is equivalentc to the Z-equivariant φ-coordinated quasi modulee category for the generalized affine vertex algebra VA(ℓ12, 0). In Section 5, e we introduce and study a category O of Vq-modules. We prove that all the modules in the category O are restricted Vq-modules.e We also give a classification of simple modules in the category O. e Throughout this paper, we denote by Z, N, Z+ and C the sets of integers, nonnegative integers, positive integers and complex numbers respectively.

2 Generalized aﬃne vertex algebras and their representations

In this section, we construct and study certain generalized affine Lie algebras g and the corresponding generalized affine vertex algebras Vg(ℓ12, 0). We also discuss the e e relations between Vg(ℓ12, 0) and affine vertex algebra Vg(ℓ2, 0). e b Definition 2.1. Let g be a Lie algebra. A 2-cocycle ψ : g × g −→ C on g is a skew-symmetric bilinear form satisfying ψ(x, [y, z]) + ψ(y, [z, x]) + ψ(z, [x, y]) = 0 for any x, y, z ∈ g. Let g be a (possibly infinite-dimensional) Lie algebra over the complex number field C. Let h , i : g×g −→ C be a symmetric invariant bilinear form on g. Suppose that ψ : g × g −→ C is a 2-cocycle on g. With respect to g, h , i, ψ, there are the following three ways that we can get affine-type Lie algebras:

−1 (1) The aﬃne Lie algebra g = g ⊗ C[t, t ] ⊕ CK2 with Lie bracket

i b j i+j [a ⊗ t , b ⊗ t ] = [a, b] ⊗ t + iha, biδi+j,0K2, (2.1)

where K2 is a central element, a, b ∈ g, i, j ∈ Z.

∗ (2) Let g = g ⊕ CK1 be the 1-dimensional central extension of g by K1 via ψ. For a, b ∈ g, λ,µ ∈ C, the Lie bracket on g∗ is given by

[a + λK1, b + µK1] = [a, b]+ ψ(a, b)K1. (2.2)

The bilinear form h , i on g can be naturally extended to a symmetric invariant ∗ ∗ ∗ bilinear form on g with hg ,K1i = hK1, g i = 0. Then we have the aﬃne Lie

4 ∗ ∗ −1 algebra g = g ⊗ C[t, t ] ⊕ CK2 with Lie bracket b ∗ i ∗ j ∗ ∗ i+j ∗ ∗ [a ⊗ t , b ⊗ t ] = [a , b ] ⊗ t + iha , b iδi+j,0K2, (2.3)

∗ ∗ ∗ −1 where K2 is a central element, a , b ∈ g , i, j ∈ Z. Clearly, K1 ⊗ C[t, t ] is also in the center of the aﬃne Lie algebra g∗.

b −1 (3) The generalized aﬃne Lie algebra g = g ⊗ C[t, t ] ⊕ CK1 ⊕ CK2 with Lie bracket e

i j i+j [a ⊗ t , b ⊗ t ] = [a, b] ⊗ t + ψ(a, b)δi+j+1,0K1 + iha, biδi+j,0K2, (2.4)

where K1, K2 are central elements, a, b ∈ g, i, j ∈ Z.

Remark 2.2. (1) The 2-cocycle ψ can be extended to a 2-cocycle ψ2 on g with

i j b ψ2(a ⊗ t , b ⊗ t )= ψ(a, b)δi+j+1,0,

ψ2(g,K2)= ψ2(K2, g)=0 for a, b ∈ g, i, j ∈ Z. Thenb g can be viewedb as a 1-dimensional central extension of the affine Lie algebrae g by K1 via ψ2. More precisely, we can C write g = g ⊕ K1 with Lie bracketb e b [a + λK1, b + µK1] = [a, b]+ ψ2(a, b)K1, b b b b b b where a, b ∈ g, λ,µ ∈ C. Note that g may not be a subalgebra of g. b b b ∗ b−1 ∼ ∼ e (2) It is easy to see that g /(K1 ⊗ C[t, t ]) = g and g/CK1 = g as Lie algebras. b b e b In the following, we consider the (general) affine Lie algebra g. For a ∈ g, form a generating function e a(x)= a(n)x−n−1, X n∈Z where a(n) stands for a ⊗ tn. The defining relation (2.4) of g can be written as e −1 x2 [a(x1), b(x2)] = [a, b](x2)x1 δ x1 −1 x2 ∂ −1 x2 + ψ(a, b)x1 δ K1 + ha, bi x1 δ K2 (2.5) x1 ∂x2 x1 for a, b ∈ g. Let ℓ1,ℓ2 be complex numbers. View C as a module for the subalgebra g⊗C[t]+ CK1 + CK2 of g with g ⊗ C[t] acting trivially and with K1, K2 acting as scalars ℓ1,ℓ2 respectively.e Then form an induced module C Vg(ℓ12, 0) = U(g) ⊗U(g⊗C[t]+CK1+CK2) , e e 5 where U(·) denotes the universal enveloping algebra of a Lie algebra. Set 1 =1⊗1 ∈ Vg(ℓ12, 0) and identify g as a subspace of Vg(ℓ12, 0) through linear map e e g −→ Vg(ℓ12, 0); a 7→ a(−1)1. e Then there is a unique vertex algebra structure on (Vg(ℓ12, 0),Y, 1) given by Y (a, x)= e a(x) for a ∈ g. Furthermore, Vg(ℓ12, 0) is a Z-graded vertex algebra: e

Vg(ℓ12, 0) = Vg(ℓ12, 0)(n), (2.6) e a e n≥0 1 r i where Vg(ℓ12, 0)(n) is spanned by the vectors a (−m1) ··· a (−mr)1 for r ≥ 0, a ∈ g, e mi ≥ 1 with m1 + ··· + mr = n. For a vector space W , set E(W ) = Hom(W, W ((x))) ⊂ (EndW )[[x, x−1]]. Deﬁnition 2.3. We say that a g-module W is restricted if for any w ∈ W , a ∈ g, a(n)w = 0 for n suﬃciently large,e or equivalently, a(x) ∈ E(W ). We say W is of level ℓ12 if the central element Ki acts as scalar ℓi, i =1, 2. As usual, there is the following result (cf. Theorem 6.2.12 and Theorem 6.2.13 of [9]).

Theorem 2.4. Let ℓ1,ℓ2 be any complex numbers. Let W be any restricted g-module of level ℓ12. Then there exists a unique Vg(ℓ12, 0)-module structure on W suche that e −n−1 for a ∈ g, YW (a, x) = aW (x)(= Z a(n)x ). Conversely, any module W Pn∈ for the vertex algebra Vg(ℓ12, 0) is naturally a restricted g-module of level ℓ12, with e aW (x)= YW (a, x) for a ∈ g. Furthermore, the Vg(ℓ12, 0)-submodulese of W coincide with the g-submodules of W . e e ∗ For affine Lie algebras g and g , we denote by Vg(ℓ2, 0) and Vg∗ (ℓ2, 0) their cor- b b responding affine vertex algebrasb b with K2 acts as ℓ2 (cf. [3], [9], etc.). Let Jg(ℓ2, 0), Jg∗ (ℓ2, 0), Jg(ℓ12, 0) be the maximal (two-sided) ideals of the ver- b b e tex algebras Vg(ℓ2, 0), Vg∗ (ℓ2, 0), Vg(ℓ12, 0) respectively. Denote by Lg(ℓ12, 0) = b b e e Vg(ℓ12, 0)/Jg(ℓ12, 0), Lg∗ (ℓ2, 0) = Vg∗ (ℓ2, 0)/Jg∗ (ℓ2, 0), and Lg(ℓ2, 0) = Vg(ℓ2, 0)/Jg(ℓ2, 0) thee correspondinge simpleb vertex algebras.b b b b b We now investigate the relations between the three types of affine vertex algebras

Vg(ℓ2, 0), Vg∗ (ℓ2, 0), and Vg(ℓ12, 0) as well as their simple quotients. b b e Remark 2.5. There is a natural surjective vertex algebra homomorphism ϕ :

Vg∗ (ℓ2, 0) −→ Vg(ℓ2, 0) with K1(−m)1 7→ 0 for m ≥ 1. Then Ker(ϕ)= hK1(−m)1, m ≥ 1ib. It induces ab vertex algebra isomorphism ∼ Vg∗ (ℓ2, 0)/hK1(−m)1, m ≥ 1i = Vg(ℓ2, 0). (2.7) b b And ∼ Lg∗ (ℓ2, 0) = Lg(ℓ2, 0) (2.8) b b as simple vertex algebras, for any ℓ2 ∈ C.

6 Clearly, if ℓ1 = 0, then

Vg(ℓ12, 0) = Vg(ℓ2, 0) and Lg(ℓ12, 0) = Lg(ℓ2, 0) e b e b as vertex algebras.

Deﬁnition 2.6. A 2-cocycle ψ : g × g −→ C is said to be trivial if there exists a linear 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 ⊕ CK1 and g = g ⊕ CK1 as Lie algebras. e b 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 ⊕ CK1 −→ g , (a,λK1) 7→ a +(µ(a)+ λ)K1, a ∈ g, λ ∈ C, is a Lie algebra isomorphism. Note that the 2-cocycle ψ2 in Remark 2.2 then is also a trivial 2-cocycle. Let i µ2 : g −→ C be a linear map deﬁned by µ2(a ⊗ t + λK2) = µ(a)δi+1,0 for any a ∈ bg, i ∈ Z, λ ∈ C. Then ψ2(a, b) = µ2([a, b]) for any a, b ∈ g. Similarly, C C f2 : g ⊕ K1 −→ g, (a,λK1) 7→ a +(b bµ2(a)+ λ)bKb1, a ∈ g, λ ∈b b, isb a Lie algebra isomorphism.b e b b b b b Then we can get the following assertion.

Proposition 2.8. Let ℓ1 be any nonzero complex number. If ψ is a trivial 2-cocycle on g. Then we have ∼ Vg(ℓ12, 0) = Vg(ℓ2, 0) e b ∼ as vertex algebras, for any ℓ2 ∈ C. Hence also Lg(ℓ12, 0) = Lg(ℓ2, 0) for any ℓ2 ∈ C. e b

Proof. Vg(ℓ2, 0) is naturally a g-module. It can be viewed as a g ⊕ CK1-module b C with K1 acts as a scalar ℓ1. Explicitly,b (a,λK1).w = a.w + λℓ1w forb a ∈ g, λ ∈ , i w ∈ Vg(ℓ2, 0). By Proposition 2.7, Vg(ℓ2,b0) is then a bg-module with ba ⊗ bt acts as ib b Z (a ⊗ t , −µ(a)δi+1,0K1), K1 acts as ℓ1, K2 acts as ℓ2, fore any a ∈ g, i ∈ . Let ϕ : Vg(ℓ12, 0) −→ Vg(ℓ2, 0) be a g-module homomorphism with ϕ(1) = 1. e b Then e ϕ(a(−1)1)= a(−1).1 = a(−1)1 − µ(a)ℓ11 for any a ∈ g. It is straightforward to check that ϕ(anb)= ϕ(a)nϕ(b) for any a, b ∈ g, n ∈ Z (note that 1nb = δn+1,0b). Hence ϕ is a vertex algebra homomorphism. i Similarly, Vg(ℓ12, 0) can be viewed as a g-module with K2 acts as ℓ2, a ⊗ t acts i e Z as a ⊗ t + µ(a)δi+1,0ℓ1 for any a ∈ g, i ∈ b . Let Φ : Vg(ℓ2, 0) −→ Vg(ℓ12, 0) be a g-module homomorphism with Φ(1) = 1. Then ϕ is a vertexb algebrae isomorphism bwith inverse Φ.

7 At the end of this section, we give two examples. The second example plays an important 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 with only ﬁnitely many nonzero entries under the usual commutator bracket. For any m, n ∈ Z, denote by Em,n the matrix whose only nonzero entry is the (m, n)-entry which is 1. Then Em,n, m, n ∈ Z, form a bases of gl∞. Let m, n, p, q ∈ Z. The Lie bracket on gl∞ is given by

[Em,n, Ep,q]= δn,pEm,q − δq,mEp,n. gl∞ is equipped with a natural symmetric invariant bilinear form:

hEm,n, Ep,qi = trace(Em,nEp,q)= δm,qδn,p.

There is a 2-cocycle ψ on the Lie algebra gl∞ deﬁned by

ψ(Em,n, En,m)=1= −ψ(En,m, Em,n) if m ≤ 0 and n ≥ 1,

ψ(Em,n, Ep,q) = 0 otherwise. ∗ The corresponding 1-dimensional central extension gl∞ and its restricted modules has been related to certain quantum vertex algebras and their φ-coordinated modules in [7].

Example 2.10. Let g = A := span{Em,n | m + n ∈ 2Z}. Then A is a subalgebra of gl∞. For α, m ∈ Z, denote by Gα,m = Em+α,m−α ∈A. Then A = span{Gα,m | α, m ∈ Z} with Lie bracket

[Gα,m,Gβ,n]= δα+β,m−nGα+β,α+n − δα+β,n−mGα+β,n−α (2.9) for α,β,m,n ∈ Z. The symmetric invariant bilinear form on gl∞ in Example 2.9 restricted to A is a symmetric invariant bilinear form on A with

hGα,m,Gβ,ni = δα+β,0δm−n,0, for α,β,m,n ∈ Z. Let ψ : A×A −→ C be a bilinear map deﬁned by

ψ(Gα,m,Gβ,n)= αδα+β,0δm−n,0. It is straightforward to check that ψ is a 2-cocycle. The Lie algebra A, vertex algebras related to A and their (equivariant) quasi modules are in connection with restricted modules ofb certain Lie algebras in [5], [6], [14], etc.

The importance and signiﬁcance of vertex algebras VA∗ (ℓ2, 0) and VA(ℓ12, 0) as well as their certain module categories show up in the studyc of the algeebra of q- diﬀerence operators as we will see in the following two sections.

8 3 AﬃneLiealgebra A∗ and the algebra of q-diﬀerence operators c

In this section, we first review the algebra of q-difference operators Vq (cf. [2], [8], ∗ etc.). Then we show that Vq is isomorphic to the Z-covariant algebra ofe A . At last, we show that restricted Vq-modulese of level ℓ12 with ℓ1 = 0 are one-to-onec correspond to equivariant quasi modulese for the affine vertex algebra VA∗ (ℓ2, 0). Let q ∈ C× be generic, i.e. q is not a root of unity. Thec algebra of q-difference operators is the universal central extension of the Lie algebra of inner derivations of ±1 ±1 the quantum torus Cq[x ,y ] ([8]). We rewrite its definition (and add E0,0 = 0) in the following way.

Deﬁnition 3.1. The algebra of q-diﬀerence operators Vq is a Lie algebra spanned 2 by Ek,l, c1, c2, where (k,l) ∈ Z , c1 and c2 are central elements,e with Lie bracket

rl−sk sk−rl [Ek,l, Er,s]=(q − q )Ek+r,l+s + δk,−rδl,−s(kc1 + lc2), (3.1) for (k,l), (r, s) ∈ Z2.

We first give a characterization of Vq as certain covariant algebra of the affine Lie algebra A∗. e For any rc∈ Z, define

σr(Gα,m)= Gα,m+r, σr(K1)= K1.

∗ It is easy to see that σr is an automorphism of A , for any r ∈ Z. LetΓ= {σr | r ∈ Z}. Then Γ =∼ Z, and Γ preserves h , i. Γ extends canonically to an automorphism group of A∗ and furthermore to an automorphism group of the Z-graded vertex algebra VAc∗ (ℓ2, 0). Consider the following linear character c × r χq : Z −→ C ; r 7→ q for r ∈ Z.

We have (cf. [4], [11]):

Proposition 3.2. The algebra of q-diﬀerence operators Vq is isomorphic to the ∗ ∗ (Z, χq)-covariant algebra A [Z] of the aﬃne Lie algebra A ewith c1 = K1 ⊗ 1, c2 = K2, and c c m Eα,m = Gα,0 ⊗ t for α, m ∈ Z.

Proof. From the deﬁnition, Vq has a basis {Eα,m,c1,c2 | α, m ∈ Z} with the given bracket relations. On the othere hand, from Proposition 4.4 of [11], A∗[Z] as a vector space is the quotient space of A∗, modulo the subspace linearly spannedc by c m −mr m ∗ σr(a) ⊗ t − q (a ⊗ t ) for a ∈A , r, m ∈ Z,

9 and its bracket relation is given by

[a ⊗ tm, b ⊗ tn] = qmr [σ (a), b] ⊗ tm+n + mhσ (a), biδ K , (3.2) Γ X r r m+n,0 2 r∈Z

m m ∗ ∗ where a ⊗ t denotes the image of a ⊗ t in A [Z] for a ∈ A , m ∈ Z, and K2 is identiﬁed with its image. Note that we have (qcis generic)

m+n (m+n)α m+n Gα+β,−α ⊗ t = q Gα+β,0 ⊗ t , (3.3)

m+n −(m+n)α m+n Gα+β,α ⊗ t = q Gα+β,0 ⊗ t , (3.4)

m+n K1 ⊗ t = δm+n,0K1 ⊗ 1. (3.5)

m ∗ We see that K2, K1 ⊗ 1 and Gα,0 ⊗ t (α, m ∈ Z) form a bases of A [Z]. Let α,β,m,n ∈ Z, then c

m n [Gα,0 ⊗ t , Gβ,0 ⊗ t ]Γ m(α+β) m+n −m(α+β) m+n = q Gα+β,α ⊗ t − q Gα+β,−α ⊗ t m+n + αδα+β,0K1 ⊗ t + mδα+β,0δm+n,0K2 mβ−nα nα−mβ m+n =(q − q )Gα+β,0 ⊗ t + αδα+β,0δm+n,0K1 ⊗ 1

+ mδα+β,0δm+n,0K2. (3.6)

∗ m It follows that Vq is isomorphic to A [Z] with Eα,m corresponding to Gα,0 ⊗ t for α, m ∈ Z, c1 correspondinge to K1 ⊗c1 and c2 corresponding to K2. For k ∈ Z, form a generating function

E (x)= E x−l−1. k X k,l l∈Z Then the deﬁning relation (3.1) can be written as

[Ek(x1), Er(x2)] k+r −k−r k k −1 q x2 −k −k −1 q x2 = q Ek+r(q x2)x1 δ − q Ek+r(q x2)x1 δ x1 x1 −1 −1 x2 ∂ −1 x2 + kδk,−rx1 x2 δ c1 + δk,−r x1 δ c2 (3.7) x1 ∂x2 x1 for k,r ∈ Z.

Deﬁnition 3.3. A Vq-module W is said to be restricted if for any w ∈ W , k ∈ Z, Ek,lw = 0 for l suﬃcientlye large, or equivalently, Ek(x) ∈ E(W ) for any k ∈ Z.

We say a Vq-module W is of level ℓ12 if the central elements c1,c2 act as scalars ℓ1,ℓ2 ∈ C. e

10 Remark 3.4. Let V be a Z-graded vertex algebra. Denote by L(0) the degree operator 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 algebra −L(0) with Rg = χ(g) g for g ∈ Γ (see [10], [11] for the details). Z By Remark 3.4, VA∗ (ℓ2, 0) becomes a Γ-vertex algebra with Γ = and Rr = −rL(0) c q σr for r ∈ Z.

Now we give a connection between aﬃne vertex algebra VA∗ (ℓ2, 0) and restricted c Vq-modules. e Theorem 3.5. Assume that q is not a root of unity and let ℓ2 ∈ C. Then for any restricted Vq-module W of level ℓ12 with ℓ1 = 0, there exists an equivariant quasi VA∗ (ℓ2, 0)-modulee structure YW (·, x) on W , which is uniquely determined by c m m YW (K1, x)=0,YW (Gα,m, x)= q Eα(q x) for α, m ∈ Z. (3.8)

On the other hand, for any equivariant quasi VA∗ (ℓ2, 0)-module (W, YW ) such that c YW (K1, x)=0, W becomes a restricted Vq-module of level ℓ12 with ℓ1 =0 and e Eα(x)= YW (Gα,0, x) for α ∈ Z. (3.9)

Proof. Let W be a restricted Vq-module of level ℓ12 with c1 acts as 0. In view of ∗ Proposition 3.2, W is a restrictede A [Z]-module with K1 ⊗ 1 acts as 0, K2 acts as ℓ2, and with c m Eα,m = Gα,0 ⊗ t for α, m ∈ Z. × Note that q is not a root of unity, χq : Z −→ C is one-to-one. By Theorem 4.9 of [11], there exists an quasi VA∗ (ℓ2, 0)-module structure YW (·, x) on W , which is uniquely determined by c Y (K , x)= K (x)(= K ⊗ tnx−n−1 = K ⊗ 1x−1)=0, W 1 1 X 1 1 n∈Z Y (G , x)= G (x)(= G ⊗ tnx−n−1) for α, m ∈ Z. W α,m α,m X α,m n∈Z For α, m ∈ Z, we have

m m m m YW (Gα,m, x)= Gα,m(x)= σm(Gα,0)(x)= q Gα,0(q x)= q Eα(q x). Hence r r YW (σr(Gα,m), x)= YW (q Gα,m, q x) for any r ∈ Z.

Consequently, there eixts an equivariant quasi VA∗ (ℓ2, 0)-module structure YW (·, x) on W such that c m m YW (K1, x)=0,YW (Gα,m, x)= q Eα(q 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 ℓ1 ∈ C if we require a less natural condition −1 that YW (K1, x)= ℓ1x (so that K1 ⊗ 1 acts as ℓ1).

11 4 Vertex algebra VA(ℓ12, 0) and Vq e e In order to naturally relate restricted Vq-modules of level ℓ12 for any complex numbers ℓ1, ℓ2 to vertex algebras and theire corresponding modules, in this section, we consider φ-coordinated quasi modules for vertex algebras. More precisely, we show that restricted Vq-module of level ℓ12 are in one-to-one correspondence to Z- equivariant φ-coordinatede quasi modules for the vertex algebra VA(ℓ12, 0). For the deﬁnitions and related results about (Z-equivariant) φ-coordinatede quasi modules, we refer to [12] and [13]. Set φ = φ(x, z) = xez ∈ C((x))[[z]], which is ﬁxed throughout this section. For k ∈ Z, we modify the generating function Ek(x) by

E (x)= xE (x)= E x−l. k k X k,l b l∈Z Then, for any k,r ∈ Z, we have

qk+rx q−k−rx [E (x ), E (x )] = E (qkx )δ 2 − E (q−kx )δ 2 k 1 r 2 k+r 2 x k+r 2 x b b b 1 b 1 x2 ∂ x2 + kδk,−rδ c1 + δk,−rx2 δ c2. (4.1) x1 ∂x2 x1 Furthermore, let m Ek,m(x)= Ek(q x) for k, m ∈ Z. Then e b q−m+n+k+r q−m+n−k−r [Ek,m(x1), Er,n(x2)] = Ek+r,n+k(x2)δ − Ek+r,n−k(x2)δ x1 x1 e e e n−m e n−m q x2 ∂ q x2 +kδk,−rδ c1 + δk,−rx2 δ c2 (4.2) x1 ∂x2 x1 for any k, m, r, n ∈ Z. For k, m ∈ Z, recall the formal operator

G (x)= (G ⊗ ti)x−i−1. k,m X k,m i∈Z

Then in A, for any k, m, r, n ∈ Z, we have e [Gk,m(x1),Gr,n(x2)] −1 x2 −1 x2 = δ−m+n+k+r,0Gk+r,n+k(x2)x1 δ − δ−m+n−k−r,0Gk+r,n−k(x2)x1 δ x1 x1 −1 x2 ∂ −1 x2 + kδk,−rδn−m,0x1 δ K1 + δk,−rδn−m,0 x1 δ K2. (4.3) x1 ∂x2 x1

12 i i For any r ∈ Z, deﬁne a linear map τr on A by τr(Gk,m⊗t )= Gk,m+r⊗t , τr(Kj)= Kj, for k, m, i ∈ Z, j =1, 2. Then Γ = {τr |er ∈ Z} =∼ Z is an automorphism group of the Lie algebra A. It then gives rise to an automorphism group of the vertex algebra VA(ℓ12, 0). e On thee one hand, we have:

Theorem 4.1. Let W be a restricted Vq-module of level ℓ12. Then there exists a Z -equivariant φ-coordinated quasi VA(ℓ12e, 0)-module structure YW (·, x) on W , which is uniquely determined by e

YW (Gk,m, x)= Ek,m(x) for k, m ∈ Z. e Z Proof. Since T = { Gk,m, 1 | k, m ∈ } generates VA(ℓ12, 0) as a vertex algebra, the uniqueness is clear. We now establish the existence.e Set

2 UW = {1W , Ek,m(x) | (k, m) ∈ Z }⊂E(W ). e Let (k, m), (r, n) ∈ Z2. From (4.2) we have

−m+n+k+r −m+n−k−r n−m 2 (x1 − q x2)(x1 − q x2)(x1 − q x2) [Ek,m(x1), Er,n(x2)]=0.(4.4) e e So UW is a quasi local subset of E(W ). In view of Theorem 5.4 of [12] or Theorem e 4.10 of [13], UW generates a vertex algebra hUW ie under the vertex operation YE with W a φ-coordinated quasi module, where YW (a(x), z)= a(z) for a(x) ∈hUW ie . By Lemma 4.13 of [13], from (4.2), we have

e Ek,m(x)nEr,n(x)=0 for n ≥ 2, e e e Ek,m(x)1Er,n(x)= δk,−rδn−m,0ℓ21W , e e e Ek,m(x)0Er,n(x)= δ−m+n+k+r,0Ek+r,n+k(x) e e e − δ−m+n−k−r,0Ek+r,n−k(x)+ kδk,−rδn−m,0ℓ11W . e Then by Borcherds’ commutator formula we have

e e [YE (Ek,m(x), x1),YE (Er,n(x), x2)] e e 1 ∂ n x = Y e(E (x)e E (x), x ) x−1δ 2 X E k,m n r,n 2 n!∂x 1 x n≥0 e e 2 1 e e −1 x2 = δ− Y (E (x), x ) − δ− − − Y (E − (x), x ) x δ m+n+k+r,0 E k+r,n+k 2 m+n k r,0 E k+r,n k 2 1 x e e 1 x ∂ x −1 2 1 −1 2 1 + kδk,−rδn−m,0x1 δ ℓ1 W + δk,−rδn−m,0 x1 δ ℓ2 W . (4.5) x1 ∂x2 x1

With (4.3), we see that hUW ie is a A-module of level ℓ12 with Gk,m(z) acting as e Z2 YE (Ek,m(x), z) for (k, m) ∈ . e e 13 Let ρ : VA(ℓ12, 0) −→ hUW ie be a A-module homomorphism with ρ(1) = 1W . e Z2 Z Then for any (k, m) ∈ , n ∈ , v ∈ VAe(ℓ12, 0), we have e e ρ((Gk,m)nv)= Ek,m(x)nρ(v)= ρ(Gk,m)nρ(v). e It follows that ρ is a homomorphism of vertex algebras. Consequently, W becomes Z2 a φ-coordinated quasi VA(ℓ12, 0)-module with YW (Gk,m, x)= Ek,m(x), (k, m) ∈ . e Moreover, for any k,m,d ∈ Z, we have e

r YW (σr(Gk,m), x)= YW (Gk,m+r, x)= Ek,m+r(x)= Ek,m(q x)= YW (Gk,m, χq(σr)x), e e and it is clear that {YW (v, x) | v ∈ T } is χq(Z)-quasi local. Then it follows from Z Lemma 4.21 of [13] that (W, YW ) is a -equivariant φ-coordinated quasi VA(ℓ12, 0)- module. e On the other hand, we have the following theorem. Z Theorem 4.2. Let W be a -equivariant φ-coordinated quasi VA(ℓ12, 0)-module. e Then W is a restricted Vq-module of level ℓ12 with e Ek,m(x)= YW (Gk,m, x) for k, m ∈ Z. e Proof. For k,m,r ∈ Z, we have

r YW (Gk,m+r, x)= YW (σr(Gk,m), x)= YW (Gk,m, χq(σr)x)= YW (Gk,m, q x).

Let (k, m), (r, n) ∈ Z2. For any r ∈ Z, s ≥ 0, there is

(σr(Gk,m))sGl,n =(Gk,m+r)sGl,n s −1 =(Gk,m+r)s(Gl,n)−11 = [Gk,m+r ⊗ t ,Gl,n ⊗ t ]1

= δs,0[Gk,m+r,Gl,n]−11 + δs,0ψ(Gk,m+r,Gl,n)ℓ11

+ s hGk,m+r,Gl,ni δs−1,0ℓ21. (4.6)

Noticing that χq is injective (q is not a root of unity), by Theorem 4.19 of [13], we

14 have

[YW (Gk,m, x1),YW (Gl,n, x2)] ∂ x0(x2 ) χq(σr)x2 = Res Y (Y (σ (G ), x )G , x )e ∂x2 δ x0 X W r k,m 0 l,n 2 x r∈Z 1 1 ∂ s qrx = Y (σ (G ) G , x ) x δ 2 X X W r k,m s l,n 2 s! 2 ∂x x r∈Z s≥0 2 1 qrx qrx = Y ([G ,G ], x )δ 2 + ψ(G ,G )δ 2 ℓ 1 X W k,m+r l,n 2 x X k,m+r l,n x 1 W r∈Z 1 r∈Z 1 ∂ qrx + hG ,G i x δ 2 ℓ 1 X k,m+r l,n 2 ∂x x 2 W r∈Z 2 1 −m+n+k+l −m+n−k−l q x2 q x2 = YW (Gk+l,n+k, x2)δ − YW (Gk+l,n−k, x2)δ x1 x1 n−m n−m q x2 ∂ q x2 + kδk,−lδ ℓ11W + δk,−lx2 δ ℓ21W . (4.7) x1 ∂x2 x1

With (4.2), we see that W is a Vq-module of level ℓ12 with Ek,m(x)= YW (Gk,m, x), for 2 2 (k, m) ∈ Z . And YW (Gk,m, x)e∈E(W ) for (k, m) ∈ Z . Therefore,e W is a restricted

Vq-module of level ℓ12. e We then wish to study the generalized affine vertex algebra VA(ℓ12, 0) and its e simple quotient LA(ℓ12, 0). For this, there is the following result. e Proposition 4.3. The 2-cocycle ψ of A in Example 2.10 gives trivial central extension of A. Similarly, the 2-cocycle ψ2 in Remark 2.2 gives trivial central extension of the affine Lie algebra A. b Proof. Let µ : A −→ C be a linear map defined by 1 µ(G )= δ m, α,m 2 α,0 for α, m ∈ Z. Then ψ(Gα,m,Gβ,n)= µ([Gα,m,Gβ,n]) for any α,β,m,n ∈ Z.

Let µ2 : A −→ C be a linear map deﬁned by b 1 µ (G ⊗ ti)= δ δ m, and µ (K )=0, 2 α,m 2 α,0 i+1,0 2 2

i j i j for α, m, i ∈ Z. Then ψ2(Gα,m ⊗ t ,Gβ,n ⊗ t ) = µ2([Gα,m ⊗ t ,Gβ,n ⊗ t ]), for any α,β,m,n ∈ Z.

∼ C Therefore, by Proposition 2.8, we have LA(ℓ12, 0) = LA(ℓ2, 0) for any ℓ1,ℓ2 ∈ . e b Vertex algebra LA(ℓ2, 0) and its modules has been studied in Section 4 of [14]. b 15 5 Category O of Vq-modules e In this section, we introduce and study a category O of Vq-modules. We prove that objects in the category O are restricted Vq-modules. We alsoe classify simple modules in the category O. e Let b = CE ⊕ Cc ⊕ Cc . M k,l 1 2 l≥0,k∈Z

Let ℓ1,ℓ2 ∈ C. Given a b-module V with c1, c2 act as scalars ℓ1,ℓ2 respectively. Consider the induced module

Indℓ12 (V )= U(Vq) ⊗U(b) V. e We ﬁrst show that under certain conditions the induced module Indℓ12 (V ) is a simple Vq-module. e Theorem 5.1. Let V be a simple b-module and assume that there exists t ∈ Z+ such that

(1) Ek,t acts injectively on V for all k ∈ Z.

(2) Ek,lV =0 for all k ∈ Z, l > t. C Then for any ℓ1,ℓ2 ∈ , the induced module Indℓ12 (V ) is a simple Vq-module. ′ e Proof. Let v be a nonzero element in V . Let k ,k1 ∈ Z, j1 ∈ Z+. It is straightforward to show (by induction) that for any i1 ∈ N we have i ′ 1 C ′ Ek ,t+jEk1,−j1 v ∈ Ek +i1k1,t+j−i1j1 v ⊂ V for all j ≥ i1j1.

Let n ≥ 1, k1,k2,...,kn ∈ Z, j1, j2,...,jn ∈ Z+. Similarly, one can get that for any i1, i2,...,in ∈ N we have

in i2 i1 E ′ E ··· E E v ∈ CE ′ n n v ⊂ V k ,t+j kn,−jn k2,−j2 k1,−j1 k +Pl=1 ilkl,t+j−(Pl=1 iljl) n for all j ≥ iljl. In particular, Pl=1 n in i i E ′ E ··· E 2 E 1 v =0 for j > i j k ,t+j kn,−jn k2,−j2 k1,−j1 X l l l=1 and in i2 i1 E ′ n E ··· E E v ∈ CE ′ n v ⊂ V. k ,t+Pl=1 iljl k1,−jn k2,−j2 k1,−j1 k +Pl=1 ilkl,t

By PBW theorem, every nonzero element v of Indℓ12 (V ) can be uniquely written in the form of a ﬁnite sum w = Ekm,jm ··· Ek2,j2 Ek1,j1 vm, where vm ∈ V . With mP≥0 above and the assumptions (1) and (2), for any w ∈ Indℓ12 (V ), we can always arrive at a nonzero element in V . The simplicity of V then tells us that Indℓ12 (V ) is a simple Vq-module. e 16 Recall that a module V over a Lie algebra L is locally ﬁnite if for any v ∈ V , dim( Lnv) < +∞. nP∈Z+ For any t ∈ Z+, denote by

(t) V = CE . q X k,l e k∈Z,l≥t

It is a subalgebra of Vq. Let O be a category of Vq-modules which are locally ﬁnite e (t) e with respect to some Vq for t ∈ Z+. The following Lemmae tells us that all the modules in the category O are restricted modules.

Lemma 5.2. Let V be a Vq-module in the category O. Then for any nonzero vector e (t) v ∈ V , there exists t ∈ Z+ such that Vq v =0. e (r) Proof. Let 0 =6 v ∈ V . By assumption, there exists r ∈ Z+ such that Vq acts (r) (r) e locally finite on V . Hence W = U(Vq )v is a finite dimensional Vq -module. Let (r) e e (r) I ⊂ Vq be the kernel of the representation map, i.e. I = {x ∈ Vq | x.v = 0}. e (r) e Then I is an ideal of Vq of finite codimension. We claim that there exists k,l ∈ Z, l ≥ r, such that Ek,l e∈ I. If not, then there exists a minimal m ∈ N such that I contains

(s) (s) (s+m) (s+m) En,m := (a1 Ek11,s +···+an1 Ek1n1 ,s)+···+(a1 Ekm1,s+m +···+anm Ekmnm ,s+m) for some s ≥ r, kij ∈ Z, i = 1,...,m, j = 1,...,n1,...,nm, and complex numbers (l) (s) (s) aj satisfying a1 =6 0, an1 =6 0 if m = 0 (and then n1 is taken to be the minimal (s+m) Z one) or anm =6 0 if m ∈ +. Then I contains [Ek11,s, En,m] which is an element of (s) (s) (s+m) the form (b2 Ek11+k12,2s + ··· + bn1 Ek11+k1n1 ,2s)+ ··· +(b1 Ek11+km1,2s+m + ··· + (s+m) Z bnm Ek11+kmnm ,2s+m). If m = 0, then it contradicts to our choice of n1. If m ∈ +, then I contains

n −1 [E 1 n1−1−i n1−1 , ··· , [Ek11+k12,2s, [Ek11,s, En,m]] ··· ], Pi=1 2 k1i+k1n1 ,2 s

(s+1) (s+1) n1 n1 which is of the form (c1 Er21,(2 −1)s+s+1 + ··· + cn2 Er2n2 ,(2 −1)s+s+1)+ ··· + (s+m) (s+m) n n (c1 Erm1,(2 1 −1)s+s+m + ··· + cnm Ermnm ,(2 1 −1)s+s+m), this contradicts to our choice of m. (2r+l) Let Ek,l ∈ I. Then l ≥ r, k ∈ Z, Ek,lv = 0. We claim then Vq ⊂ I. For any (2r+l) e Em,n ∈ Vq . Then n ≥ 2r + l. We have e k(n−r)−lm lm−k(n−r) [Em−k,n−l−r, Ek,l]=(q − q )Em,n−r.

17 Case 1: If k(n − r)= lm. Take a nonzero integer s with ns =6 mr (for latter use). Then sl −sl [Em−k−s,n−l−r, Ek,l]=(q − q )Em−s,n−r gives that Em−s,n−r ∈ I. And then

ns−mr mr−ns [Em−s,n−r, Es,r]=(q − q )Em,n gives that Em,n ∈ I. Case 2: If k(n − r) =6 lm, then Em,n−r ∈ I. And then if m =6 0,

−mr mr [Em,n−r, E0,r]=(q − q )Em,n gives that Em,n ∈ I. For m =0. If k =6 0, then

kn −kn [E−k,n−l, Ek,l]=(q − q )E0,n gives that E0,n ∈ I. For m = 0 and k = 0, we are in Case 1. Hence the result holds.

As we need, we introduce the following notion. If a Vq-module V is generated by a vector 0 =6 v ∈ V with Ek,lv = 0 for all k ∈ Z and l>e0, then V is called a highest weight module. Harish-Chandra modules in [15] are highest weight modules.

Finally, we give a classiﬁcation of all simple Vq-module in the category O. e Theorem 5.3. Let S be a simple Vq-module in the category O. Then S is a highest weight module, or there exists ℓ1,ℓ2e∈ C, t ∈ Z+ and a simple b-module V such that ∼ both conditions (1) and (2) of Theorem 5.1 are satisﬁed and S = Indℓ12 V . Proof. By Lemma 5.2, for any nonzero v ∈ S, there exists some j ∈ N such that v is annihilated by all Ek,l, k ∈ Z, l > j. Consider the following vector space

Nj = {v ∈ S | Ek,lv = 0 for all k ∈ Z,l>j}.

We have Nj =6 0 for some j by the above argument. If N0 =6 0, then S is an irreducible highest weight module. Assume j ≥ 1. Thus we can ﬁnd a smallest positive integer, say t, with V := Nt =6 0. It is easy to check that V is a b-module. Note that c1,c2 act as scalars on V , say ℓ1,ℓ2 respectively. It follows from t is smallest that assumption (1) of Theorem 5.1 is satisﬁed. Then there is a canonical surjective (since S is simple) morphism

π : Indℓ12 V −→ S, 1 ⊗ v 7→ v, ∀v ∈ V. It remains to show that π is also injective. Let K = Ker(π). If K =6 0. Take

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