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Harish-Chandra modules over the generalized W -algebra W (2 , 2)

: In this paper, using the theory of A -cover developed in [2], we completely classify all simple Harish-Chandra modules for any generalized W -algebra W (2 , 2). As a byproduct, we also obtain the classiﬁcation of simple Harish-Chandra modules over the classical W -algebra W (2 , 2) studied in [7,12,17].


Introduction
In the representation theory of infinite-dimensional Lie algebras, there is a very important class of weight modules called Harish-Chandra modules (namely, weight modules with finitedimensional weight spaces). It is well-known that the classification of simple Harish-Chandra modules over the Virasoro algebra (also called N = 0 superconformal algebra), conjectured by Kac (see [13]), was given in [20]. Combining [23] with [27], a new method was presented to obtain this classification. After that, a lot of general versions of the Virasoro algebra have been investigated by some authors. Those include, but are not limited to, the generalized Virasoro algebra (see, e.g., [11,19,22,25,[28][29][30][31]), the generalized Heisenberg-Virasoro algebra (see [10,15,18]), the W -algebra W (2, 2) (see [7,12,17]), the loop-Virasoro algebra (see [12]), and so on. To classify all simple Harish-Chandra modules over the Lie algebra W n of vector fields on n-dimensional torus, Billig and Futorny developed a very useful technique called A-cover theory in [2]. The new result gained here was a generalized version of Mathieu's classification theorem for the Virasoro algebra. From then on, the A-cover theory was used in some other Lie (super)algebras (see, e.g., [3,4,7,32]).
The W -algebra W (2, 2) was introduced in [33] by Zhang and Dong for the investigated the classification of simple vertex operator algebras generated by two weight 2 vectors. The centerless W -algebra W (2, 2) W[Z] can be obtained from the point of view of non-relativistic analogues of the conformal field theory. By using the "non-relativistic limit" on a pair of commuting algebras vect(S 1 ) ⊕ vect(S 1 ) (see [26]) via a group contraction, one has the following generators L q = −t q+1 d dt − (q + 1)t q y d dy − (q + 1)σt q − q(q + 1)ηt q−1 y, where q ∈ Z, σ and η are respectively the scaling dimension and a free parameter. Then is the Lie algebra with the basis {L q , W q | q ∈ Z} and the non-vanishing commutators as follows [L q , L q ′ ] = (q ′ − q)L q+q ′ , [L q , W q ′ ] = (q ′ − q)W q+q ′ for q, q ′ ∈ Z. It is easy to see that W[Z] is an infinite-dimensional extension of an algebra called either non-relativistic or conformal Galilei algebra CGA(1) ∼ = L ±1,0 , W ±1,0 (see [9]). As everyone knows, the relationship with conformal algebras makes it widely studied in string theory (see [6]). Furthermore, W[Z] can be realized by the semidirect product of the Witt algebra V[Z] and the V[Z]-module A 0,−1 of the intermediate series in [14], that What we concern most is the approach of realizing W[Z] from a truncated loop-Witt algebra (see, e.g., [11,16]). The detailed description on it will be shown in Section 3.1, which is closely associated with the usage of A-cover theory. The aim of this paper is to present a completely classification of simple Harish-Chandra modules over the generalized W -algebra W (2, 2) (see Section 2.1). In particular, we classify the simple Harish-Chandra modules over the high rank W -algebra W (2, 2) (i.e., rank k W -algebra W (2, 2)), which reobtains the classification result of classical W -algebra W (2, 2) (when k = 1) studied in [7,12,17]. The method here mainly depends on the theory of A-cover given in [2].
The paper is organized as follows. In Section 2, we introduce some notations and definitions related to generalized W -algebra W (2, 2) and Harish-Chandra modules. We also recall some known classification theorems over several related Lie algebras for later use. In Section 3, we give a classification of all simple Harish-Chandra modules for any generalized W -algebras W (2, 2). We first use the theory of A-cover developed in [2] to determine the simple Harish-Chandra modules over the higher rank W -algebra W (2, 2), which give the classification result over the classic W -algebra W (2, 2) as a byproduct (also see [7,12,17]). Note that the method we employed here is conceptional and non-computational. At last, by the similar job in [10,11], we can generalize this result to the general case.
Throughout the present article, we denote by C, R, Z, N and Z + the sets of complex numbers, real numbers, integers, nonnegative integers and positive integers, respectively. All vector spaces and Lie algebras are over C. All simple modules are considered to be non-trivial. For a Lie algebra L, we use U(L) to denote the universal enveloping algebra.

Preliminaries
2.1 The generalized W -algebra W (2, 2) and its cuspidal module Let Γ be an additive subgroup of C. We first present the definition of generalized W -algebra W (2, 2) which satisfies the following Lie brackets

Generalized highest weight modules
In this subsection, a general class of Lie algebras are considered. Assume that H = α∈Γ H α is a Γ-graded Lie algebra such that H 0 is abelian. And the gradation of H is the root space decomposition with respect to H 0 .
Let g be a subgroup of Γ such that Γ = g ⊕ Zµ for some µ ∈ Γ. We define the subalgebra of H as follows Let K be a simple H g -module. Then K can be extended to an (H g + H + g )-module by defining H + g K = 0. Now we can define the generalized Verma module V g,µ,K for H as It is easy to know that V g,µ,K has a unique simple quotient module for H and we write it as P H g,µ,K . Then P H g,µ,K is called a simple highest weight module. As far as we know, the generalized Verma module (or generalized highest weight module) was introduced and investigated in some other references (see, e.g., [5,8,21]). Now we introduce an arbitrary H with Γ ∼ = Z k for k ∈ Z + . Fix a basis of Γ and identify . . , k}, then the element v ∈ M is called a generalized highest weight vector.
The following general result of cut H-modules appeared in Theorem 4.1 of [24].
Assume that M is a simple weight module over H, which is neither dense nor trivial. If M contains a generalized highest weight vector, then M is a cut module.

The know results
The classification theorems of simple Harish-Chandra modules over the classical W -algebra W (2, 2) and generalized Virasoro algebra will be recalled in this subsection.
It is well know that simple Harish-Chandra modules over the Virasoro algebra and Q-Virasoro algebra were respectively classified in [20] and [22]. These results were extended to arbitrary generalized Virasoro algebras (see [11,19,30]), and we summarize them as follows. g,µ,K for some µ ∈ Γ * , a subgroup g of Γ with Γ = g ⊕ Zµ and a non-trivial simple intermediate series g,γ,K be the simple module defined in Theorem 2.4. Then for any subgroup Γ of Γ and λ ∈ Supp(P ), the V[ Γ] module P λ+ Γ is cuspidal if and only if Γ ⊆ g.
Using Lemma 2.5 and similar proof in [15], we get the following result for the generalized W -algebra W (2, 2).
g,γ,K is the simple module defined in Section 2.2 with Γ ∼ = Z k for some k ∈ Z + . Then for any subgroup Γ of Γ and λ ∈ Supp(P ), the W[ Γ] module P λ+ Γ is cuspidal if and only if Γ ⊆ g.

AW[Γ]-module
In the following, we shall describe the structure of cuspidal W[Γ]-modules that admit a compatible action of the commutative unital algebra A. (3.1) Let M be a weight module over AW [Γ]. From (3.1), we see that the action of A is compatible with the weight grading of M: We suppose that AW[Γ]-module M has a weight space decomposition, and one of the weight spaces is finite-dimensional. Based on all non-zero homogeneous elements of A are invertible, we know that all weight spaces of M have the same dimension. Then M is also a free A-

module of a finite rank. It is clear that AW[Γ]-module M is cuspidal (as W[Γ]-modules).
Assume that M is a cuspidal AW[Γ]-module. Let W = M g for g ∈ Γ and dim(W ) < ∞. From M is a free A-module, we can write Proof. For any β, n ∈ Γ, by (3.1), we have Then there is a homomorphism of algebras χ : 1 ⊗ x → C such that 1 ⊗ x acts on M as So the action of W β t n on M can be written as nµt n+β , where µ ∈ C. Now from we get µ = 0 by taking n = 0, α = β, namely, (W β t n )M = 0. Putting this into (3.2), one has W α (t n v) = t n (W α v) for v ∈ M. The lemma has been proved.
Consider m j = 0. Then from the linearly independence of ǫ 1 , . . . , ǫ j , one has j−1 i=1 m i ǫ i − m j ǫ j = 0. By the induction assumption, we conclude that D( j i=1 m i ǫ i ) is a polynomial in m 1 , . . . , m j , where 1 < j ≤ k. Choosing j = k, one can see that D(m) is a polynomial in m 1 , . . . , m k . By the definition of operator D(m), one has D(0) = gId for g ∈ C. We complete the proof. Now we can write D(m) in the form (also see [1,3,15] where i! = k j=1 i j ! and only has a finite number of the nonzero operators D ( i) ∈ End(W ).
Note that D(0) = D ( 0) . For m, s ∈ Γ, by (3.4), we have Comparing the coefficients of s i m j i! j! in (3.6), we check that Then G = p∈N G p is a Z-graded Lie algebra (also see [1,3,15]). According to (3.7), we know that G 0 = span{D (ǫ i ) | i = 1, . . . , k} is a subalgebra of G, whose Lie algebra structure is presented as where i, j ∈ {1, 2, . . . , k}. It is easy to get that [G 0 , G 0 ] is nilpotent. Hence, G 0 is a solvable Lie algebra. Now we define the following one-dimensional G-module V (h) = Cv = 0 for any h ∈ C: (3.8) The following lemma was proved in [15].
Lemma 3.5. Assume that T and W are finite-dimensional simple modules over G 0 and G, respectively. Then (c) G p W = 0 for all p ∈ Z + and W ∼ = V (h) as a G 0 -module for some h ∈ C.
Theorem 3.6. Any simple cuspidal AW[Γ]-module is isomorphic to a module of intermediate series M (g, h; Γ) for some g, h ∈ C.

A-cover of a cuspidal W[Γ]-module
We first recall the definitions of coinduced module and A-cover (see [2]).    Then there exists r ∈ Z + such that for all α, β, γ ∈ Γ the following two operators
Since M is an A-module, we see that one of its weight spaces is finite-dimensional. For a fixed weight β ∈ Γ, we shall show that M β is finite-dimensional. Obviously, the space M β is spanned by the set Define a norm on Γ as follows For all q ∈ Γ and w ∈ M β+q , we have φ(L α−q , w) and φ(W α−q , w) in M α+β . Now we prove this claim by induction on q . If |q i | ≤ r 2 for all i ∈ {1, . . . , k}, the result clears. On the other hand, suppose |q i | > r 2 for some i ∈ {1, . . . , k}. We may assume q i < − r 2 , and the other case q i > − r 2 can be proved by the similar method. It is easy to see that q + jǫ i < q for all j ∈ {1, . . . , r}. We only need to give the proof for β + q = 0. From the action of L 0 on M β+q is a nonzero scalar, we can write w = L 0 v, where v ∈ M β+q . In the following, we shall verify in M . Based on Definition 3.8 and Lemma 3.10, for any m ∈ Γ we deduce Thus, one has By induction assumption the right hand sides of (3.12) and (3.13) belong to M α+β , and so do φ(L α−q , w), φ(W α−q , w). This proves the claim. Therefore, M α+β is finite-dimensional. The proposition follows.
The Claim 1 can also be described as follows.

Non-cuspidal modules
In this subsection, we will determine the simple weight modules with finite-dimensional weight spaces which are not cuspidal for the higher rank W -algebras W (2, 2). These modules are called generalized highest weight modules exactly and defined in Section 2.2. At last, the results in Section 3.1 and this subsection will be extended to the arbitrary generalized W -algebras W (2, 2). Now we let Γ ∼ = Z k for some k ∈ Z + .
The results of generalized Virasoro algebras appeared in Theorem 2.4 play a key role in the following proof. g,µ,K , where µ ∈ Γ * , g is a subgroup of Γ such that Γ = g ⊕ Zµ and K is a non-trivial simple cuspidal W[Γ] g -module.
Proof. Suppose that M is not a cuspidal module over W [Γ]. Let us recall that M = α∈Γ M α where M α = {w ∈ M | L 0 w = (g + α)w}. By Theorem 2.7, we see that the statement holds for any Γ of k = 1. Now suppose k ≥ 2. View M as a V[Γ]-module. Then based on Theorem 2.4(c), we obtain that the action of the central element C on M is trivial. Hence M can be seen as a W[Γ]-module. We fix a Z-basis of Γ, which is also suitable for Z k . For any σ ∈ R k and g ∈ Γ, we have the inner product (σ|g Since M is not cuspidal, we can find some rank k − 1 direct summand Γ of Γ such that M Γ is not cuspidal. Without loss of generality, we may assume that Γ is spanned by {ǫ 1 , ǫ 2 , . . . , ǫ k } \ {ǫ j }, where j ∈ {1, 2, . . . , k}. Then there exists some α ∈ Γ such that (3.14) For simplicity, we denote ξ j = α + ǫ j and ξ i = α + ǫ j + ǫ i for any i ∈ {1, 2, . . . , k} \ {j}. Then it is easy to check that the linear transformation sending each ǫ i to ξ i for any i ∈ {1, . . . , k}, has determinant 1 and therefore {ξ i | i = 1, . . . , k} is also a Z-basis of Γ. According to (3.14), there exists some nonzero element w ∈ M − α such that L ξ i w = W ξ i w = 0 for all i ∈ {1, . . . , k}. Thus, w is a generalized highest weight vector associated with the Z-basis {ξ i | i = 1, . . . , k}.
Based on Theorems 3.14 and 3.15, we give a classification of simple Harish-Chandra modules over the higher rank W -algebra W (2, 2). where Γ is an additive subgroup of C and Γ ∼ = Z k for some k ∈ Z + . Remark 3.19. It is clear that Theorem 3.18(1) is just Theorem 3.16 for k = 1 and Theorem 3.18(2) is just Lemma 3.17. Moreover, we remark that the method of the proof of Theorem 3.18(3) is exactly the same as the proof of Theorem 2.4(3). Hence, according to Lemma 2.6, Theorem 2.8 and Theorem 3.16, we transplant the proof of [11] here by replacing V[Γ] with W[Γ]. We omit the detailed process.