WEIGHT MODULES OVER EXP-POLYNOMIAL LIE ALGEBRAS

Yuly Billig^{a} and Kaiming Zhao^{b}

aSchool of Mathematics and Statistics Carleton University Ottawa, Ontario, K1S 5B6 Canada Email: billig@math.carleton.ca

and

bInstitute of Mathematics Academy of Mathematics and System Sciences Chinese Academy of Sciences Beijing 100080, P. R. China Email: kzhao@mail2.math.ac.cn

Abstract. In this paper, we generalize a result by Berman and Billig on weight modules over Lie algebras with polynomial multiplication. More precisely, we show that a highest weight module with an exp-polynomial “highest weight” has finite dimensional weight spaces. We also get a class of irreducible weight modules with finite dimensional weight spaces over generalized Virasoro algebras which do not occur over the classical Virasoro algebra.

0. Introduction.

Representations of affine Lie algebras and the Virasoro algebra have many important applications in mathematics and physics. One of the main ingredients of these theories is the construction of the highest weight modules. Recently there has been substantial activity in developing representation theories for higher rank

2000 Mathematics Subject Classification. 17B10, 17B65, 17B70. Key words and phrases. Lie algebra with exp-polynomial multiplication, Weight module. Research supported by Natural Sciences and Engineering Research Council of Canada (Y.B.)

and Hundred Talents Program of Chinese Academy of Sciences (K.Z.).

Typeset by AMS-TEX

infinite dimensional Lie algebras, in particular toroidal Lie algebras, generalized Virasoro algebras and quantum torus algebras (see [1], [2], [3], [4], [8], [9], [10], [11], [13], [15]).

Unlike rank one algebras (affine and Virasoro), the higher rank infinite dimensional Lie algebras do not possess a triangular decomposition, which makes the standard construction of the highest weight modules inapplicable. Nonetheless, there have been found several explicit realizations of representations for these algebras using the vertex operator approach.

In the vertex constructions the highest weight is replaced with a loop-like module for the subalgebra of degree zero (in general, this subalgebra is not commutative). Let us describe in brief these representations from the perspective of the highest weight modules.

Let G be a ℤ × ℤ^{n}-graded Lie algebra and let G = G^{−} ⊕ G^{(0)} ⊕ G^{+} be a decomposition of G relative to ℤ-grading. The subalgebra G^{(0)} is an infinite-dimensional Lie algebra of rank n. We take some natural module V for G^{(0)} (usually V is either ℤ^{n}-graded or finite dimensional). Parallel to the construction of a highest weight module, we let G^{+} act on V trivially, and introduce the induced module

M̃(V) = Ind_{G^{(0)}+G^{+}}^{G}V ≃ U(G^{−}) ⊗_{ℂ} V.

If V is ℤ^{n}-graded then M̃(V) inherits a ℤ × ℤ^{n}-grading, and it is ℤ-graded when V is finite-dimensional.

The difficulty here is that M̃(V) will have infinite-dimensional homogeneous components (and thus will not have a character). Nonetheless the explicit vertex operator constructions show that in some cases M̃(V) has quotients with finitedimensional homogeneous components. This situation has been clarified in [1], where it was proved that M̃(V) has a graded factor-module M(V) with finitedimensional components provided that G is a polynomial Lie algebra and V is a polynomial module. The polynomiality condition means that the structure constants of the Lie algebra and of the module are given by polynomial expressions (see Section 1 for the precise definitions). However this left out some important examples, notably quantum torus Lie algebras. In this paper we expand the class of Lie algebras and modules for which the theorem is applicable. We prove that M(V) has finite-dimensional homogeneous components when G is an exp-polynomial Lie algebra and V is an exp-polynomial module.

We illustrate our definitions and theorems with a sequence of 24 examples.

The paper is organized as follows. In Section 1 we define exp-polynomial Lie algebras, exp-polynomial modules and give the statement of the main results (Theorems 1.5 and 1.7). We provide numerous examples (old and new) of such Lie algebras and modules as well as give examples of applications of the main theorems. In Section 2 we first establish the extended Vandermonde determinant formula, and then give the proof of Theorems 1.5 and 1.7. In Section 3 we show a similar but stronger result for generalized Virasoro algebras to give a class of irreducible weight modules with finite dimensional weight spaces over generalized Virasoro algebras which do not occur over the classical Virasoro algebra.

1. Exp-polynomial Lie algebras and exp-polynomial modules.

Let ℂ be the complex number field. We assume that all Lie algebras and vector spaces are over ℂ in this paper, although ℂ can be replaced by any field of characteristic 0.

Definition 1.1. The algebra of exp-polynomial functions in r variables, n_{1}, …, n_{r}, is the algebra of functions f(n_{1}, …, n_{r}) : ℤ^{r} → ℂ generated as an algebra by functions n_{j} and a^{n_{j}}, where a ∈ ℂ^{∗} = ℂ \ {0}, j = 1, …, r.

An exp-polynomial function may be written as a finite sum

f(n_{1}, …, n_{r}) = ∑_{k∈ℤ_{+}^{r}} ∑_{a∈(ℂ^{∗})^{r}} c_{k,a}n_{1}^{k_{1}} … n_{r}^{k_{r}}a_{1}^{n_{1}} … a_{r}^{n_{r}},

where c_{k,a} ∈ ℂ, k = (k_{1}, …, k_{r}) with k_{j} ≥ 0, and a = (a_{1}, …, a_{r}).

An exp-polynomial function f(n) has the property that the function f(n + m) is also exp-polynomial as a function of two variables.

Definition 1.2. Let G = ⊕_{α∈ℤ^{n}}G_{α} be a ℤ^{n}-graded Lie algebra. Then G is said to be an exp-polynomial Lie algebra if G has a homogeneous spanning set {g_{k}(α) | k ∈ K, α ∈ ℤ^{n}} with g_{k}(α) ∈ G_{α}, and there exists a family of exppolynomial functions {f_{k,r}^{s}(α, β) | k, r, s ∈ K} in the 2n variables α_{j}, β_{j} and where for each k, r the set {s|f_{k,r}^{s}(α, β) ≠ 0} is finite, such that

[g_{k}(α), g_{r}(β)] = ∑_{s∈K} f_{k,r}^{s}(α, β)g_{s}(α + β), for k, r ∈ K, α, β ∈ ℤ^{n}.  (1.1)

This homogeneous spanning set {g_{k}(α) | k ∈ K, α ∈ ℤ^{n}} is call a distinguished spanning set.

If the functions f_{k,r}^{s}(α, β) in (1.1) are in fact polynomials, we say that G is a polynomial Lie algebra (cf. Definition 1.6 in [1]).

Example 1. Let 𝔤 be a finite dimensional Lie algebra with a basis B = {g_{k}}_{k∈K}. Then the toroidal Lie algebra G = ℂ[t_{1}^{±}, …, t_{n}^{±}]⊗𝔤 is a polynomial Lie algebra with the distinguished spanning set g_{k}(α) = t^{α}g_{k}, α ∈ ℤ^{n}, where here and elsewhere in this paper t^{α} = t_{1}^{α_{1}} … t_{n}^{α_{n}}. The Lie bracket in G is

[g_{k}(α), g_{r}(β)] = [g_{k}, g_{r}](α + β).

Example 2. Let R_{n} = ℂ[t_{1}^{±1}, …, t_{n}^{±1}]. The Witt algebra (or Cartan type W Lie algebra) is the Lie algebra W_{n} = Der(R_{n}) = Span {t^{α}∂_{i}|α ∈ ℤ^{n}, 1 ≤ i ≤ n}, where ∂_{i} = t_{i} \frac{∂}{∂t_{i}}. The bracket in W_{n} is given by

[t^{α}∂_{i}, t^{β}∂_{j}] = β_{i}t^{α+β}∂_{j} − α_{j}t^{α+β}∂_{i}.

Thus W_{n} is a polynomial Lie algebra.

Example 3. The following family of algebras plays an important role in the representation theory of toroidal Lie algebras. Let Ω_{n}^{1} be the space of 1-forms on

a torus: Ω_{n}^{1} = ∑_{p=1}^{n} R_{n}k_{p}, where k_{p} = t_{p}^{−1}dt_{p}. We define a 2-parametric family of

algebras

V (μ, ν) = W_{n} ⊕ Ω_{n}^{1}/dR_{n}.

The distinguished spanning set is

{d_{j}(α), k_{j}(α)|α ∈ ℤ^{n}, j = 1, … n},

where d_{j}(α) = t^{α}∂_{j}, k_{j}(α) = t^{α}k_{j}. The Lie bracket in V (μ, ν) is given by

[d_{i}(α), k_{j}(β)] = β_{i}k_{j}(α + β) + δ_{ij} ∑_{p=1}^{n} α_{p}k_{p}(α + β),

[d_{i}(α), d_{j}(β)] = β_{i}d_{j}(α + β) − α_{j}d_{i}(α + β) + (μβ_{i}α_{j} + να_{i}β_{j}) ∑_{p=1}^{n} β_{p}k_{p}(α + β),

[k_{i}(α), k_{j}(β)] = 0.

Note that the distinguished spanning set for this polynomial Lie algebra is not a basis because of the linear dependencies between k_{j}(α): α_{1}k_{1}(α)+…+α_{n}k_{n}(α) = 0.

Next we will give an example of a Lie algebra which is exp-polynomial, but not polynomial.

Example 4. Consider an associative quantum torus

ℂ_{q} = ⊕_{i∈ℤ,α∈ℤ^{n}} ℂt_{0}^{i}t^{α}

generated by the variables t_{0}^{±}, t_{1}^{±}, …, t_{n}^{±}, where t_{1}, …, t_{n} commute and t_{0} does not commute with t_{1}, …, t_{n}, but satisfies the relations: t_{j}t_{0} = q_{j}t_{0}t_{j}, for some q_{1}, …, q_{n} ∈ ℂ^{∗}. The Lie algebra which is obtained from the associative algebra C_{q} is an exp-polynomial Lie algebra with the distinguished set t_{0}^{i}(α) = t_{0}^{i}t^{α}, and the Lie bracket given by [t_{0}^{i}(α), t_{0}^{j}(β)] = (q^{jα}−q^{iβ})t_{0}^{i+j}(α + β). Here q^{jα} = q_{1}^{jα_{1}} … q_{n}^{jα_{n}}.

Definition 1.3. Let G = ⊕_{α∈ℤ^{n}}G_{α} be an exp-polynomial Lie algebra. A G module V = ⊕_{α∈ℤ^{n}}V_{α} is called a ℤ^{n}-graded exp-polynomial module if V has a basis {v_{j}(α)}_{j∈J,α∈ℤ^{n}}, and there exists a family of exp-polynomial functions h_{k,j}^{s}(α, β) for k ∈ K, j, s ∈ J such that

g_{k}(α)v_{j}(β) = ∑_{s∈J} h_{k,j}^{s}(α, β)v_{s}(α + β),

where {g_{k}(α)}_{k∈K} is the distinguished spanning set for G, and for each k, j the set {s|h_{k,j}^{s}(α, β) ≠ 0} is finite. The homogeneous components of the ℤ^{n}-grading on V = ⊕_{α∈ℤ^{n}} V_{α} are given by V_{α} = Span {v_{j}(α)}_{j∈J}.

Note that if G is a Lie algebra with polynomial multiplication and all h_{k,j}^{s} are polynomial functions, then the module V is actually a polynomial module defined in [1], Definition 1.8.

Example 5. Let V_{1}, …, V_{k} be finite dimensional modules for a finite dimensional Lie algebra 𝔤. Fix q_{1}, …, q_{k} ∈ (ℂ^{∗})^{n}. We define on the space V = R_{n} ⊗ V_{1} ⊗ … ⊗ V_{k} the structure of an exp-polynomial module for the toroidal Lie algebra G = R_{n} ⊗ 𝔤 by

g(α)v_{1} ⊗ … ⊗ v_{k}(β) = ∑_{p=1}^{k} q_{p}^{α_{p}}v_{1} ⊗ … (gv_{p}) ⊗ … ⊗ v_{k}(α + β),

where v_{1} ⊗ … ⊗ v_{k}(β) = t^{β}v_{1} ⊗ … ⊗ v_{k}. It is easy to see that in general V is an exp-polynomial module and not a polynomial module.

Example 6. A tensor module for the Witt algebra W_{n} is a polynomial module. Let V_{0} be a finite-dimensional gl_{n} module. Then the tensor module V = R_{n} ⊗ V_{0} is a module for W_{n} under the action

t^{α}∂_{j}v(β) = β_{j}v(α + β) + ∑_{p=1}^{n} α_{p}(E_{pj}v)(α + β),

where v(β) = t^{β} ⊗ v, and E_{pj} are the elementary matrices in gl_{n}. The families {v_{i}(β)} with {v_{i}} being a basis of V_{0}, form a distinguished basis in V, for which the structure constants are polynomials.

We extend Definition 1.6 from [1] to give the following

Definition 1.4. Let G be a ℤ^{n}-graded exp-polynomial Lie algebra. We call this algebra ℤ^{n}-extragraded if G has another ℤ-gradation

G = ⊕_{i∈ℤ} G^{(i)},  (1.2)

and the set K is a disjoint union of finite subsets K_{i},

K = ∪_{i∈ℤ} K_{i},

such that the elements of the homogeneous spanning set {g_{k}(α) | k ∈ K_{i}, α ∈ ℤ^{n}} are homogeneous of degree i under this new ℤ-gradation and span G^{(i)}.

Many important infinite-dimensional Lie algebras are in fact ℤ^{n}-extragraded exppolynomial Lie algebras. Here we give some examples.

Example 8. We slightly modify Example 1 to get an extragraded algebra. Adding an extra variable t_{0}, we get an n + 1 toroidal Lie algebra R_{n+1} ⊗ 𝔤, where R_{n+1} = ℂ[t_{0}^{±}, t_{1}^{±}, …, t_{n}^{±}]. We consider a ℤ^{n}-grading on this algebra by degrees in t_{1}, …, t_{n}, and a ℤ grading by degree in t_{0}. The distinguished spanning set is

t_{0}^{i}g_{k}(α) = t_{0}^{i}t^{α}g_{k},

with i ∈ ℤ, α ∈ ℤ^{n}, and {g_{k}} being a basis of 𝔤.

The previous example can be generalized in two ways:

Example 9. Let G^{(0)} be an exp-polynomial Lie algebra with G_{α}^{(0)} being finite dimensional for all α ∈ ℤ^{n}. Then G = ℂ[t_{0}, t_{0}^{−1}] ⊗ G^{(0)} is an extragraded exppolynomial Lie algebra.

Example 10. Let 𝔤 be a ℤ-graded Lie algebra with finite-dimensional homogeneous components. Then G = 𝔤⊗R_{n} is an extragraded exp-polynomial Lie algebra.

Remark. In the definition of an exp-polynomial Lie algebra we can relax the requirement that each family {g_{k}(α)} be defined for all α ∈ ℤ^{n}. We may instead require that {g_{k}(α)} is defined for α in some sublattice L_{k} ⊂ ℤ^{n}. Of course in this case we should have a restriction so that the expression in the right hand side of (1.1) is well defined.

Example 11. With this relaxed definition we may consider the following as an extragraded exp-polynomial Lie algebra:

G = 𝔤 ⊗ R_{n+1} ⊕ Der ℂ[t_{0}, t_{0}^{−1}]

for a finite dimensional Lie algebra 𝔤 with the distinguished spanning set {t_{0}^{i}g_{k}(α), t_{0}^{i}∂_{0}|i ∈ ℤ, α ∈ ℤ^{n}, g_{k} ∈ B}. Here the sublattice that corresponds to the elements t_{0}^{i}∂_{0} is L = (0).

Example 12. Introducing an extra variable t_{0} we can construct an extragraded version of the Lie algebra from Example 3:

V (μ, ν) = DerR_{n+1} ⊕ Ω_{n+1}^{1}/dR_{n+1}.

The ℤ-grading is by degree in t_{0}, and the distinguished spanning families are t_{0}^{i}d_{j}(α) = t_{0}^{i}t^{α}∂_{j} and t_{0}^{i}k_{j}(α) = t_{0}^{i}t^{α}k_{j}.

Example 13. The exp-polynomial Lie algebra from Example 4 is actually extragraded. The ℤ-grading on it is by degree in t_{0}.

Example 14. Let R_{n+1} = ℂ[t_{0}^{±1}, t_{1}^{±1}, …, t_{n}^{±1}], W_{n+1} = Der(R_{n+1}) = Span {t_{0}^{i}t^{α}∂_{k}, t_{0}^{i}t^{α}∂_{0}|i ∈ ℤ, α ∈ ℤ^{n}, 1 ≤ k ≤ n}. The Lie bracket in W_{n+1} is given by

[t_{0}^{i}t^{α}∂_{k}, t_{0}^{j}t^{β}∂_{s}] = t_{0}^{i+j}t^{α+β}(β_{k}∂_{s} − α_{s}∂_{k}), 1 ≤ k, s ≤ n,
[t_{0}^{i}t^{α}∂_{0}, t_{0}^{j}t^{β}∂_{s}] = t_{0}^{i+j}t^{α+β}(j∂_{s} − α_{s}∂_{0}), 1 ≤ s ≤ n,

[t_{0}^{i}t^{α}∂_{0}, t_{0}^{j}t^{β}∂_{0}] = (j − i)t_{0}^{i+j}t^{α+β}∂_{0}.

It can be easily seen that W_{n+1} is a ℤ^{n}-extragraded Lie algebra.

Cartan type S Lie algebras S_{n+1} are also ℤ^{n}-extragraded polynomial Lie algebras. However we do not know whether Cartan type H or K Lie algebras are ℤ^{n}-extragraded polynomial Lie algebras.

Example 15. The Virasoro-like algebra L over ℂ is the Lie algebra with a ℂ-basis {L_{x} | x ∈ ℤ^{2}} and subject to the following commutator relations

[L_{x}, L_{y}] = det ( \begin{matrix}y \\ x \end{matrix} ) L_{x+y}, ∀ x, y ∈ ℤ^{2},

where x = (x^{(1)}, x^{(2)}), y = (y^{(1)}, y^{(2)}), ( \begin{matrix}y \\ x \end{matrix} ) = ( \begin{matrix}y^{(1)} & y^{(2)} \\ x^{(1)} & x^{(2)} \end{matrix} ). It can be easily seen

that L is a ℤ^{1}-extragraded Lie algebra.

From now on we assume that G is a ℤ^{n}-extragraded Lie algebra with gradations (1.1) and (1.2), i.e., G = ⊕_{i∈ℤ, α∈ℤ^{n}}G_{α}^{(i)} is a ℤ^{n+1}-graded Lie algebra which has a homogeneous spanning set {g_{k}^{(i)}(α) | k ∈ K_{i}, (i, α) ∈ ℤ^{n+1}} with g_{k}^{(i)}(α) ∈ G_{α}^{(i)}, and there exists a family of exp-polynomial functions {f_{k,m,i,j}^{s}(α, β)} in the 2n variables α_{p}, β_{p} where k ∈ K_{i}, m ∈ K_{j}, s ∈ K_{i+j} and where for each k, m, i, j the set {s|f_{k,m,i,j}^{s}(α, β) ≠ 0} is finite, such that

[g_{k}^{(i)}(α), g_{m}^{(j)}(β)] = ∑_{s∈K_{i+j}} f_{k,m,i,j}^{s}(α, β)g_{s}^{(i+j)}(α + β), for α, β ∈ ℤ^{n}.  (1.3)

Let G^{+} = ⊕_{i≥1} G^{(i)}, G^{−} = ⊕_{i≤−1} G^{(i)}. Then we have the decomposition

G = G^{−} ⊕ G^{(0)} ⊕ G^{+}.  (1.4)

Note that G^{(0)} is a ℤ^{n}-graded exp-polynomial Lie algebra.

Following the construction in [1], we now introduce our ℤ^{n+1}-graded module over ℤ^{n+1}-graded Lie algebra G.

Assume V = ⊕_{α∈ℤ^{n}}V_{α} is a ℤ^{n}-graded G^{(0)} module with exp-polynomial action as defined in Definition 1.3. We can define the action of G^{+} on V by G^{+}V = 0 and then consider the induced module

M̃(V) = Ind_{G^{(0)}+G^{+}}^{G}V ≃ U(G^{−}) ⊗_{ℂ} V.  (1.5)

It is clear that M̃(V) is a ℤ^{n+1}-graded module over G and

M̃(V) = ⊕_{i≤0,α∈ℤ^{n}} M̃(V)_{α}^{(i)},  (1.6)

where M̃(V)_{α}^{(i)} is naturally defined, for example, M̃(V)_{α}^{(0)} = V_{α}. In general, the homogeneous components M̃(V)_{α}^{(i)} with i < 0 are infinite-dimensional.

It is easy to see that M̃(V) has a unique maximal proper ℤ^{n+1}-graded submodule M̃^{rad} which intersects trivially with V. Let

M(V) = M̃(V)/M̃^{rad}.  (1.7)

Then we have the induced ℤ^{n+1}-gradation

M(V) = ⊕_{i≤0,α∈ℤ^{n}} M(V)_{α}^{(i)}.  (1.8)

The main result of this paper is the following theorem.

Theorem 1.5. Assume that G is a ℤ^{n}-extragraded Lie algebra with grading (1.3), V = ⊕_{α∈ℤ^{n}}V_{α} is a ℤ^{n}-graded exp-polynomial G^{(0)} module as defined in Definition 1.3 with J being finite. Then the ℤ^{n+1}-graded G module M(V) defined in (1.7) has finite-dimensional homogeneous spaces, i.e., dim M(V)_{α}^{(i)} < ∞, for all i ∈ ℤ, α ∈ ℤ^{n}.

If G^{(0)} is a polynomial Lie algebra and V is a polynomial ℤ^{n}-graded G^{(0)} module, the claim of Theorem 1.5 was proved in [1], Theorem 1.12.

Example 16. Let us consider an n + 1-toroidal Lie algebra G = R_{n+1} ⊗ 𝔤 as defined in Example 8. Its component G^{(0)} is an n-toroidal Lie algebra. We consider a module V = R_{n} ⊗ V_{1} ⊗ … ⊗ V_{k} for G^{(0)} which was described in Example 5. By the above theorem, the G module M(V) has finite-dimensional homogeneous components. Such modules were studied by Chari [5] and Rao [7].

Example 17. Let us consider the Virasoro-like algebra L as defined in Example 15. Let L^{(i)} = ⊕_{k∈ℤ}ℂL_{i,k}. Fix an exp-polynomial function f(k). Then V = ⊕_{j∈ℤ}ℂv_{j} becomes an exp-polynomial L^{(0)} module via

L_{0,k}v_{j} = f(k)v_{j+k}.

Then M(V) has finite dimensional homogeneous spaces.

Example 18. Consider a Witt algebra W_{n+1} = Der ℂ[t_{0}^{±}, t_{1}^{±}, …, t_{n}^{±}], which we view as an extragraded polynomial Lie algebra with ℤ grading given by degree in t_{0}. The zero component with respect to this grading is W_{n} ⊕ R_{n}∂_{0}. Consider a tensor module V = R_{n} ⊗ V_{0} for W_{n} as discussed in Example 6. We let R_{n}∂_{0} act upon it by shifts t^{α}∂_{0} · t^{β} ⊗ v = d t^{α+β} ⊗ v for some fixed constant d ∈ ℂ. By Theorem 1.5 the module M(V) is a weight module with finite dimensional weight spaces.

Example 19. Let V (μ, ν) be the extragraded Lie algebra from Example 12. Its zero component with respect to ℤ grading is

W_{n} ⊕ R_{n}∂_{0} ⊕ (∑_{p=0}^{n} R_{n}k_{p}/dR_{n}) .

Consider a tensor module V from the previous example on which we define the action of 1-forms of degree zero (in t_{0}) as follows:

t^{α}k_{p} · t^{β} ⊗ v = 0, p = 1, …, n,
t^{α}k_{0} · t^{β} ⊗ v = ct^{α+β} ⊗ v, for some c ∈ ℂ.

Again by Theorem 1.5 the module M(V) is a weight module with finite dimensional weight spaces. These modules and their irreducible quotients were studied in [1], [3], [4], [8], [13].

Example 20. Let G be the quantum torus Lie algebra from Example 4. We consider a module V = ℂ[t_{1}^{±}, … t_{n}^{±}] for G^{(0)} = ℂ[t_{1}^{±}, … t_{n}^{±}] with the action defined as follows: t^{α} · t^{β} = f(α, β)t^{α+β}, where f(α, β) is some fixed exp-polynomial function. By Theorem 1.5 the module M(V) has finite-dimensional homogeneous components.

Example 21. Let 𝔤 be a simple finite-dimensional Lie algebra with the triangular decomposition 𝔤 = 𝔤_{−} ⊕𝔥⊕𝔤_{+}. We consider a finite ℤ-grading on 𝔤 compatible with this decomposition. As explained in Example 10 above, the algebra G = R_{n} ⊗ 𝔤 is an extragraded polynomial Lie algebra. Its zero component is the abelian algebra G^{(0)} = R_{n} ⊗𝔥. Consider a G^{(0)} module V = ℂ[t_{1}^{±}, … t_{n}^{±}] with the action t^{α}h_{k} ·t^{β} = f_{k}(α, β)t^{α+β}, where {h_{k}} forms a basis of 𝔥 and {f_{k}(α, β)} are some fixed exppolynomial functions. Applying again Theorem 1.5 we conclude that M(V) is a weight module with finite dimensional weight spaces.

We would like to discuss now a finite-dimensional version of the exp-polynomial modules.

Definition 1.6. Let G = ⊕_{α∈ℤ^{n}}G_{α} be an exp-polynomial Lie algebra as defined in Definition 1.2. A G-module V is called a finite-dimensional exp-polynomial module if V has a finite basis {v_{j}}_{j∈J}, and there exists a family of exp-polynomial functions h_{k,j}^{s}(α) for k ∈ K, j, s ∈ J such that

g_{k}(α)v_{j} = ∑_{s∈J} h_{k,j}^{s}(α)v_{s},

where {g_{k}(α)}_{k∈K,α∈ℤ^{n}} is the distinguished spanning set for G.

Let G be an extragraded exp-polynomial Lie algebra and let V be a finitedimensional exp-polynomial module for G^{(0)}. Just as in (1.5) and (1.7) we define a ℤ-graded G module M̃(V) and its ℤ-graded factor-module M(V) = M̃(V)/M̃^{rad}.

Theorem 1.7. Let G be a ℤ^{n}-extragraded Lie algebra with grading (1.3), V be a finite-dimensional exp-polynomial G^{(0)} module. Then the ℤ-graded G module M(V) has finite dimensional homogeneous spaces, i.e., dim M(V)^{(i)} < ∞, for all i ∈ ℤ.

Let us now give some examples of the applications of this theorem.

Example 22. Let G be an n + 1-toroidal Lie algebra G = R_{n+1} ⊗ 𝔤 as defined in Example 8. Its component G^{(0)} is an n-toroidal Lie algebra. Let V_{1}, …, V_{k} be finite-dimensional modules for 𝔤, and let q_{1}, …, q_{k} ∈ (ℂ^{∗})^{n}. We define a structure of a G^{(0)} module on the finite-dimensional space V = V_{1} ⊗ … ⊗ V_{k} in the following way:

g(α)v_{1} ⊗ … ⊗ v_{k} = ∑_{p=1}^{k} q_{p}^{α_{p}}v_{1} ⊗ … (gv_{p}) ⊗ … ⊗ v_{k}.

It is easy to see that V is a finite-dimensional exp-polynomial module for G^{(0)}. The induced module M̃(V) will have infinite-dimensional homogeneous components, however, by Theorem 1.7, the homogeneous components of its factor M(V) are finite-dimensional.

The next two examples are modifications of Examples 20 and 21.

Example 23. Let G be the quantum torus Lie algebra from Example 4. We consider a one-dimensional module ℂ for G^{(0)} = ℂ[t_{1}^{±}, … t_{n}^{±}] with the action defined as follows: t^{α}·1 = f(α)1, where f(α) is some fixed exp-polynomial function (highest weight). By Theorem 1.7 the module M(ℂ) has finite-dimensional homogeneous components.

Example 24. Let 𝔤 be a simple finite-dimensional Lie algebra with the triangular decomposition 𝔤 = 𝔤_{−} ⊕ 𝔥 ⊕ 𝔤_{+}. As explained in Example 21 above, the algebra G = R_{n} ⊗ 𝔤 is an extragraded polynomial Lie algebra. Its zero component is the abelian algebra G^{(0)} = R_{n} ⊗𝔥. Let {h_{1}, …, h_{ℓ}} be a basis of 𝔥. Fix exp-polynomial functions f_{1}(α), …, f_{ℓ}(α) (highest weight) and consider a one-dimensional G^{(0)} module ℂ with the action t^{α}h_{k} · 1 = f_{k}(α)1. Applying again Theorem 1.7 we conclude that M(ℂ) is a ℤ-graded module with finite dimensional homogeneous components.

2. Proof of the main theorems.

In this section we shall prove Theorems 1.5 and 1.7. The key step in the proof of Theorem 1.12 in [1] was the Vandermonde determinant argument. Here we would need a generalization of the Vandermonde determinant formula for the exppolynomial functions.

Lemma 2.1. Let a_{1}, …, a_{m} be elements of a field, s_{1}, s_{2}, …, s_{m} ∈ ℕ with s_{1} + … + s_{m} = s. Consider the following sequence of s exp-polynomial functions in one integer variable: f_{1}(n) = a_{1}^{n}, f_{2}(n) = na_{1}^{n}, …, f_{s_{1}}(n) = n^{s_{1}−1}a_{1}^{n}, f_{s_{1}+1}(n) = a_{2}^{n}, …, f_{s_{1}+s_{2}}(n) = n^{s_{2}−1}a_{2}^{n}, …, f_{s}(n) = n^{s_{m}−1}a_{m}^{n}. Let V = (v_{pk}) be the square s × s matrix where v_{pk} = f_{k}(p − 1), p, k = 1, … s. Then

det(V) = Π_{j=1}^{m}(s_{j} − 1)!! aj^{\frac{s_{j} (s_{j} −1)}{2}} Π_{1≤i<j≤m}(a_{j} − a_{i})^{s_{i}s_{j}}.  (2.1)

Here we use the notation m!! = m! × (m − 1)! × … × 2! × 1! with the convention 0!! = 1.

Proof. This elementary lemma may be proved by induction on s using elementary row and column transformations of the matrix. We will give here just an outline of the proof. The basis of induction case s = 1 is trivial. To establish the inductive step, one begins by applying elementary row operations to the matrix, the same as in the proof of the ordinary Vandermonde determinant formula: subtract from the last row the preceding row number s − 1 multiplied by a_{1}, then subtract from row s − 1 the preceding row s − 2 multiplied by a_{1}, and so on. This will produce a matrix that has 1 as the top entry of the first column, and the rest of the entries in the first column being zeros.

Next we expand this determinant along the first column, which will yield an (s − 1) × (s − 1) matrix with the same determinant. Finally, applying elementary column operations, it is possible to bring this (s − 1) × (s − 1) matrix to the form corresponding to the sequence of functions a_{1}a_{1}^{n}, 2a_{1}na_{1}^{n}, 3a_{1}n^{2}a_{1}^{n}, …, (s_{1} −1)a_{1}n^{s_{1}−2}a_{1}^{n}, (a_{2} −a_{1})a_{2}^{n}, (a_{2} −a_{1})na_{2}^{n}, …, (a_{2} −a_{1})n^{s_{2}−1}a_{2}^{n}, (a_{3} −a_{1})a_{3}^{n}, …, (a_{m} − a_{1})n^{(s_{m}−1)}a_{m}^{n}.

Pulling out the factors a_{1}, 2a_{1}, 3a_{1}, …, (s_{1} − 1)a_{1} from the first s_{1} − 1 columns, and (a_{j} − a_{1}) from the remaining columns, we bring the matrix to the extended Vandermonde form of rank s − 1. To establish the inductive step, we only need to multiply the expression for the extended Vandermonde determinant of rank s − 1 given by the induction assumption by the factors we pulled out of the columns. □

Corollary 2.2. In the notations of the previous lemma, let a_{1}, …, a_{m} be distinct non-zero elements of a field of characteristic greater or equal to the maximum of s_{1}, … s_{m}, or of characteristic 0. Then the set of exp-polynomial functions f_{1}(n) = a_{1}^{n}, …, f_{s}(n) = n^{s_{m}−1}a_{m}^{n} is linearly independent.

Proof. From the determinant formula (2.1) we see that the vectors of values of the functions (f_{j}(0), f_{j}(1), …, f_{j}(s − 1)), j = 1, …, s, are linearly independent. □

Corollary 2.3. Let the exp-polynomial functions f_{1}(n), … f_{s}(n) satisfy the conditions of Lemma 2.1 and Corollary 2.2. Let {c_{k}} be a sequence with only finitely many non-zero terms. The sequence {c_{k}} satisfies an infinite system of linear equations

∑_{k} (∑_{j=1}^{s} d_{kj}f_{j}(n)) c_{k} = 0 for all n ∈ ℤ  (2.2)

if and only if it satisfies a finite system

∑_{k} d_{kj}c_{k} = 0 for all j = 1, …, s.  (2.3)

Proof. Since

∑_{k} (∑_{j=1}^{s} d_{kj}f_{j}(n)) c_{k} = ∑_{j=1}^{s} f_{j}(n) ∑_{k} d_{kj}c_{k},

we see that (2.3) implies (2.2).

Suppose now that (2.2) holds. Evaluating (2.2) at n = 0, …, s − 1, we get that

∑_{j=1}^{s} f_{j}(n) ∑_{k} d_{kj}c_{k} = 0 for n = 0, …, s − 1.  (2.4)

Now s equations of (2.4) are linear combinations of s equations of (2.3). The change of basis matrix from (2.3) to (2.4) is an extended Vandermonde matrix and is invertible, since by Lemma 2.1 it has a non-zero determinant. Hence, the equations of (2.3) are linear combinations of equations of (2.4) and thus (2.3) holds. □

The last Corollary also admits a straightforward multi-variable generalization.

Corollary 2.4. Let f_{1}(n_{1}, …, n_{r}), … f_{s}(n_{1}, …, n_{r}) be a set of distinct exppolynomial functions of the form f_{j}(n_{1}, …, n_{r}) = n_{1}^{p_{1}} … n_{r}^{p_{r}}b_{1}^{n_{1}} … b_{r}^{n_{r}}, such that exponents p_{i}’s are less than the characteristic of the field, in case if the field is of finite characteristic. Let {c_{β}}_{β∈ℤ^{m}} be a set with only finitely many non-zero terms. The set {c_{β}} satisfies an infinite system of linear equations

∑_{β∈ℤ^{m}} (∑_{j=1}^{s} d_{βj}f_{j}(n_{1}, …, n_{r})) c_{β} = 0 for all (n_{1}, …, n_{r}) ∈ ℤ^{r}  (2.5)

if and only if it satisfies a finite system

∑_{β∈ℤ^{m}} d_{βj}c_{β} = 0 for all j = 1, …, s.  (2.6)

Now we are ready to give a proof of Theorem 1.5.

Proof of Theorem 1.5. For the ℤ^{n}-graded G^{(0)} module V = ⊕_{α∈ℤ^{n}}V_{α} as defined in Definition 1.3, we have used g_{i}(α) to denote g_{i}^{(0)}(α). We stress here that both {k | g_{k}^{(i)}(α) ≠ 0} for any fixed (i, α) ∈ ℤ^{n+1} and J in Definition 1.3 are finite sets. The proof of the theorem will amount to proving two claims.

Claim 1. Let us fix i_{1}, i_{2}, …, i_{s} ∈ ℕ, k_{1}, k_{2}, …, k_{s}, with k_{p} ∈ K_{−i_{p}}, k ∈ J and α ∈ ℤ^{n}. There exist exp-polynomial functions f_{1}(β), …, f_{d}(β) in ns variables β such that a linear combination

∑_{β=(β_{1},... ,β_{s})∈ℤ^{ns}} b_{β} g_{k_{1}}^{(−i_{1})}(β_{1})g_{k_{2}}^{(−i_{2})}(β_{2}) … g_{k_{s}}^{(−i_{s})}(β_{s})v_{k}(α − β_{1} − … − β_{s}) (2.7)

belongs to M̃^{rad} if and only if the set {b_{β}}_{β∈ℤ^{ns}} (with finitely many non-zero elements) satisfies a finite system of linear equations:

∑_{β∈ℤ^{ns}} b_{β}f_{p}(β) = 0, for p = 1, …, d.

Proof of Claim 1. Denote the sum in (2.7) by x. We have that x ∈ M̃^{rad} if and only if

g_{m_{1}}^{(j_{1})}(γ_{1}) … g_{m_{r}}^{(j_{r})}(γ_{r})x = 0

for all j_{1}, …, j_{r} ∈ ℕ with j_{1} + … + j_{r} = i_{1} + … + i_{s}, m_{1} ∈ K_{j_{1}}, …, m_{r} ∈ K_{j_{r}}, γ_{1}, …, γ_{r} ∈ ℤ^{n}.

From the Poincaré-Birkhoff-Witt argument and the fact that the Lie algebra G and the module V are exp-polynomial, it follows that

g_{m_{1}}^{(j_{1})}(γ_{1}) … g_{m_{r}}^{(j_{r})}(γ_{r})x = ∑_{β∈ℤ^{ns}} b_{β} ∑_{ℓ∈J} h_{ℓ}(β, γ)v_{ℓ}(α + γ_{1} + … γ_{r}).

The functions h_{ℓ} are exp-polynomial in β = (β_{1}, … β_{s}) and γ = (γ_{1}, …, γ_{r}), and depend on j_{1}, …, j_{r}; m_{1}, …, m_{r}. We get that x ∈ M̃^{rad} if and only if {b_{β}} satisfies the system of equations

∑_{β∈ℤ^{ns}} b_{β} ∑_{ℓ∈J} h_{ℓ}(β, γ) = 0  (2.8)

for all γ ∈ ℤ^{nr}, ℓ ∈ J, j_{1}, …, j_{r} ∈ ℕ with j_{1} + … + j_{r} = i_{1} + … + i_{s}, m_{1} ∈ K_{j_{1}}, …, m_{r} ∈ K_{j_{r}}. Since j_{1}, …, j_{r} ∈ ℕ are bounded by the condition j_{1}+…+j_{r} = i_{1} + … + i_{s} and m_{1}, …, m_{r} belong to finite sets, we conclude that only finitely many functions h_{ℓ}(β, γ) appear in the system (2.8). Nonetheless the system (2.8) has infinitely many equations because γ has an infinite range. Our goal is to reduce (2.8) to a system with finitely many equations.

A exp-polynomial function h_{ℓ}(β, γ) could be expanded in γ:

h_{ℓ}(β, γ) = ∑_{p} f_{p}(β)a_{p}^{γ}γ^{δ_{p}},

where a_{p} ∈ ℂ^{nr}, δ_{p} ∈ ℤ^{nr} and the summation in p is finite. By Corollary 2.4, the system (2.8) is equivalent to a finite system of linear equations:

∑_{β∈ℤ^{ns}} b_{β}f_{p}(β) = 0,

with a finite number (say, d) of exp-polynomial functions f_{p}(β). This establishes Claim 1.

Claim 2. In the notations of Claim 1, the set

{g_{k_{1}}^{(−i_{1})}(β_{1}) … g_{k_{s}}^{(−i_{s})}(β_{s})v_{k}(α − β_{1} − … − β_{s})}_{β∈ℤ^{ns}}

spans a subspace of dimension less or equal to d in M_{α}^{(−i_{1}−...−i_{s})}(V).

Proof of Claim 2. To prove this claim we need to show that any d + 1 vectors in this set are linearly dependent. Let B be a subset in ℤ^{ns} of size d + 1. The homogeneous system of d linear equations

∑_{β∈B} b_{β}f_{p}(β) = 0

in d + 1 variables {b_{β}}_{β∈B} has a non-trivial solution. By Claim 1, the set

{g_{k_{1}}^{(−i_{1})}(β_{1}) … g_{k_{s}}^{(−i_{s})}(β_{s})v_{k}(α − β_{1} − … − β_{s})}_{β∈B}

is linearly dependent modulo M̃^{rad}. Claim 2 is now proved.

Theorem 1.5 now follows immediately from Claim 2 since the space M_{α}^{(−i)} is spanned by the elements

{g_{k_{1}}^{(−i_{1})}(β_{1}) … g_{k_{s}}^{(−i_{s})}(β_{s})v_{k}(α − β_{1} − … − β_{s})}

with i_{1} + … + i_{s} = i; i_{1}, …, i_{s} ∈ ℕ. Thus there are finitely many possibilities for (i_{1}, …, i_{s}), and the indices k_{1}, …, k_{s} and k run over finite sets K_{−i_{1}}, …, K_{−i_{s}} and J. □

Proof of Theorem 1.7. The proof of Theorem 1.7 may be derived from the proof of Theorem 1.5 by forgetting ℤ^{n}-grading on the module V and replacing expressions v_{k}(α − β_{1} − … − β_{s}) etc., simply with v_{k}. □

3. Weight modules over generalized Virasoro algebras.

The Virasoro algebra Vir := Vir[ℤ] over ℂ is a Lie algebra with a ℂ-basis {c, d_{i} | i ∈ ℤ} and subject to the following commutator relations

\begin{matrix}[c, d_{i}] = 0, \\ [d_{i}, d_{j}] = (j − i)d_{i+j} + δ_{i,−j} \frac{i^{3} − i}{12}c, for i, j ∈ ℤ. \end{matrix}

The structure and representation theory for the Virasoro algebra has been well developed. For details, we refer the readers to [14] and the book [12] and the references therein.

Generalized Virasoro algebras Vir[M] in characteristic 0 were introduced by Patera and Zassenhaus in [16], which are Lie algebras obtained from the Virasoro algebra by replacing the index group ℤ with an arbitrary nonzero subgroup M of the base field ℂ, which is also the 1-dimensional universal central extension of some generalized Witt algebras [6]. Representations for Generalized Virasoro algebras Vir[M] have been studied in several references. In [17], all weight modules with weight multiplicity 1 were determined, which are some so-called intermediate series modules. In [15], it was proved that all irreducible Harish-Chandra modules over the generalized Virasoro algebra Vir[ℚ] over ℂ are intermediate series modules (where ℚ is the rational number field). In [11], the irreducibility of Verma modules over the generalized Virasoro algebra Vir[M] over ℂ was completely determined.

In this section we assume that M = ℤ⊕M_{0} ⊂ ℂ where M_{0} is a nonzero subgroup of ℂ. We simply denote L = Vir[M]/ℂc. For any i ∈ ℤ, we denote

L_{i} = ⊕_{a∈M_{0}}ℂd_{i+a},
L_{+} = ⊕_{i∈ℤ_{+}}L_{i}, L_{−} = ⊕_{i<0}L_{i}.

Then L_{0} ≃ Vir[M_{0}]/ℂc.

For any α, β ∈ ℂ, we have the L_{0} module V (α, β) = ⊕_{a∈M_{0}}ℂv_{a} subject to the actions

d_{a}v_{b} = (b + α + aβ)v_{a+b}, for a, b ∈ M_{0}.  (3.1)

We extend L_{0} module structure on V (α, β) to L_{+} +L_{0} module structure by defining

L_{+}V (α, β) = 0.  (3.2)

Then we obtain the induced L module

Ṽ = Ṽ(α, β) = Ind_{L_{+}+L_{0}}^{L}V (α, β) = U(L) ⊗_{U(L_{+}+L_{0})} V (α, β),

where U(L) is the universal enveloping algebra of the Lie algebra L. It is clear that, as vector spaces, Ṽ ≃ U(L_{−}) ⊗_{ℂ} V (α, β). The module Ṽ has a unique maximal proper submodule J trivially intersecting with V (α, β). Then we obtain the quotient module

V̄ = V̄ (α, β) = Ṽ/J.  (3.3)

It is clear that V̄ is uniquely determined by the constants α, β, and V̄ = ⊕_{i∈ℤ_{+}} V̄_{−i+α+M_{0}} where

V̄_{−i+α+M_{0}} = ⊕_{a∈M_{0}} V̄_{−i+α+a}, V̄_{−i+α+a} = {v ∈ V̄ | d_{0}v = (−i + α + a)v}. (3.4)

We can similarly define Ṽ_{i+α+M_{0}} and Ṽ_{−i+α+a}. It is easy to see that

dim Ṽ_{−i+α+a} = ∞, for i ∈ ℕ, a ∈ M_{0}.

It will be different for V̄_{−i+α+a}.

Note that if M_{0} ≃ ℤ^{n}, then L is a ℤ^{n}-extragraded Lie algebra, V (α, β) is a L module with polynomial action, and V̄ (α, β) = M(V (α, β)) as defined in Sect.2. In this case, from Theorem 1.5, we know that V̄ (α, β) has finite dimensional weight spaces , i.e., dim V̄_{−i+α+a} < ∞ for all i ∈ ℕ, a ∈ M_{0}. More generally, we have

Theorem 3.1 The module V̄ (α, β) defined in (3.3) over Vir[M] has finite dimensional weight spaces, more precisely, dim V̄_{−i+α+a} ≤ (2i+1)!! for all i ∈ ℕ, a ∈ M_{0}.

Proof. Since L_{−} is generated by L_{−1} , and L_{+} is generated by L_{1}, we deduce that

L_{−1}V̄_{−i+α+M_{0}} = V̄_{−i−1+α+M_{0}}, for i ∈ ℤ_{+},

and, if v ∈ V̄_{−i+α+M_{0}} where i ∈ ℕ, satisfies L_{1}v = 0 then v = 0. We also know that, for any n ∈ ℕ, a ∈ M_{0},

V̄_{−n+α+a} = span{d_{−1+a_{n}}d_{−1+a_{n−1}}...d_{−1+a_{1}}v_{a_{0}} | a_{i} ∈ M_{0},
with a_{0} + a_{1} + ... + a_{n} = a}.

Claim 1. For any n ∈ ℤ_{+}, and b_{i} ∈ ℂ, α_{i} ∈ M_{0}, (only finitely many b_{i} are not zero),

(a) if

∑_{i∈ℤ} α_{i}^{k}b_{i} = 0, for 0 ≤ k ≤ 2n + 2,  (3.5)

then

∑_{i∈ℤ} b_{i}d_{−1+α_{i}}d_{−1+a_{n}}...d_{−1+a_{1}}v_{a_{0}−α_{i}} = 0;  (3.6)

(b) if

∑_{i∈ℤ} α_{i}^{k}b_{i} = 0, for 0 ≤ k ≤ 2n + 1,  (3.5′)

then

∑_{i∈ℤ} b_{i}d_{α_{i}}d_{−1+a_{n}}...d_{−1+a_{1}}v_{a_{0}−α_{i}} = 0,  (3.7)

for all a_{0}, a_{1}, ..., a_{n} ∈ M_{0}, where all the sums are finite.

Now we first consider n = 0. Suppose (3.5′) holds for n = 0. We have

∑_{i∈ℤ} b_{i}d_{α_{i}}v_{a_{0}−α_{i}} = ∑_{i∈ℤ} b_{i}(a_{0} − α_{i} + α + βα_{i})v_{a_{0}} = 0,

which is (3.7) for n = 0. Now suppose (3.5) holds for n = 0. For any γ ∈ M_{0}, we deduce

d_{1+γ}(∑_{i∈ℤ} b_{i}d_{−1+α_{i}}v_{a_{0}−α_{i}}) = ∑_{i∈ℤ} b_{i}(−2 + α_{i} − γ)d_{γ+α_{i}}v_{a_{0}−α_{i}}

= ∑_{i∈ℤ} b_{i}(−2 + α_{i} − γ)(a_{0} − α_{i} + α + β(γ + α_{i}))v_{a_{0}+γ} = 0,

to yield ∑_{i∈ℤ} b_{i}d_{−1+α_{i}}v_{a_{0}−α_{i}} = 0, which is (3.6) for n = 0.

Suppose our claim holds for any n ≤ m for a fixed m ∈ ℤ_{+}. Now we consider the claim for n = m + 1. To get (3.7) for n = m + 1, we assume (3.5′) for n = m + 1. For any γ ∈ M_{0}, we compute

d_{1+γ}(∑_{i∈ℤ} b_{i}d_{α_{i}}d_{−1+a_{m+1}}...d_{−1+a_{1}}v_{a_{0}−α_{i}})

= ∑_{i∈ℤ} b_{i}(−1 − γ + α_{i})d_{1+γ+α_{i}}d_{−1+a_{m+1}}...d_{−1+a_{1}}v_{a_{0}−α_{i}}

+ ∑_{i∈ℤ} b_{i}d_{α_{i}}d_{1+γ}d_{−1+a_{m+1}}...d_{−1+a_{1}}v_{a_{0}−α_{i}}.  (3.8)

Since d_{1+γ}d_{−1+a_{m+1}}...d_{−1+a_{1}}v_{a_{0}−α_{i}} ∈ (L_{−1})^{m}v_{a_{0}−α_{i}} and the coefficients are independent of α_{i}, by inductive hypothesis and (3.5′), we see that the second term on the right hand side of (3.8) is 0. So

d_{1+γ}(∑_{i∈ℤ} b_{i}d_{α_{i}}d_{−1+a_{m+1}}...d_{−1+a_{1}}v_{a_{0}−α_{i}})

= ∑_{i∈ℤ} b_{i}(−1 − γ + α_{i})d_{1+γ+α_{i}}d_{−1+a_{m+1}}...d_{−1+a_{1}}v_{a_{0}−α_{i}}

= ∑_{i∈ℤ} b_{i}(−1 − γ + α_{i})(−2 − γ − α_{i} + a_{m+1})d_{γ+α_{i}+a_{m+1}}d_{−1+a_{m}}...d_{−1+a_{1}}v_{a_{0}−α_{i}}

+...+d_{−1+a_{m+1}}...d_{−1+a_{2}} ∑_{i∈ℤ} b_{i}(−1−γ +α_{i})(−2−γ −α_{i} +a_{1})d_{γ+α_{i}+a_{1}}v_{a_{0}−α_{i}}. (3.9)

From inductive hypothesis and (3.5′), we know that each sum on the right hand side of (3.9) is 0. Thus, for all γ ∈ M_{0},

d_{1+γ}(∑_{i∈ℤ} b_{i}d_{α_{i}}d_{−1+a_{m+1}}...d_{−1+a_{1}}v_{a_{0}−α_{i}}) = 0,

which gives

∑_{i∈ℤ} b_{i}d_{α_{i}}d_{−1+a_{m+1}}...d_{−1+a_{1}}v_{a_{0}−α_{i}} = 0.

So (3.7) holds for n = m + 1.

To verify (3.6) for n = m + 1, we suppose (3.5) holds for n = m + 1. By using (3.7) for n = m + 1, for any γ ∈ M_{0} we deduce that

d_{1+γ} ∑_{i∈ℤ} b_{i}d_{−1+α_{i}}d_{−1+a_{m+1}}...d_{−1+a_{1}}v_{a_{0}−α_{i}}

= ∑_{i∈ℤ} b_{i}(−2 + α_{i} − γ)d_{γ+α_{i}}d_{−1+a_{m+1}}...d_{−1+a_{1}}v_{a_{0}−α_{i}} = 0,

which implies (3.6) for n = m + 1. Thus Claim 1 holds for n = m + 1. By inductive principle, Claim 1 follows.

Fix p ∈ M_{0} \ {0}. Let P_{i} = {kp|0 ≤ k ≤ 2i} for any i ∈ ℕ. For any n ∈ ℕ, a ∈ M_{0}, α_{1}, ...α_{n} ∈ ℤp with α_{i} ∈ P_{i}, let

W (a, α_{1}, α_{2}, ..., α_{n})

= span {d_{−1+α_{n+1}}d_{−1+α_{n}}...d_{−1+α_{1}}v_{a−α_{1}−α_{2}−...−α_{n+1}} | α_{n+1} ∈ P_{n+1}} .

Claim 2. For any β ∈ M_{0}, we have

d_{−1+β}d_{−1+a_{n}}...d_{−1+a_{1}}v_{a−α_{1}−...−α_{n}−β} ∈ W (a, α_{1}, α_{2}, ..., α_{n}).

It is clear that we can find nontrivial b_{α_{n+1}} ∈ ℂ for α_{n+1} ∈ P_{n+1} (consider the following as a linear system of 2n + 3 unknowns with 2n + 3 equations) such that:

β^{k} + ∑_{α_{n+1}∈P_{n+1}} α_{n}^{k}b_{α_{n+1}} = 0, for 0 ≤ k ≤ 2n + 2,

which is an equality of (3.5). Then applying (3.6), we see that Claim 2 follows.

From Claim 2 we know that

dim V̄_{−n+α+a} = {d_{−1+α_{n}}d_{−1+a_{n−1}}...d_{−1+a_{1}}v_{a−α_{1}−α_{2}−...−α_{n}} :
α_{i} ∈ {kp|0 ≤ k ≤ 2i}} ≤ 1 · 3 · ... · (2n + 1), for n ∈ ℤ_{+}, a ∈ M_{0}.

Note that dim V̄_{α+a} = 1 for all a ∈ M_{0}. Thus our theorem follows. □

It is clear that V̄ (α, β) is an irreducible Vir[M] module if and only if V (α, β) defined by (3.4) is an irreducible Vir[M_{0}] module. Thus, if we start with the irreducible Vir[M_{0}] module V′(α, β) (which is the irreducible subquotient of V (α, β) and some of which are no longer exp-polynomial modules), instead of V (α, β), we get an irreducible Vir[M] module V̄′(α, β), which can be also realized by taking the irreducible subquotient of V̄ (α, β) for all α, β ∈ ℂ.

The module V̄ (α, β) contains height weight Vir[ℤ] (the classical Virasoro algebra) modules U(Vir[ℤ])v_{a} for any a ∈ M_{0}. Thus not all its weight multiplicities are 1, which indicates the modules V̄ (α, β) are not modules of intermediate series.

It is natural to ask the following questions: Are Vir[ℤ] modules U(Vir[ℤ])v_{a} irreducible ? Is it true that U(Vir[ℤ])v_{a} = ⊕_{i∈ℤ_{+}} V̄_{i+a} ? Are there any other types of irreducible weight modules with finite dimensional weight spaces over Vir[M] ?

It is important to calculate the character formula for the modules V̄ (α, β). We know the following about dim V̄_{−i+α+a} for i ∈ ℤ_{+}, a ∈ M_{0}.

Corollary 3.2. For any i ∈ ℤ_{+}, a_{1}, a_{2} ∈ M_{0}, if (−i + a_{1})(−i + a_{2}) ≠ 0, we have

dim V̄_{−i+α+a_{1}} = dim V̄_{−i+α+a_{2}}.  (3.10)

Proof. Suppose a_{1} ≠ a_{2}. Let a = a_{1} − a_{2}, G = Vir[ℤa] = ⊕_{i∈ℤ}ℂd_{ia}. Then G is isomorphic to the classical centerless Virasoro algebra. Consider the G module W = ⊕_{x∈ℤa}V̄_{−i+α+a_{2}+x}. From Theorem 3.1, we know that W is a uniformly bounded G module. By a well-known result, we see that all the dimensions of V̄_{−i+α+a_{2}+x} are equal except for V̄_{0}. The corollary follows. □

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