Journal of Lie Theory Volume 9 (1999) 361–367 \begin{matrix}c \\ ◯ \end{matrix} 1999 Heldermann Verlag

The supports of simple modules over toroidal algebras

Volodymyr Mazorchuk

Communicated by G. I. Olshanski

Abstract. We present a description of possible supports for simple modules over toroidal Lie algebras associated with 𝔰𝔩(2, ℂ). This description is analogous to that known for finite-dimensional simple Lie algebras and affine Lie algebras.

1. Introduction

The problem to describe the support of a simple weight module over a Lie algebra with triangular decomposition is very popular and has been studied in many cases. The final results were obtained for complex simple finite-dimensional algebras in [1], for superalgebras in [2], for the A_{1}^{(1)} case in [4], for all affine Lie algebras by I. Penkov (work in progress), for rank two generalized Witt algebras in [6] and for Harish-Chandra modules over higher rank Virasoro algebras in [7].

In the present paper we give the complete answer on the formulated question in the case of an arbitrary toroidal Lie algebra that can be obtained from 𝔰𝔩(2) by the method described in [3]. In fact, the final result states that any simple weight module over such an algebra is either dense (i.e. for any weight λ and root β an element λ + β is again a weight) or cut (i.e. its support is a subset of the support of some induced module).

The paper is organized as follows: In Section 1 we collect all necessary preliminaries and formulate the main result of this paper. In Section 3 we will prove some auxiliary lemmas that will be used in the proof of the main theorem presented in Section 4. Finally, in Section 5 we construct an example of a simple cut 𝔊-module without semi-primitive elements.

2. Toroidal algebras and main theorem

Let ℂ denote the set of complex numbers, ℤ denote the set of integers, ℤ_{+} denote the set of non-negative integers and ℕ denote the set of positive integers. For a Lie algebra L we will denote by U(L) its universal enveloping algebra.

Fix n ∈ ℕ. Let 𝔄 = 𝔰𝔩(2, ℂ) be the Lie algebra of 2 × 2 complex matrix with zero trace and A = ℂ[t_{1}^{±1}, …, t_{n}^{±1}] be the algebra of Laurent polynomials

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag with complex coefficients. Let e, f, h be the standard basis of 𝔄. Consider the Lie algebra 𝔊_{A} = 𝔄⊗_{ℂ}A with bracket [x⊗a, y⊗b] = [x, y]⊗ab, x, y ∈ 𝔄 and a, b ∈ A. Let 𝔊̂ be the universal covering algebra for 𝔊_{A} ([5]). The algebra 𝔊̂ is usually called toroidal. To get the representation theory of 𝔊̂ reasonable we factor out the central ideal consisting of the span of all homogeneous elements of non-zero degree in Ω_{A}/dA and obtain the algebra 𝔊̃. Then for our convenience we will extend 𝔊̃ by commuting differentials d_{1}, …, d_{n} such that [d_{i}, x⊗t_{1}^{k_{1}} … t_{n}^{k_{n}}] = k_{i}x⊗t_{1}^{k_{1}} … t_{n}^{k_{n}}, x ∈ 𝔄 (d_{i} commutes with Ω_{A}/dA) to form the Lie algebra 𝔊.

Let H = ⟨h⟩ be the standard Cartan subalgebra of 𝔄. Then

ℌ = (H ⊗ 1) ⊕ Ω_{A}/dA ⊕ ⟨d_{1}, …, d_{n}⟩

is the standard Cartan subalgebra of 𝔊. Let Δ ⊂ ℌ^{∗} be the root system of 𝔊 with respect to ℌ. For β ∈ Δ let 𝔊_{β} denote the corresponding root space in 𝔊. In a standard way Δ can be decomposed into the disjoint union Δ = Δ_{ℜ} ∪ Δ_{ℑ}, where Δ_{ℜ} is the set of roots of elements of the form e⊗a or f ⊗a, a is a monomial in A and Δ_{ℑ} = Δ \ Δ_{ℜ}. We will also denote by Δ_{ℜ}^{+} (Δ_{ℜ}^{−}) the set of roots of the elements of the form e⊗a (f ⊗a) and by 𝔊_{+} (𝔊_{−}) the corresponding subalgebras in 𝔊. A root α ∈ Δ_{ℑ} will be called simple if α/k is not a root for any k ∈ ℕ. It follows easily from the definition of 𝔊 that the sum of two elements from Δ_{ℜ}^{+} (or Δ_{ℜ}^{−}) is never a root and the sum of any element form Δ_{ℜ}^{+} (or Δ_{ℜ}^{−}) with any element from Δ_{ℑ} is always a root.

For a 𝔊-module V and λ ∈ ℌ^{∗} let V_{λ} = {v ∈ V | hv = λ(h)v for all h ∈ ℌ} denote the weight subspace of V corresponding to a weight λ. A module V is said to be a weight module provided it can be decomposed into a direct sum of its weight subspaces. For a weight 𝔊-module V let supp V be the set of all weights of V with non-trivial weight subspaces. This set will be called the support of V .

Let P denote the abelian group spanned by Δ. One can easily see that P ≃ ℤ^{n+1}. For any order ⩽ on P (here and on by an order we mean an order on an abelian group, hence, we assume that this order is compatible with the group structure) we will denote P_{+}^{⩽} = {p ∈ P | 0 ⩽ p, p ⩽̸ 0}, P_{−}^{⩽} = {p ∈ P | p ⩽ 0, 0 ⩽̸ p}, P_{0}^{⩽} = {p ∈ P | 0 ⩽ p, p ⩽ 0} and set Δ_{i}^{⩽} = P_{i}^{⩽} ∩ Δ, i = 0, +, −. We will say that ⩽ is non-trivial if P_{+}^{⩽} is not empty.

Clearly, supp V ⊂ λ + P for any simple weight module V and any λ ∈ supp V . A weight module V is called dense if supp V = λ + P, λ ∈ supp V and cut if supp V ⊂ λ + P_{−}^{⩽} for a non-trivial order ⩽ and some λ ∈ ℌ^{∗}.

For any weight module V and any v ∈ V_{λ} (λ ∈ supp V ) we will say that a subset S ⊂ Δ is an annihilating v-set provided 𝔊_{β}v = 0 for any β ∈ S. A non-zero element v ∈ V_{λ} will be called semi-primitive if there exists a non-trivial order ⩽ on P such that Δ_{+}^{⩽} is an annihilating v-set. Any subset S ⊂ Δ can be enlarged by adding to it all the roots of the form α + β, where α, β ∈ S and at least one of them belongs to Δ_{ℜ}. Starting from S we can enlarge it to the set S_{1}, then we can enlarge the obtained set to the set S_{2} and so on. The set \overline{S} = ∪_{i=1}^{∞}S_{i} will be called the additive closure of S. The following lemma follows easily from the fact that dim 𝔊_{δ} = 1 for all δ ∈ Δ_{ℑ}. Lemma 2.1. Let V be a weight 𝔊-module, λ ∈ supp V , v ∈ V_{λ} and S be an annihilating v-set. Then \overline{S} is an annihilating v-set.

Now we can formulate the main result of this paper:

Theorem 2.2. Let V be a simple weight 𝔊-module. Then V is either dense or cut.

We have to remark that in the case n = 1 our algebra 𝔊 is an affine Lie algebra. In this case theorem 2.2 was obtained in [4].

3. Preliminary lemmas

For a simple weight 𝔊-module V we will denote by P (V) the set λ + P, λ ∈ supp V . Clearly, P (V) does not depend on the choice of λ. We also set s(V) = P (V) \ supp V . During this section we fix a non-trivial non-dense simple weight 𝔊-module V . For a fixed μ ∈ ℌ^{∗} we will write H_{μ} for the set {μ} ∪ μ + Δ. We note that s(V) is not empty since V is not dense.

A non-zero element v of a weight module V will be called bounded of type (β_{1}, …, β_{n}), where β_{i} ∈ Δ_{ℑ}, 1 ⩽ i ⩽ n are linearly independent simple roots provided there exist α^{±} ∈ Δ_{ℜ}^{±} such that {α^{±} + k_{1}β_{1} + ⋯ + k_{n}β_{n} | k_{i} ∈ ℤ_{+}, i = 1, …, n} is an annihilating v-set.

Lemma 3.1. There exists μ ∈ s(V) and λ ∈ supp V ∩H_{μ} such that μ−λ ∈ Δ_{ℜ}.

Proof. Clearly, there exists μ_{1} ∈ s(V) such that supp V ∩ H_{μ_{1}} is not empty. If μ_{1} + Δ_{ℜ} ⊂ s(V) then one can take μ ∈ μ_{1} + Δ_{ℜ} and λ ∈ supp V ∩ H_{μ_{1}} .

Lemma 3.2. Suppose that v ∈ V_{λ} is a non-zero element. Then either 𝔊_{+}v ≠ 0 (𝔊_{−}v ≠ 0) or v is semi-primitive.

Proof. Follows from the fact that by setting Δ_{+}^{⩽} = Δ_{ℜ}^{+} one defines a non-trivial order ⩽ on P.

Lemma 3.3. Let v ∈ V_{λ} and S ⊂ Δ be an annihilating v-set. Suppose that g ∈ 𝔊_{β} and S_{1} ⊂ S such that β + S_{1} ∩ Δ ⊂ S. Then S_{1} is an annihilating gv-set.

Proof. Let α ∈ S_{1} and x ∈ 𝔊_{α}. One has xgv = [x, g]v + gxv. The right hand side of this equality will be zero as soon as [x, g] = 0 since xv = 0. Moreover, if [x, g] ≠ 0 it follows that [x, g] ∈ 𝔊_{α+β} with α + β ∈ S. Hence xgv = 0 and the statement follows.

Lemma 3.4. Let V be a simple weight module and 0 ≠ v ∈ V be a bounded element of type (β_{1}, …, β_{n}). Then any non-zero element in V is bounded of the same type.

Proof. Let 0 ≠ w ∈ V . Since V is simple there exists u ∈ U(𝔊) such that w = uv. Thus it is sufficient to prove the statement of the lemma for any element of the form xv where x ∈ 𝔊_{γ}, γ ∈ Δ.

Consider an r-dimensional Euclidian space X and fix linearly independent x_{1}, …, x_{r} ∈ X. For y ∈ X set C_{y} be the cone consisting of all elements of the form y + s_{1}x_{1} + ⋯ + s_{r}x_{r} where s_{i} are non-negative for 1 ⩽ i ⩽ r. Since X is rdimensional it follows immediately that for any two cones C_{y_{1}} and C_{y_{2}} there exists y ∈ X such that C_{y} ⊂ C_{y_{1}} ∩ C_{y_{2}} . The statement now follows from Lemma 3.3.

Lemma 3.5. Let V be a non-trivial simple weight 𝔊-module and v ∈ V be a non-zero bounded element of type β_{1}, …, β_{n}. Suppose that V is not dense and μ ∈ s(V). Then μ + s_{1}β_{1} + ⋯ + s_{n}β_{n} ∈ s(V) for s_{i} ∈ ℕ, 1 ⩽ i ⩽ n.

Proof. Suppose that λ = μ + s_{1}β_{1} + ⋯ + s_{n}β_{n} ∈ supp V for some s_{i} ∈ ℕ, 1 ⩽ i ⩽ n and 0 ≠ w ∈ V_{λ}. Then w is a bounded element of type β_{1}, …, β_{n} by Lemma 3.4. Let T denotes the corresponding annihilating w-set. Then the additive closure of T ∪ {μ − λ} is an annihilating w-set by Lemma 3.3. Using previous lemmas one can show that this closure coincides with Δ and thus w generates a trivial 𝔊-submodule in V and we obtain a contradiction.

Lemma 3.6. Let V be a simple weight 𝔊-module. Suppose that there exists α ∈ Δ_{ℜ}, 0 ≠ X_{α} ∈ 𝔊_{α}, k ∈ ℕ and 0 ≠ v ∈ V such that X_{α}^{k}v = 0. Then X_{α} acts locally nilpotent on V . Proof. Follows from the fact that adX_{α} is nilpotent on 𝔊.

Lemma 3.7. Let V be a simple weight non-dense 𝔊-module, λ ∈ supp V , μ ∈ s(V) such that μ−λ = kα for some α ∈ Δ_{ℜ} and k ∈ ℕ. Then λ+lα ∉ supp V for all l ⩾ k. Proof. Let 0 ≠ v ∈ V_{λ} and 0 ≠ X_{α} ∈ 𝔊_{α}. Then X_{α}^{k}v = 0 and thus X_{α} is locally nilpotent on V by Lemma 3.6. Suppose that λ + lα ∈ supp V for some l ⩾ k. Then V_{λ+lα} ≠ 0 and 𝔊_{−α}^{l−k}V_{λ+lα} = 0 and thus X_{−α} is also locally nilpotent on V . Since 𝔊_{±α} generate an 𝔰𝔩(2)-subalgebra of 𝔊 it follows that 𝔰𝔩(2)-module ⊕_{m∈ℤ}V_{λ+mα} contains two finite dimensional subquotients and their supports have empty intersection. The last is impossible and thus we obtain the statement of the Lemma.

4. Proof of the main theorem

Suppose that V is a simple weight non-dense 𝔊-module and n > 1 (for n = 1 our proof reduces to that from [4]). Clearly, the existence of a semi-primitive vector in V will imply the statement of the main theorem. It is impossible to prove the existence of a semi-primitive element in the general case, so, in fact we will prove the following statement: Let V be a non-dense simple 𝔊-module, then either there exists a semi-primitive element in V or V is cut. Thus we can suppose that there were no semi-primitive elements in V . Our first goal is to prove that there exist some bounded element in V .

Consider the elements μ ∈ s(V) and λ ∈ supp V given by Lemma 3.1. Without loss of generality we can assume that μ − λ ∈ Δ_{ℜ}^{+}. Let v ∈ V_{λ} be a non-zero element. Then the set S = {μ − λ} is an annihilating v-set.

Since v is not semi-primitive, by Lemma 3.2 there exist α̂ ∈ Δ_{ℜ}^{+} and g ∈ 𝔊α̂ such that v_{1} = gv ≠ 0. Let δ be a simple root such that μ − λ − α̂ = Nδ for some N ∈ ℕ. One can choose α̂ such that 𝔊_{α̂+kδ}v = 0, 0 < k < N.

By Lemma 3.3 we have that S is an annihilating v_{1}-set. Moreover, we can enlarge S to an annihilating v_{1}-set S_{1} by elements α̂ + kδ, 0 < k < N, Nδ and thus by μ − λ + kNδ, k ∈ ℕ. Applying 𝔊_{δ} to v_{1} we can find an element 0 ≠ v_{2} ∈ V_{μ+N_{1}δ} such that S_{2} = {δ} ∪ {μ − λ + kδ | k ∈ ℕ} is an annihilating v_{2}-set.

Since v_{2} is not semi-primitive, using Lemma 3.2 and the same procedure as above one can find an element α ∈ Δ_{ℜ}^{+} and g_{1} ∈ 𝔊_{α} such that v_{3} = g_{1}v_{2} ≠ 0. Suppose that α = μ − λ + kδ for some k ∈ ℤ and it is impossible to choose α that is not of this form. Then Δ_{ℜ}^{+} \ {μ − λ + kδ | k ∈ (ℤ \ ℕ)} is an annihilating v_{3}-set. Applying to v_{3} any non-zero element x from 𝔊_{β} where β ∈ Δ_{ℑ} such that β and δ are linearly independent we immediately obtain that either v_{3} or xv_{3} ≠ 0 is semi-primitive.

Thus we can choose α such that α ≠ μ−λ+kδ for all k ∈ ℤ. Let α_{1} be the weight of v_{3}. In this case an additive closure T of {μ − λ + kδ | k ∈ ℕ} ∪ {μ − α_{1}} is an annihilating v_{3}-set by Lemma 2.1 and Lemma 3.3. Moreover, one can see that α can be chosen such that it would be possible to find simple β_{1}, β_{2} ∈ Δ_{ℑ} and γ^{±} ∈ Δ_{ℜ}^{±} such that T contains all γ^{±} + s_{1}β_{1} + s_{2}β_{2} for s_{,}s_{2} ∈ ℤ_{+}.

Applying the same procedure to v_{3} with the use of elements from Δ_{ℜ}^{−} and then again from Δ_{ℜ}^{+} and so on, one can construct a bounded element 0 ≠ w ∈ V of type β_{1}, …, β_{n} for some simple linearly independent β_{i} ∈ Δ_{ℑ}, 1 ⩽ i ⩽ n. Thus, any element of V should be bounded of type β_{1}, …, β_{n} by Lemma 3.4. By Lemma 3.5 we also obtain that μ+s_{1}β_{1} +⋯+s_{n}β_{n} ∈ s(V) for s_{i} ∈ ℕ, 1 ⩽ i ⩽ n.

Now, using Lemma 3.4 and the same arguments as in the preceding paragraph it is easy to see that there exists a non-trivial order ⩽_{ℑ} on Δ_{ℑ} and ξ ∈ (μ+Δ_{ℑ})∪{μ} such that ξ +P_{ℑ}^{+} ⊂ s(V), where P_{ℑ}^{+} = {α ∈ Δ_{ℑ} | α ⩽̸_{ℑ} 0, 0 ⩽_{ℑ} α}. Indeed, fixing the support of V and assuming that this statement is false it follows with the same arguments as above that s(V) can be enlarged. Clearly, we can choose ξ minimal with respect to ⩽.

The last observation together with Lemma 3.4 immediately implies the following: if μ′ ∈ supp V and α ∈ Δ_{ℑ} such that μ′ + α ∈ s(V) then there exists minimal ξ′ ∈ μ′ + Δ_{ℑ} such that ξ′ + P_{ℑ}^{+} ⊂ s(V). Otherwise, one can easy find 0 ≠ v ∈ V such that 𝔊v = 0.

Consider the subsets H_{±} of H_{μ} defined as follows: H_{±} = μ + Δ_{ℜ}^{±}. Suppose that H_{−} ⊂ supp V (H_{+} ⊂ supp V ). Then H_{+} ⊂ s(V) (H_{−} ⊂ s(V)) by Lemma 3.7 and we obtain that V contains a semi-primitive element. Hence we can fix elements ξ′ ∈ H_{−} and ξ as above. Set β = ξ − ξ′ ∈ Δ_{ℜ}^{+}.

Consider the non-trivial order ⩽ on Δ (and thus on P) such that Δ_{+}^{⩽} = P_{ℑ}^{+} ∪ ±β + P_{ℑ}^{+}. Now Lemma 3.7 guarantees that ξ + P_{+}^{⩽} ⊂ s(V) which completes the proof of the theorem.

5. Examples

Example of cut 𝔊-modules with semi-primitive elements can easily be constructed as the unique simple quotients of Verma modules using partitions of Δ as it was done for example in [4, 8]. At the same time, examples of dense modules also can be constructed by using the standard technique. Unlike the classical case of affine Kac-Moody Lie algebras we can not state that any cut 𝔊-module contains a semi-primitive element. The aim of this section is to construct an example of a simple cut module without semi-primitive elements.

Let P_{ℑ} = ℤΔ_{ℑ} and consider an order ⩽ on P_{ℑ} that satisfies the following condition: for any 0 ⩽ x ⩽ y, x, y ∈ P_{ℑ} there exists k ∈ ℕ such that y ⩽ kx. Let P_{+} denote the set {x ∈ P_{ℑ} | 0 ⩽ x, x ⩽̸ 0}. Fix α ∈ Δ_{ℜ}^{+} and set Δ_{+} = P_{+} ∪ {α} ∪ α + P_{+} ∪ −α + P_{+}, Δ_{−} = −Δ_{+}. Let 𝔊 = 𝔊_{+} ⊕ ℌ ⊕ 𝔊_{−} be the corresponding decomposition of 𝔊. Consider ℂ as the trivial 𝔊_{+} ⊕ ℌ-module and form a module M as follows:

M = U(𝔊) ⨂_{U(𝔊_{+}⊕ℌ)} ℂ.

Let P (−) be the semigroup generated by Δ_{−}∪{0}. Clearly, M is a weight module and supp M = P (−). Let N be the subspace of M generated by all M_{λ}, λ ≠ 0. Proposition 5.1. 1. N is a 𝔊-submodule of M.

2. N is simple.

3. N contains no semi-primitive elements. Proof. The first statement follows from the fact that N is the kernel of the canonical epimorphism of M onto trivial 𝔊-module. The last one follows from the second and the description of supp N. Thus we only need to prove the second statement.

Let v denote a canonical generator of M. First we note that it is enough to show that for any w ∈ N the module N_{w} = U(𝔊)w contains an element of the form X_{β}v for some β ∈ Δ_{−}. Fix a non-trivial w. The order ⩽ trivially induces an order on Δ_{−} which we will also denote by ⩽. Thus by PBW theorem w can be written as a linear combination of the monomials and, moreover, each monomial is a product of X_{β} for β ∈ Δ_{−}. By the length of a monomial we will mean the number of multiplicands occurring in it.

Fix the set S(w) of monomials of maximal length in w and choose the smallest β_{1} that occurs as a multiplicand in these monomials. Applying the elements X_{β_{2}} to w such that −β_{2} is smaller than arbitrary multiplicand of a monomial from S(w) but β_{1} one can easily show that supp N_{w} = supp N and, moreover, that N_{w} contains an element w_{1} such that |S(w_{1})| = 1. Thus we can assume that |S(w)| = 1.

Now we can consider the sets S_{1} = {−β_{1} + ℤα} ∩ Δ, S_{2} = S_{1} + β for some β ∈ Δ_{+} small enough and S_{3} = S_{1} ∪ S_{2}. Consider the elements X_{γ}w, γ ∈ S_{3}. One can see that β can be chosen in such way that at least one of the elements X_{γ}w is non-zero. Now it is easy to obtain that any monomial occurring in this non-zero element has length smaller than |S(w)|. Trivial induction completes our proof.

6. Acknowledgments

The author would like to thank Professor V. Futorny for stimulating discussions and to CRDF (grant UM1-327) for financial support.

References

[1] Cylke, A., and V. Futorny, S. Ovsienko, On the support of irreducible weight modules, Preprint 96-011, Bielefeld University.

[2] Dimitrov, I., and O. Mathieu, I. Penkov, On the structure of weight modules, to appear in Transactions of the Amer. Math. Soc.

[3] Fabbri, M. A., and R. V. Moody, Irreducible representations of Virasoro-Toroidal Lie algebras, Commun. Math. Phys. 159 (1994), 1–13.

[4] Futorny, V., Irreducible non-dense A_{1}^{(1)}-modules, Pac. J. Math., 172 (1996), 83–99.

[5] Kassel, C., Kähler differentials and coverings of complex simple Lie algebras extended over a commutative algebra, J. Pure Appl. Algebra 34 (1985), 265–275.

[6] Mazorchuk, V., Futorny theorem for generalized Witt algebras of rank 2, Comm. Alg. 25 (1997), 533–541.

[7] —, On the support of simple Harish-Chandra modules over higher rank Virasoro algebra, to appear.

[8] —, Verma modules over generalized Witt algebras, to appear in Compositio Math.

Mechanics and Mathematics Kyiv Taras Shevchenko University 64, Volodymyrska st. 252033 Kyiv Ukraine e-mail: mazorchu@uni-alg.kiev.ua

Received May 1, 1998 and in final form August 14, 1998