Journal of Lie Theory Volume 14 (2004) 141–150 \begin{matrix}c \\ ◯ \end{matrix} 2004 Heldermann Verlag

An Equivalence between Categories of Modules for Generalized Kac-Moody Lie Algebras

Elizabeth Jurisich

Communicated by F. Knop

Abstract. An equivalence between categories of modules for a generalized Kac-Moody algebra and modules for an appropriate parabolic subalgebra is shown. In particular, properties such as the irreducibility and complete reducibility of a module whose weights satisfy certain conditions can be determined by restriction to a subalgebra.

1. Introduction

Generalized Kac-Moody algebras are Lie algebras determined by certain symmetrizable matrices as in [11], and [1]. Let 𝔤 be a generalized Kac-Moody algebra. In [10] some examples of irreducible highest weight modules for 𝔤 which are generalized Verma modules are given. These generalized Verma modules are modules that are induced from an irreducible highest weight module for a “parabolic” subalgebra 𝔭 = (𝔤_{S} + 𝔥) ⊕ 𝔲^{+} , where 𝔤_{S} is a generalized Kac-Moody subalgebra of 𝔤 (often 𝔤_{S} can be chosen to be a Kac-Moody or semi-simple Lie algebra). In this paper, we show that the converse is also true, that in fact any irreducible 𝔤 module whose weights satisfy the appropriate conditions is a generalized Verma module for a parabolic subalgebra. More generally, it is shown that a subcategory of the category O of 𝔤-modules is equivalent to a subcategory of the category O of the Lie algebra 𝔯 = 𝔤_{S} + 𝔥.

This result is most useful in cases where the subalgebra 𝔤_{S} can be chosen to be a semi-simple or Kac-Moody Lie algebra, so that some of the representation theory for the generalized Kac-Moody algebra can be reduced to that of the semisimple or Kac-Moody subalgebra.

In §2 of this paper, we review the definition of generalized Kac-Moody algebra, introduce notation and define the category of 𝔤 modules that we use. Section 3 contains the statement and proof of the main theorem. The equivalence of categories is proven by using the functor determined by inducing a module for the parabolic subalgebra 𝔭 to 𝔤, and the functor determined by considering the set of elements of a 𝔤-module V annihilated by 𝔲^{+} . The method of proof is similar to that in [4] (see also [5] and [12]). Section 4 contains some applications.

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag

2. The Setting

We recall the definition of a generalized Kac-Moody algebra. The definition given here is similar to Borcherds’ original definition [1] and agrees with the definition in [9]. A variety of other definitions exist, for example [2], [11], [13]. We refer the reader to [11] and [13] for general definitions involving Lie algebras defined by symmetrizable matrices, of which generalized Kac-Moody algebras are an example. Definitions and proofs regarding the basic structure of generalized Kac-Moody algebras are presented in [1],[8], [9] and [10]. For the convenience of the reader, we include the definitions that differ from the definitions for the more familiar Kac-Moody algebras.

We define a generalized Kac-Moody algebra over ℂ by specifying generators and relations. Let ℤ_{+} denote the nonnegative integers. Let I be a countable index set, A = (a_{ij})_{i,j∈I} a matrix with entries in ℝ satisfying the conditions:

1. A is symmetrizable.

2. If a_{ii} > 0, then a_{ii} = 2.

3. For all k ≠ j a_{jk} ≤ 0 and for all i ∈ I such that a_{ii} > 0, a_{ij} ∈ −ℤ_{+} for all j ∈ I. Let 𝔤(A)′ be the Lie algebra with generators e_{i}, f_{i}, h_{i}, i ∈ I and defining

relations: For all i, j ∈ I

[h_{i}, h_{j}] = 0, [e_{i}, f_{j}] = δ_{ij}h_{i}
[h_{i}, e_{j}] = a_{ij}e_{j}, [h_{i}, f_{j}] = −a_{ij}f_{j}.

For all i ∈ I such that a_{ii} > 0

(ad e_{i})^{−a_{ij}+1}e_{j} = 0, (ad f_{i})^{−a_{ij}+1}f_{j} = 0.

Finally, for all i, j ∈ I such that a_{ij} = a_{ji} = 0

[e_{i}, e_{j}] = 0, [f_{i}, f_{j}] = 0.

Define degree derivations ∂_{i}, i ∈ I, on 𝔤(A)′ by taking ∂_{i}(e_{j}) = δ_{ij}, ∂_{i}(f_{j}) = −δ_{ij} and ∂_{i}(h_{j}) = 0. Let 𝔡 be the abelian Lie algebra generated by the ∂_{i}, i ∈ I.

Definition 1. The generalized Kac-Moody algebra 𝔤 = 𝔤(A) is defined to be the semidirect product 𝔤(A)′ ⋉ 𝔡.

We also call any Lie algebra of the form 𝔤(A)/𝔠 where 𝔠 is a central ideal a generalized Kac-Moody algebra. Borcherds’ Monster Lie algebra is an example of this kind.

The Lie algebra 𝔤 has a triangular decomposition 𝔤 = 𝔫^{−} ⊕ 𝔥 ⊕ 𝔫^{+} , where 𝔫^{±} is the subalgebra generated by the e_{i} (resp. the f_{i}), and the Cartan subalgebra 𝔥 is the abelian subalgebra spanned by the h_{i} and the ∂_{i}. There is a symmetric invariant bilinear form on 𝔤, and on 𝔥^{∗}, which will be denoted by (·, ·) in both cases.

The roots of 𝔤 are the nonzero elements φ of 𝔥^{∗} such that 𝔤^{φ} = {x ∈ 𝔤|[h, x] = φ(h)x for all h ∈ 𝔥} is nonzero. Denote the set of roots of 𝔤 by Δ. There is a system of positive roots Δ_{+} and Δ = Δ_{+} ∪−Δ_{+} . Roots with positive square norm are called real, and roots with non-positive square norm are called imaginary. Denote the set of real roots by Δ_{R}. The set of simple roots, denoted {α_{i}}_{i∈I} ⊂ 𝔥^{∗}, are defined by the conditions

[h, e_{i}] = α_{i}(h)e_{i} for all h ∈ 𝔥.

Note that α_{j}(h_{i}) = a_{ij}, and α_{j}(∂_{i}) = δ_{ij} for all i, j ∈ I and that the α_{i} for i ∈ I are linearly independent. The bilinear form is chosen so that (α_{i}, α_{j}) = a_{ij} for all i, j ∈ I. Note that for i ∈ I (α_{i}, α_{i}) = a_{ii} may be non-positive, so that simple roots may be imaginary.

Given an element μ ∈ 𝔥^{∗} of the form λ − ∑_{i∈I} n_{i}α_{i}, n_{i} ∈ ℤ_{+} , we define the depth d_{λ}(μ) = ∑_{i∈I} n_{i}.

Given a 𝔤-module and λ ∈ 𝔥^{∗} let

M^{λ} = {v ∈ M|h · v = λ(h)v, for all h ∈ 𝔥}.

The elements λ ∈ 𝔥^{∗} for which M^{λ} ≠ 0 are called the weights of M. Given a 𝔤-module M let P (M) denote the set of weights of M. A weight λ ∈ (𝔥)^{∗} is dominant if (λ, α_{i}) ∈ ℝ and (λ, α_{i}) ≥ 0 for all i ∈ I. A weight λ is called integral if λ(h_{i}) ∈ ℤ_{+} for all i ∈ I such that a_{ii} > 0. Denote by P_{+} the set of dominant integral weights.

Definition 2. A 𝔤-module M is a standard module if M is a highest weight module with highest weight μ ∈ P_{+} and highest weight vector v such that:

1. for i ∈ I, if (μ, α_{i}) = 0 then f_{i} · v = 0;

2. if α_{i} (i ∈ I) is real then f_{i}^{n_{i}+1} · v = 0, where n_{i} = 2(μ, α_{i})/(α_{i}, α_{i}) (necessarily a nonnegative integer). Given any μ ∈ P_{+} there is a unique (up to isomorphism) standard

irreducible highest weight 𝔤-module of highest weight μ, denote this module by L_{𝔤}(μ) [9].

Fix S a distinguished subset of I containing {i ∈ I|α_{i} ∈ Δ_{R}}. Denote by 𝔤_{S} the generalized Kac-Moody subalgebra of 𝔤 associated to the matrix (a_{ij})_{i,j∈S}. Assume that S is chosen so that the resulting matrix (a_{ij})_{i,j∈S} is indecomposible. Note that if S = {i ∈ I|α_{i} ∈ Δ_{R}} then 𝔤_{S} is a Kac-Moody algebra. In the case of the monster Lie algebra we can choose 𝔤_{S} = 𝔰𝔩_{2} ; (see §4 below).

We make the following definitions: Δ^{S} = Δ∩∐_{i∈S} ℤα_{i}, Δ_{+}^{S} = Δ_{+} ∩Δ^{S} and Δ_{−}^{S} = Δ_{−} ∩ Δ^{S}. Denote by 𝔥_{S} the span of the h_{i}, i ∈ S. There is a root space decomposition

𝔤_{S} = ∐_{φ∈Δ_{+}^{S}} 𝔤^{φ} ⊕ 𝔥_{S} ⊕ ∐_{φ∈Δ_{−}^{S}} 𝔤^{φ}.

Define the following subalgebras of 𝔤:

𝔲^{+} = ∐_{φ∈Δ_{+}\Δ_{+}^{S}} 𝔤^{φ}; 𝔲^{−} = ∐_{φ∈Δ_{−}\Δ_{−}^{S}} 𝔤^{φ}; 𝔯 = 𝔤_{S} + 𝔥.

Let 𝔭 = 𝔯 ⊕ 𝔲^{+} , the “parabolic” subalgebra of 𝔤 determined by S. Then

𝔤 = 𝔲^{−} ⊕ 𝔯 ⊕ 𝔲^{+} = 𝔲^{−} ⊕ 𝔭.

If M is a 𝔯-module we define a 𝔭-module which we also call M by letting 𝔲^{+} act as zero on M. Note that if M is irreducible, or completely reducible as a 𝔤_{S} or 𝔭-module, then the 𝔭-module M is irreducible, or completely reducible as well.

We recall the definition of generalized Verma module of [6]. Let λ ∈ P_{+} , and consider the standard (irreducible) highest weight 𝔯-module L(λ) associated to λ (this is a standard module for 𝔤_{S}). The highest weight space of L(λ) (as a 𝔤_{S} -module) is a weight space for 𝔥, with weight λ. Let 𝔲^{+} act trivially on L(λ); this gives L(λ) the structure of an irreducible 𝔭-module. Define the generalized Verma module V^{L(λ)} to be the induced module U(𝔤) ⊗_{U(𝔭)} L(λ).

Definition 3. Let O_{𝔤} denote the category of 𝔤-modules that are weight modules whose weight spaces are finite dimensional, and whose set of weights lies in a finite union of sets of the form

D(λ) = {λ − ∑_{i∈I} n_{i}α_{i}|λ ∈ 𝔥^{∗}}.

The morphisms are 𝔤-module homomorphisms.

Definition 4. Let O_{𝔤}^{S} be the full subcategory of O_{𝔤} whose objects are 𝔤modules V ∈ O_{𝔤} such that every weight μ ∈ P (V) satisfies (μ, α_{i}) > 0 for all i ∈ I\S.

Let M ∈ O_{𝔤}. Recall that a highest weight series for M is an increasing filtration

(0) ⊂ M_{1} ⊂ M_{2} ⊂ ⋯

of submodules of M satisfying ⋃_{i=0}^{∞} M_{i} = M. For M_{i+1} ≠ M_{i} the module M_{i+1}/M_{i} is a highest weight module. We extend a well known result of [6] to modules in O_{𝔤}, where the index set I may be countably infinite.

Proposition 2.1. Let M ∈ O_{𝔤}. Then M has a highest weight series {M_{i}}_{i∈ℤ}.

Proof. Let λ_{1}, λ_{2}, … λ_{r} be a set of weights of such that

P (M) ⊂ D(λ_{1}) ∪ D(λ_{2}) … ∪ D(λ_{r}).

With out loss of generality we may assume that the sets D(λ_{i}) are distinct. Then any μ ∈ P (M) is of the form μ = λ_{j} − β for some j 1 ≤ j ≤ r, with β = ∑ n_{i}α_{i} where n_{i} ∈ ℤ_{+} . Define d(μ) = ∑ in_{i}. Since d(μ) = N implies n_{i} = 0 for sufficiently large i the set {μ ∈ P (M)|d(μ) = N} is finite for all N ∈ ℤ_{+} . Now a filtration of M can be constructed using the same argument as in the case where I is finite [6]: Let μ_{1} be a weight in P (M) with minimal d(μ_{1}). Choose a nonzero weight vector x_{1} ∈ M^{μ}. By the minimality of d(μ_{1}) the vector x_{1} is a highest weight vector. Denote by M_{1} the highest weight module generated by x_{1} . To construct a module M_{2} with (0) ⊂ M_{1} ⊂ M_{2} and M_{2}/M_{1} a highest weight module repeat the above argument for the module M/M_{1} . The series d(μ_{k}) is increasing, and can only remain at one value for finitely many k. Thus eventually M^{μ} = M_{n}^{μ} = M_{n+1}^{μ} … for every μ ∈ P (M). This construction gives the desired highest weight series.

In order to prove the main theorem we will use the following result from category theory [7].

Lemma 2.2. Let A and B be categories. Let F be a functor from A to B. The categories are equivalent if and only if

1. F is full, that is, for all objects V_{1}, V_{2} of A the mapping

Hom_{A}(V_{1}, V_{2}) → Hom_{B}(F (V_{1}), F (V_{2}))

described by f ↦ F (f) is surjective.

2. F is faithful, that is, the above mapping is injective.

3. For every object M of B there is an object V of A such that M is isomorphic to F (V).

3. The equivalence of categories

We will show that the category O_{𝔤}^{S} is equivalent to the category O_{𝔯}^{S}.

If V ∈ O_{𝔤}^{S} let V^{𝔲^{+}} denote the the set of elements of V annihilated by the action of 𝔲^{+} . The space V^{𝔲^{+}} is an 𝔤_{s} -module since U(𝔤_{s})V^{𝔲^{+}} ⊂ V^{𝔲^{+}} . Likewise V^{𝔲^{+}} is an 𝔯-module and a 𝔭-module. Note that the subspace V^{𝔲^{+}} considered as an 𝔯-module is in O_{𝔯}^{S}.

The functors between our categories O_{𝔤}^{S} and O_{𝔯}^{S} are simply restriction and induction.

Definition 5. The functor R : O_{𝔤}^{S} → O_{𝔯}^{S} is defined as follows: R(V) = V^{𝔲^{+}} for V ∈ O_{𝔤}^{S} and given a pair V, U of modules in O_{𝔤}^{S} and a module homomorphism f : V → U define R(f) = f̄ to be the restriction of the homomorphism f to V^{𝔲^{+}} .

Definition 6. The functor Ind_{𝔭}^{𝔤} : O_{𝔯}^{S} → O_{𝔤}^{S} is defined by Ind_{𝔭}^{𝔤}(N) = U(𝔤) ⊗_{U(𝔭)} N for N an object of O_{𝔯}^{S}, considered as a 𝔭-module with 𝔲^{+} acting as multiplication by 0, and Ind_{𝔭}^{𝔤}(f) = 1⊗f for f a morphism in O_{𝔯}^{S}, considered as a morphism in O_{𝔭}^{S}.

Lemma 3.1. Let M ∈ O_{𝔯}^{S}. Then (Ind_{𝔭}^{𝔤}M)^{𝔲^{+}} = 1 ⊗ M. Proof. By definition M is annihilated by 𝔲^{+} as a 𝔭-module, so 1 ⊗ M ⊂ (Ind_{𝔭}^{𝔤}M)^{𝔲^{+}} .

Using the Poincare-Birkhoff-Witt theorem, we have an isomorphism Ind_{𝔭}^{𝔤}M ≃ U(𝔲^{−}) ⊗_{ℂ} M. Consider U(𝔲^{−}) ⊗_{ℂ} M = (U(𝔲^{−})𝔲^{−} ⊗_{ℂ} M) ⊕ (1 ⊗ M).

Let w ∈ U(𝔲^{−})𝔲^{−}⊗_{ℂ}M be nonzero. Since M is a weight module, we may assume w is a linear combination of vectors of the form f_{i_{1}}f_{i_{2}} … f_{i_{j}} ⊗ v, where v ∈ M is a weight vector of weight λ, and the f_{i_{1}}f_{i_{2}} ⋯ f_{i_{j}}, i_{1} ≤ i_{2} ≤ ⋯ ≤ i_{j} is an element of the Poincare-Birkhoff-Witt basis of U(𝔲^{−}) ⊂ U(𝔫^{−}). We will consider the elements f_{i_{1}}f_{i_{2}} ⋯ f_{i_{j}} ⊗ v, i_{1} ≤ i_{2} ≤ ⋯ ≤ i_{j} to be basis vectors of the vector space U(𝔲^{−})𝔲^{−} ⊗ v. Let w = f_{i_{1}}f_{i_{2}} … f_{i_{j}} ⊗ v. Not all of the indices i_{1}, i_{2}, … i_{j} are in S, by the definition of 𝔲^{−}. Assume that i_{k} ∈ I\S, we will show that e_{i_{k}}w ≠ 0. The vector w can be rewritten as w = f_{i_{1}}f_{i_{2}} ⋯ f_{i_{k−1}}f_{i_{k}}^{n}f_{i_{k+1}} ⋯ f_{i_{j}} ⊗ v where n ≥ 1, i_{i} ≤ i_{2} ≤ ⋯ ≤ i_{k−1} < i_{k} < i_{k+1} ≤ ⋯ ≤ i_{j}, renaming the indices if necessary. Using the commutation relations of the Lie algebra,

e_{i_{k}}w = f_{i_{1}} ⋯ f_{i_{k−1}} (e_{i_{k}}f_{i_{k}}^{n}) ⋯ f_{i_{j}} ⊗ v

= f_{i_{1}} ⋯ f_{i_{k−1}} (f_{i_{k}}^{n}e_{i_{k}} + nf_{i_{k}}^{n−1}h_{i_{k}} + −a_{i_{k}i_{k}} \frac{(n − 1)n}{2}f_{i_{k}}^{n−1}) f_{i_{k+1}} ⋯ f_{i_{j}} ⊗ v

= f_{i_{1}} ⋯ f_{i_{k}}^{n}e_{i_{k}}f_{i_{k+1}} ⋯ f_{i_{j}} ⊗ v + nf_{i_{1}} ⋯ f_{i_{k}}^{n−1}h_{i_{k}}f_{i_{k+1}} ⋯ f_{i_{j}} ⊗ v

+ −a_{i_{k}i_{k}} \frac{(n − 1)n}{2}f_{i_{1}} ⋯ f_{i_{k}}^{n−1}f_{i_{k+1}} ⋯ f_{i_{j}} ⊗ v.

Since v is in M it is annihilated by e_{i_{k}} ∈ 𝔲^{+} , and one has

f_{i_{1}} ⋯ f_{i_{k}}^{n}e_{i_{k}}f_{i_{k+1}} ⋯ f_{i_{j}} ⊗ v = f_{i_{1}} ⋯ f_{i_{k}}^{n}f_{i_{k+1}} ⋯ f_{i_{j}}e_{i_{k}} ⊗ v = 0.

It is also true that

h_{i_{k}}f_{i_{k+1}} ⋯ f_{i_{j}} ⊗ v = f_{i_{k+1}} ⋯ f_{i_{j}}h_{i_{k}} ⊗ v + ∑_{s=1}^{j−k} −a_{i_{k}i_{k+s}}f_{i_{k+1}} ⋯ f_{i_{j}} ⊗ v.

Thus

e_{i_{k}}w

= nf_{i_{1}} ⋯ f_{i_{k}}^{n−1}f_{i_{k+1}} ⋯ f_{i_{j}}h_{i_{k}} ⊗ v + n ∑_{s=1}^{j−k} −a_{i_{k}i_{k+s}}f_{i_{1}} ⋯ f_{i_{k}}^{n−1}f_{i_{k+1}} ⋯ f_{i_{j}} ⊗ v

− a_{i_{k}i_{k}} \frac{(n − 1)n}{2}f_{i_{1}} ⋯ f_{i_{k}}^{n−1}f_{i_{k+1}} ⋯ f_{i_{j}} ⊗ v

= nλ(h_{i_{k}})f_{i_{1}} ⋯ f_{i_{k}}^{n−1}f_{i_{k+1}} ⋯ f_{i_{j}} ⊗ v − n ∑_{s=1}^{j−k} a_{i_{k}i_{k+s}}f_{i_{1}} ⋯ f_{i_{k}}^{n−1}f_{i_{k+1}} ⋯ f_{i_{j}} ⊗ v

− a_{i_{k}i_{k}} \frac{(n − 1)n}{2}f_{i_{1}} ⋯ f_{i_{k}}^{n−1}f_{i_{k+1}} ⋯ f_{i_{j}} ⊗ v

= [nλ(h_{i_{k}}) + n ∑_{s=1}^{j−k} −a_{i_{k}i_{k+s}} + −a_{i_{k}i_{k}} \frac{(n − 1)n}{2}] f_{i_{1}} ⋯ f_{i_{k}}^{n−1}f_{i_{k+1}} ⋯ f_{i_{j}} ⊗ v.

≠ 0.

The above expression is nonzero because λ(h_{i_{k}}) > 0, a_{i_{k}i_{k+s}} ≤ 0 for all 1 ≤ s ≤ j − k and a_{i_{k}i_{k}} ≤ 0 for i_{k} ∈ I\S. Let w′ = f_{i′_{1}} ⋯ f_{i′_{l}} ⊗ v, i′_{1} ≤ i′_{2} ≤ ⋯ i′_{l} be a basis vector not equal to w. If f_{i_{k}} does not occur in w′ we have e_{i_{k}}w′ = 0, and if f_{i_{k}} does occur in w′ then the lists i_{1} ≤ i_{2} ≤ ⋯ ≤ i_{j} and i′_{1} ≤ i′_{2} ≤ ⋯ ≤ i′_{l} do not agree, so by the above computation e_{i_{k}}w is not in the linear span of e_{i_{k}}w′. Now consider an element w ∈ U(𝔲^{−})𝔲^{−} ⊗ v that is a linear combination of the basis vectors f_{i_{1}}f_{i_{2}} … f_{i_{j}} ⊗v, i_{1} ≤ i_{2} ⋯ ≤ i_{j}, choose an index i_{k} ∈ I \S that appears in the first term of w. Then e_{i_{k}}w is a linear combination of the basis vectors, with nonzero first term, so is nonzero. We conclude that w is not annihilated by 𝔲^{+} , that (Ind_{𝔭}^{𝔤}M)^{𝔲^{+}} ⊂ 1 ⊗ M, and finally that (Ind_{𝔭}^{𝔤}M)^{𝔲^{+}} = 1 ⊗ M.

Proposition 3.2. Let V ∈ O_{𝔤}^{S} be a highest weight module. Then V = U(𝔤) ⊗_{U(𝔭)} V^{𝔲^{+}} .

Proof. Let λ be the highest weight of V , with highest weight vector v_{0} . The induced module U(𝔤) ⊗_{U(𝔭)} V^{𝔲^{+}} is also a highest weight module for 𝔤. There is a natural map μ : U(𝔤) ⊗_{U(𝔭)} V^{𝔲^{+}} → U(𝔤)V^{𝔲^{+}} = V , which we will show is an isomorphism. The map μ is surjective because V^{𝔲^{+}} generates the module V , as V^{𝔲^{+}} contains the highest weight vector v_{0} . Let W = Ker μ. If W is nontrivial then W contains a weight vector w ≠ 0 whose weight is of minimal depth. Applying an element of 𝔲^{+} to w ∈ W reduces the depth of w, so that w is annihilated by 𝔲^{+} . By Lemma 3.1 w ∈ 1 ⊗ V^{𝔲^{+}} , which contradicts the fact that μ is injective when restricted to 1 ⊗ V^{𝔲^{+}} .

Proposition 3.3. Let V ∈ O_{𝔤}^{S}, then V = U(𝔤) ⊗_{U(𝔭)} N for some N ∈ O_{𝔯}^{S}.

Proof. Consider the highest weight series

(0) = V_{0} ⊂ V_{1} ⊂ V_{2} ⊂ ⋯

where ⋃_{i=0}^{∞} V_{i} = V . Note ⋃_{i=0}^{∞} V_{i}^{𝔲^{+}} = (⋃_{i=0}^{∞} V_{i})^{𝔲^{+}} = V^{𝔲^{+}} .

We will prove the existence of an isomorphism by induction on i ∈ ℤ_{+} . The module V_{1} is by assumption a highest weight module so Proposition 3.2 shows the proposition holds in this case. Similarly, the result holds for the highest weight module V_{i+1}/V_{i} any i ∈ ℤ_{+} . Assume that V_{i} = U(𝔤) ⊗ V_{i}^{𝔲^{+}} . Applying the five lemma to the diagram

0 → U(𝔤) ⊗ V_{i}^{𝔲^{+}} ⟶ U(𝔤) ⊗ V_{i+1}^{𝔲^{+}} ⟶ U(𝔤) ⊗ (V_{i+1}/V_{i})^{𝔲^{+}} → 0
≈ ↓  ↓  ≈ ↓
0 →  V_{i}  ⟶  V_{i+1}  ⟶  V_{i+1}/V_{i}  → 0

implies V_{i+1} = U(𝔤) ⊗ V_{i+1}^{𝔲^{+}} , so by induction V_{i} = U(𝔤) ⊗ V_{i}^{𝔲^{+}} for all i ∈ ℤ_{+} . Thus

V = ⋃ U(𝔤) ⊗ V_{i}^{𝔲^{+}} = U(𝔤) ⊗ ⋃ V_{i}^{𝔲^{+}} = U(𝔤) ⊗ V^{𝔲^{+}}.

Theorem 3.4. The category O_{𝔤}^{S} is equivalent to the category O_{𝔯}^{S}.

Proof. First we show that the functor Ind_{𝔭}^{𝔤} : O_{𝔯}^{S} → O_{𝔤}^{S} is full. All objects V_{1}, V_{2} of O_{𝔤}^{S} are, by Proposition 3.2, of the form V_{i} = U(𝔤) ⊗_{U(𝔭)} V_{i}^{𝔲^{+}} and V_{2} = U(𝔤) ⊗_{U(𝔭)} V_{2}^{𝔲^{+}} . Then any 𝔤-homomorphism g : V_{1} → V_{2} is determined by its action on V_{1}^{𝔲^{+}}

g(U(𝔤) ⊗_{U(𝔭)} V_{1}^{𝔲^{+}}) = U(𝔤) ⊗_{U(𝔭)} ḡ(V_{1}^{𝔲^{+}}).

Hence g = 1 ⊗ ḡ.

Since 1 ⊗ f = 1 ⊗ g if and only if f = g the functor Ind_{𝔭}^{𝔤} is faithful. Finally, Proposition 3.3 says that for every object V of O_{𝔤}^{S} there is an

N ∈ O_{𝔯}^{S} with Ind_{𝔭}^{𝔤}(N) = V . The conditions of Lemma 2.2 are satisfied and the categories are equivalent.

4. Applications

Our first corollary includes Proposition 4.2 of [10].

Corollary 4.1. Fix S ⊂ I. For μ ∈ P_{+} satisfying (μ, α_{i}) > 0 for i ∈ I\S the irreducible highest weight module L_{𝔤}(μ) is a generalized Verma module:

L_{𝔤}(μ) = Ind_{𝔭}^{𝔤}L(μ),

here L(μ) is the irreducible highest weight 𝔭-module of highest weight μ, which is a standard irreducible module for the generalized Kac-Moody algebra 𝔤_{S}. Conversely, any generalized Verma module V^{L(μ)} = Ind_{𝔭}^{𝔤}L(μ) is an irreducible highest weight 𝔤-module.

The following complete reducibility result appears in [14](where he uses the additional assumption that I is finite). For a Kac-Moody algebra 𝔩, μ ∈ 𝔥^{∗}, and an 𝔩-module M in O_{𝔩} define the multiplicity [M : L(μ)] of the irreducible 𝔩-module L(μ) in M to be the number of proper factors of type L(μ) in any local composition series of M at μ see [13].

Corollary 4.2. Let 𝔤 be a generalized Kac-Moody algebra. Take S = {i ∈ I|a_{ii} > 0}, so that 𝔤_{S} is a Kac-Moody algebra. Let V be a module in the category O_{𝔤}^{S}. Then V is completely reducible and

V = ⨁_{μ∈P_{+}}[V^{𝔲^{+}}, L(μ)]L_{𝔤}(μ).

Proof. Since 𝔤_{S} is a Kac-Moody algebra and V^{𝔲^{+}} is an integrable 𝔤_{S} -module the complete reducibility of integrable modules of a Kac-Moody algebra implies

V^{𝔲^{+}} = ⊕_{μ∈P_{+}}[V^{𝔲^{+}}, L_{𝔯}(μ)]L_{𝔯}(μ).

Applying the main theorem gives

Ind_{𝔭}^{𝔤}V^{𝔲^{+}} = ⊕_{μ∈P_{+}}[V^{𝔲^{+}}, L_{S}(μ)]Ind_{𝔭}^{𝔤}L_{𝔯}(μ)

So by Corollary 4.1 and Proposition 3.2

V = ⊕_{μ∈P_{+}}[V^{𝔲^{+}}, L_{𝔯}(μ)]L_{𝔤}(μ).

For a module V ∈ O_{𝔤} recall the formal character is defined as

ch(V) = ∑_{λ∈𝔥^{∗}} dimV_{λ}e^{λ}.

Using Theorem 3.4, we can compute the character of a module V ∈ O_{𝔤}^{S} by computing the product of the 𝔥 weight modules U(𝔲^{−}) and V^{𝔲^{+}} .

Corollary 4.3. Let V ∈ O_{𝔤}^{S}. Then

ch(V) = ch(U(𝔤) ⊗_{U(𝔭)} V^{𝔲^{+}}) = ch(U(𝔲^{−})) · ch(V^{𝔲^{+}})

Depending upon the structure of the Lie algebra, ch(U(u^{−})) and V^{𝔲^{+}} may be computed more readily than ch(V). For example, choose S so that a_{ij} ≠ 0 for all i, j ∈ I\S. The subalgebra 𝔤_{S} of 𝔤 and hence its universal enveloping algebra U(𝔤_{S}) act via the adjoint action on 𝔤. Identify the root −α_{j} ∈ −Δ_{+} of 𝔤 with the weight of the highest weight vector f_{j} of the standard 𝔤_{S} -module L_{𝔤_{S}}(−α_{i}) = U(𝔤_{S}) · f_{j} ⊂ 𝔤. By Theorem 5.1 in [9] 𝔤 = 𝔲^{+} ⊕ (𝔤_{S} + 𝔥) ⊕ 𝔲^{−}, where 𝔲^{−} is the free Lie algebra over the vector space ∑_{j∈I\S} L(−α_{i}).

Then

ch(U(𝔲^{−})) = 1 − ∑_{j∈I\S} ch(L_{S}(−α_{j})) = ∏_{φ∈Δ_{+}\Δ_{+}^{S}}(1 − e^{−φ})^{dim F (V)^{−φ}}

and

ch(V) = ∏_{φ∈Δ_{+}\Δ_{+}^{S}}(1 − e^{−φ})^{dim F (V)^{−φ}}ch(V^{𝔲^{+}}).  (1)

In the case of the monster Lie algebra, one can choose S so that 𝔤_{S} = 𝔰𝔩_{2} . Then each L(−α_{i}) is a finite dimensional 𝔰𝔩_{2} -module of highest weight −α_{1} . For this case equation (1) specializes to Borcherds’ product formula for the modular function j(τ) [2]. Formula (1) appears in [8],[9] and is used in [10] to further elucidate a portion of the monstrous moonshine phenomenon already studied in [2].

Computing the homology H_{n}(𝔤, V) for V ∈ O_{𝔤}^{S} can also be reduced to computing the homology groups for modules for the smaller Lie algebra 𝔤_{S}.

Corollary 4.4. For a g -module V in O_{𝔤}^{S}

H_{n}(𝔤, V^{t}) ≃ H_{n}(𝔭, (V^{u^{+}})^{t}).

Proof. This corollary follows from Theorem 3.4 and Proposition 4.2 of [3].

References

[1] Borcherds, R. E., Generalized Kac-Moody algebras, J. Algebra 115 (1988), 501–512.

[2] —, Monstrous moonshine and monstrous Lie superalgebras, Invent. Math. 109 (1992), 405–444.

[3] Cartan, H. and S. Eilenberg, “Homological Algebra,” Princeton University Press 1956.

[4] Cox, B., Structure of the nonstandard category of highest weight modules, Modern trends in Lie algebra representation theory (Kingston, ON, 1993) Queen’s Univ., Kingston, ON (1994), 35–47.

[5] Enright, T., and B. Shelton, Categories of highest weight modules: applications to classical Hermitian symmetric pairs, Memoirs of the American Mathematical Society, 67, 1987, 367, iv+94 pp.

[6] Garland, H., and J. Lepowsky, Lie algebra homology and the Macdonald-Kac formulas, Invent. Math. 34 (1976), 37–76.

[7]  Jacobson, N., “Basic Algebra,” 2, W. H. Freeman, New York, 1989.

[8] Jurisich, E., Generalized Kac-Moody Lie algebras, free Lie algebras and the structure of the Monster Lie algebra, Journal of Pure and Applied Algebra 126 (1998), 233–266.

[9] —, An exposition of generalized Kac-Moody algebras in: “Lie algebras and their representations,” Amer. Math. Soc., Contemporary Mathematics 194, 121–160.

[10] Jurisich, E., J. Lepowsky, and R. Wilson, Realizations of the Monster Lie algebra, Selecta Mathematica, New Series 1 (1995), 129–161.

[11] Kac, V., “Infinite Dimensional Lie Algebras,” Cambridge University Press, 3rd Edition, 1990.

[12] Lepowsky, J., and G. W. McCollum, On the determination of irreducible modules by restriction to a subalgebra, Trans. Amer. Math. Soc. 176 (1973), 217–228.

[13] Moody, R., and A. Pianzola, “Lie algebras with triangular decompositions,” Wiley-Interscience, 1995.

[14] Naito, S., The strong Bernstein-Gelfand-Gelfand resolution for generalized Kac-Moody algebras, I, The existence of the resolution, Publ. of the Research Institute for Mathematical Sciences, 29 (1993), 709–730.

Elizabeth Jurisich Department of Mathematics The College of Charleston Charleston SC 29424

Received October 7, 2002 and in final form June 4, 2003