Realizations of -modules in category

By Fulin Chen, Yun Gao, and Shaobin Tan

Abstract

In this paper, we give an explicit realization of all irreducible modules in ChariтАЩs category for the affine Kac-Moody algebra by using the idea of free fields. We work on a much more general setting which also gives us explicit realizations of all simple weight modules for certain current algebra of with finite weight multiplicities, including the polynomial current algebra , the loop algebra and the three-point Lie algebra arisen in the work by Kazhdan-Lusztig.

1. Introduction

Let be any multiplicatively closed subset of the polynomial ring in one variable, and let be the localization of at . We will construct representations for the current Lie algebra and the affine Kac-Moody algebra . By specializing , this gives us representations for the polynomial current algebra , the loop algebra and the three-point Lie algebra appearing in the study of the tensor structure of affine Kac-Moody algebras by Kazhdan-Lusztig Reference 34.

The purpose of this paper is two-fold. The first one is to present an explicit realization of all irreducible -modules in the category . The category , introduced by Chari Reference 8, is an analog of the BGG category Reference 3 corresponding to the natural Borel subalgebra of . The category appeared naturally in the study of level modules for affine Kac-Moody Lie algebras Reference 8Reference 10Reference 14. The irreducible integrable objects in the category for were classified in Reference 8 and then realized in Reference 13, which exhaust all irreducible level integrable -modules with finite weight multiplicities. Moreover, it was proved in Reference 30Reference 31 that any level unitarizable highest weight module for (without derivations) must be an irreducible module induced from the natural Borel subalgebra. In contrast to the irreducible highest weight modules in the category of Reference 32, the irreducible modules in the category have both finite and infinite weight multiplicities and much more complicated structure Reference 8Reference 26. A character formula for the irreducible -modules in the category with finite weight multiplicities was obtained in Reference 40. In Section 6 of this paper, we give a free field(-like) realization of all (non-integrable) irreducible -modules in the category .

Free field realization of modules for affine Kac-Moody algebras plays an important role in representation theory and conformal field theory. In Reference 30, Jakobsen-Kac gave a free field construction for certain level Verma type -modules, which is referred as imaginary Verma modules Reference 25. The free field realization of the imaginary Verma -modules at an arbitrary level was given by Bernard-Felder in Reference 2 and then extended in Reference 15 to the case of . To prove the Kac-Kazhdan conjecture on the characters of irreducible highest weight -modules at the critical level, Wakimoto gave in Reference 39 a remarkable free field construction of -modules at an arbitrary level. Since then, the Wakimoto modules for general affine Lie algebras have been extensively studied Reference 17Reference 19Reference 20Reference 22Reference 23Reference 27Reference 28Reference 29Reference 33Reference 36Reference 38. In this paper, for the purpose of realizing the irreducible -modules in the category with finite weight multiplicities, we introduce a free field(-like) construction of level -modules.

The free field constructions of -modules given in Reference 2Reference 30Reference 39 are all realized on the polynomial rings in infinitely many variables, in terms of infinite sums of partial differential operators. In contrast to the constructions given in Reference 2Reference 30Reference 39, in this paper we realize a class of -modules on the polynomial rings in finitely many variables, in terms of finite sums of partial differential operators that are glued together by certain Lagrange interpolation polynomials. Explicitly, for any and with , we prove in Section 4.2 that the map

defines an -module structure on the polynomial ring , where stand for the fundamental Lagrange interpolation polynomials of degree at the points , and

Especially, by applying Chari-PressleyтАЩs loop module construction Reference 13Reference 21, we construct in this way an inverse system

of level -modules in the category with finite weight multiplicities. We emphasize that if , then the last term in Equation 1.3 is irreducible.

Besides the algebra , the free field constructions for with have been given in Reference 15 (see also Reference 17). By combining the above construction with that given in Reference 15, one can also realize a class of irreducible -modules in the category with finite weight multiplicities. In particular, the character formula of such -modules will be obtained, which are not known in general. Details for general will be given in another paper.

The second goal of this paper is to provide an explicit realization of all Harish-Chandra modules for the current algebra . An -module is said to be Harish-Chandra if it is irreducible and decomposes into finite dimensional common eigenspaces with respect to . Recently, Lau gave in Reference 37 the classification of Harish-Chandra modules for the general current algebra , where is a reductive Lie algebra and is a finitely generated commutative algebra. In Section 7, we give a free field realization of all Harish-Chandra -modules. By taking , this leads to a free field realization of all Harish-Chandra -modules. Besides the loop algebras which appeared naturally in the affine Kac-Moody algebra theory, there are some other current algebras of particular importance, including the polynomial current algebras and the -point Lie algebras. Motivated by its relationship with the representation theory of affine and quantum affine algebras, the representation theory of the polynomial current algebra is now extensively studied Reference 1Reference 9Reference 11Reference 12Reference 24. By choosing , we obtain a realization of all Harish-Chandra -modules. On the other hand, by taking , we obtain a realization of all Harish-Chandra modules for the -point Lie algebras Reference 5Reference 6. The three and four point Lie algebras appeared naturally in the work of Kazhdan-Lusztig Reference 34 on the tensor structure of modules over affine Kac-Moody algebras. And, by generalizing WakimotoтАЩs construction, the free field realizations of modules for the three and four point Lie algebras were given respectively in Reference 18 and Reference 16.

We now outline the structure of the paper. In Sections 2 and 3, we collect some elementary facts on the highest weight -modules. In Section 4, we give a free field realization of certain quasi-finite highest weight -modules. Using this and a result of Jakobsen-Kac, we give an explicit realization of all irreducible highest weight -modules in Section 5. As applications, in Section 6 we present an explicit realization of all irreducible -modules in ChariтАЩs category , and in Section 7 we give an explicit realization of all Harish-Chandra -modules.

In this paper, let , , , and be the set of complex numbers, nonzero complex numbers, integers, nonnegative integers and positive integers, respectively.

2. Highest weight theory for current algebras of

Throughout this section, let be a unital commutative associative algebra over and let be a -valued linear function on .

2.1. Highest weight -modules

For any Lie algebra over , let be the current algebra of , with the commutator relations given by

where and . Let

be the Lie algebra of traceless -matrices over . We fix a basis of as in Equation 1.2. Let be the standard triangular decomposition of , where and . Then we have the triangular decomposition:

A -module is said to be a weight module if , where . For a weight -module , we denote by

the set of weights on .

Definition 2.1.

(i) A weight -module is called quasi-finite if for every , and is called Harish-Chandra if is in addition irreducible.

(ii) A -module is called a highest weight module with highest weight if there exists a nonzero vector , called the highest weight vector, such that

Note that any highest weight -module with highest weight is a weight module and the weights have the form , where and is the root of defined by .

Remark 2.2.

We define a notion of lowest weight -module with lowest weight by replacing with in Definition 2.1(ii). Extend the Chevalley involution

to an involution of , still denoted as , such that for . Let be a highest weight -module with highest weight . By twisting the involution , one obtains a new -module structure on , denoted as . One can check that is a lowest weight -module with lowest weight , and is irreducible provided that is irreducible.

2.2. Verma type highest weight -modules

Let be the one dimensional -module defined by for . We extend to an -module with acting trivially. Form the induced -module

Note that any highest weight -module with highest weight is a quotient of , and the -module is quasi-finite if and only if . Thus, if , then every highest weight -module is quasi-finite.

We view the dual space of as an -module under the natural action

For any , denote by

the annihilator ideal of associated to . Set

a linear isomorphism of vector spaces.

Remark 2.3.

Since and is generated by , it follows that any proper -submodule of must intersect trivially. Thus, there is a unique maximal proper -submodule, denoted as , of .

We have:

Lemma 2.4.

One has that

Proof.

Let . It is obvious that if and only if . Moreover, for any , one has

Thus we have if and only if .

тЦа

2.3. Irreducible quasi-finite highest weight -modules

We denote by

the irreducible quotient of the Verma type -module . In this subsection, we give a sufficient and necessary condition for to be quasi-finite.

Let be another commutative associative algebra, and let be an algebra homomorphism. For any -module , the pull back of yields a natural -module structure on with

We denote by the resulting -module. The following result is obvious.

Lemma 2.5.

Let be a highest weight -module with highest weight . Assume that the homomorphism is surjective. Then is a highest weight -module with highest weight

Set . Since vanishes on , it induces a linear map

In view of Lemma 2.5, one has the following -module isomorphism

where is the quotient map. We denote by

Proposition 2.6.

The irreducible highest weight -module is quasi-finite if and only if .

Proof.

If , then the -module is quasi-finite and so is the -module (see Equation 2.3). Conversely, if the -module is quasi-finite, then one can conclude from Lemma 2.4 that