Irreducible Representations of the Lie-Algebra of the Diffeomorphisms of ad-Dimensional Torus

Abstract

In this paper we investigate the irreducibility of certain modules for the Lie-algebra of diffeomorphisms of torus T . The Lie-algebra of diffeomorw x Ž phisms can be described as the derivations of A s C t , . . . t see 1 d w x. w x RSS . We denote this Lie-algebra by Der A. Larsson L1 constructed a functor from gl -modules to Der A modules. In this paper we prove that d the image of a finite-dimensional irreducible gl module is most often d irreducible. The only exceptions are fundamental modules and the onedimensional modules. In these cases we describe the sub-modules and the Ž quotients. Surprisingly the same class of modules image of finite dimen. w x sional gl -module are also constructed in E without any reference to d gl -modules. They are motivated by the vertex operator constructions in d w x Ž . EM more details near the end of this introduction . We will now describe our results in more detail. Let gl be the d Lie-algebras of d = d matrices over complex numbers C. Let gl s sl [ d d C Id where sl is the finite-dimensional simple Lie-algebra of trace zero d matrices and Id, the identity matrix which is central. It is well known that finite dimensional irreducible modules and dominant integral weights are in one]one correspondence for a finite dimensional simple Lie-algebra. So Ž . let V c be finite dimensional irreducible sl module corresponding to a d dominant integral weight c . Let Id, the central element, act by a complex Ž . number b and denote the corresponding gl module by V c , b . d Ž . a Ž d d. Larsson defined see 1.6 a functor F for a belonging to C rZ from w x gl modules to Der A modules in L1 . These modules are weight modules d