Structure of some categories of representations of infinite-dimensional lie algebras

Abstract

The highest weight representations of a finite-dimensional complex semisimple Lie algebra 9 have been studied extensively and a number of interesting results have been obtained in the last few years (an extensive bibliography is given in [7]). In [2, 3] a category cY of representations of g was introduced which turns out to be the proper setting for considering various questions regarding highest weight representations. The aim of this paper is to extend the study of this notion to a Kac-Moody algebra, or, more generally, to a contragredient Lie algebra G(A) corresponding to a square matrix A over C. The study of such algebras was begun independently in [8, 14] and subsequently a number of interesting results were proved. These results show that many important properties from the finite-dimensional setup (i.e., from the context of 9) can be proved in the general setup involving G(A). In particular, the notion of category • was extended in [9] to the case of Kac-Moody algebras and used in the proof of combinatorial identities.