Representation Theory of the Witt Algebra

Abstract

1.1. Let g be a finite-dimensional restricted Lie algebra over an algebraically closed field of characteristic p ) 0. Every simple g-module is U Ž w finite-dimensional and therefore admits a character x g g see SF, x. U Theorem 5.2.5 . Conversely, for any such linear form x g g , there exists Ž . a finite-dimensional algebra u g , x which is a quotient of the universal Ž . enveloping algebra U g and whose simple modules are exactly the simple modules for g with character x . Because of this fact, these algebras play an important role in studying the representations of g. In particular, the Ž . restricted g-representations coincide with the u g , 0 -modules. Every finite-dimensional restricted simple Lie algebra over an algebraically closed field of characteristic p ) 7 is either classical or of Cartan w x type BW . For a classical Lie algebra g , the representation theory of Ž . w x u g , x was first studied by Kac and Weisfeiler WD, KW , and further