Generalized Cartan TypeSLie Algebras in Characteristic Zero

Abstract

Ž . Generalized Witt algebras W s W A, T , w over a field F of characterw x istic 0 were introduced by N. Kawamoto 8 about 10 years ago, and have Ž w x. been studied by several authors since that time see 2, 6, 7, 9]12 . Let us just mention here that A is a torsion-free abelian group, T a vector space over F, and w : T = A a F a map which is linear in the first variable and additive in the second one. As a vector space, W s FA m T , where FA is F the group algebra of A over F with basis t , x g A. We write t ­ instead of t x m ­ for x g A and ­ g T. For the definition of the Lie bracket see the next section. The Lie algebra W is A-graded with homogeneous components W s t T , x g A. x For simplicity, we shall assume in this introduction that the Lie algebra W is simple, i.e., that w is nondegenerate in the sense defined in the next w x section. In our previous paper 2 , we have determined the derivation Ž . algebra Der W of W, we have constructed all isomorphisms W a W9 between two simple generalized Witt algebras, and we have computed the 2Ž . second cohomology group H W, F . One can introduce several classes of simple Lie subalgebras of W. One such class, so-called algebras of generalized Cartan type W, has been w x introduced and investigated in our recent paper 4 . In the present paper we introduce yet another class, to which we refer as the algebras of