The first cohomology groups of infinite dimensional Lie algebras

Abstract

Let y be a finite dimensional vector space. We denote by D(V) the Lie algebra consisting of all formal vector fields over V. Let L be a Lie subalgebra of D(V). We are interested in the first cohomology group H(L) of a Lie algebra L with adjoint representation. Let L be an infinite dimensional transitive simple Lie algebra, that is, L is one of D(V), Lgt, Lip, or Lct. (For a notation, see §2.) It is known in T. Morimoto [5] that H(D(V)) = H(Lct) = Q, and dim H (Lg[) = dim H (L9J = l. In this paper we will treat the following two types of infinite dimensional Lie algebras: