Second cohomology of generalized Cartan type S Lie algebras in characteristic zero

Abstract

Let W = W ( A, T, ϑ) be a simple generalized Witt algebra over a field F of characteristic 0 built from an abelian torsion-free group A, an F-vector space T, and a pairing ϑ: T × A → F. For simplicity, assume that dim T ≥ 4. Let L = t z S, z gE A, be the simple subalgebras of W of generalized Cartan type S, as introduced in our earlier paper (1997). We show that H 2( L, F) is isomorphic to the space of skew-symmetric bilinear forms T × T → F.