On the low-dimensional cohomology of some infinite-dimensional simple Lie algebras

Stephen Berman

Vol. 83 (1979), No. 1, 27–36

DOI: 10.2140/pjm.1979.83.27

Abstract

The two dimensional cohomology, with values in the base field K of characteristic 0, of a simple Lie algebra attached to any non-Euclidean indecomposable Cartan matrix is computed. We find that dim(H²(ℒ,K)) equals the nullity of the Cartan matrix which defines ℒ. We also show that there is an invariant 3-cocycle of ℒ if and only if the matrix defining ℒ is symmetrizable. This yields cohomological interpretations for all the known isomorphism class invariants of these algebras.

Mathematical Subject Classification 2000

Primary: 17B56

Secondary: 17B65

Milestones

Received: 31 July 1978

Revised: 2 January 1979

Published: 1 July 1979