On the structure of symmetric self-dual Lie algebras

Abstract

A finite-dimensional Lie algebra is called (symmetric) self-dual, if it possesses an invariant nondegenerate (symmetric) bilinear form. Symmetric self-dual Lie algebras have been studied by Medina and Revoy, who have proven a very useful theorem about their structure. In this paper we prove a refinement of their theorem that has wide applicability in conformal field theory, where symmetric self-dual Lie algebras start to play an important role due to the fact that they are precisely the Lie algebras that admit a Sugawara construction. We also prove a few corollaries that are important in conformal field theory.