On the structure of symmetric self-dual Lie algebras

TL;DR

This paper refines the structural theorem of symmetric self-dual Lie algebras, highlighting their significance in Conformal Field Theory through the Sugawara construction and related corollaries.

Contribution

It provides a refined structural theorem for symmetric self-dual Lie algebras and explores their applications in Conformal Field Theory.

Findings

Refined the Medina-Revoy theorem for symmetric self-dual Lie algebras.

Identified symmetric self-dual Lie algebras as those admitting a Sugawara construction.

Derived corollaries relevant to Conformal Field Theory.

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 which 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 which admit a Sugawara construction. We also prove a few corollaries which are important in Conformal Field Theory. (This paper provides mathematical details of results used, but only sketched, in the companion paper hep-th/9506151.)