Higher-order simple Lie algebras

TL;DR

This paper introduces higher-order simple Lie algebras derived from cocycles on simple Lie algebras, providing a classification, a constructive method, and a unified BRST operator framework.

Contribution

It defines higher-order Lie algebras using cocycles, offers a classification scheme, and constructs a complete BRST operator for each simple algebra.

Findings

Higher-order Lie algebras satisfy generalized Jacobi identities.

A classification of higher-order simple Lie algebras is provided.

A constructive procedure and a unified BRST operator are introduced.

Abstract

It is shown that the non-trivial cocycles on simple Lie algebras may be used to introduce antisymmetric multibrackets which lead to higher-order Lie algebras, the definition of which is given. Their generalised Jacobi identities turn out to be satisfied by the antisymmetric tensors (or higher-order `structure constants') which characterise the Lie algebra cocycles. This analysis allows us to present a classification of the higher-order simple Lie algebras as well as a constructive procedure for them. Our results are synthesised by the introduction of a single, complete BRST operator associated with each simple algebra.