An application of free Lie algebras to current algebras and their representation theory

TL;DR

This paper employs free Lie algebras to analyze current algebras of Kac-Moody algebras, revealing new representation constructions and connections to classical modules, with implications for understanding their structure and extensions.

Contribution

It introduces a novel realization of current algebras via free Lie algebras, enabling new insights into their representation theory and classical limits.

Findings

Every ad-invariant ideal yields a current algebra representation.

Classical limits of Kirillov-Reshetikhin modules are explicitly constructed.

Extensions in the category of finite-dimensional modules are studied.

Abstract

We realize the current algebra of a Kac-Moody algebra as a quotient of a semi-direct product of the Kac-Moody Lie algebra and the free Lie algebra of the Kac-Moody algebra. We use this realization to study the representations of the current alg ebra. In particular we see that every ad-invariant ideal in the symmetric algebra of the Kac-Moody algebra gives rise in a canonical way to a representation of the current algebra. These representations include certain well-known families of representations of the current algebra of a simple Lie algebra. Another family of examples, which are the classical limits of the Kirillov-Reshe tikhin modules, are also obtained explicitly by using a construction of Kostant. Finally we study extensi ons in the category of finite dimensional modules of the current algebra of a simple Lie algebra.