Representations of Loop Kac-Moody Lie Algebras

TL;DR

This paper investigates the structure and properties of representations of Loop Kac-Moody Lie algebras, focusing on finite-dimensional weight spaces and their graded variants, with applications to toroidal Lie algebras.

Contribution

It introduces a comprehensive study of representations of Loop Kac-Moody Lie algebras, including finite-dimensional weight spaces and graded modules, extending to toroidal Lie algebras.

Findings

Characterization of representations with finite-dimensional weight spaces

Development of graded module frameworks

Connections established to toroidal Lie algebras

Abstract

We study representations of the Loop Kac-Moody Lie algebra g \otimes A, where g is any Kac-Moody algebra and A is a ring of Laurent polynomials in n commuting variables. In particular, we study representations with finite dimensional weight spaces and their graded versions. When we specialize g to be a finite dimensional or affine Lie algebra we obtain modules for toroidal Lie algebras.