Representations of Affine and Toroidal Lie Algebras

TL;DR

This paper studies the structure and properties of level zero integrable modules of loop algebras, exploring their homological aspects, irreducible modules, and connections to finite-dimensional algebra representations.

Contribution

It introduces a detailed analysis of the category of level zero modules, including universal Weyl modules and their homological properties, extending understanding of non-semisimple Lie algebra representations.

Findings

Characterization of irreducible level zero modules

Development of local and global Weyl modules

Connections established with finite-dimensional algebra representations

Abstract

We discuss the category $\cal I$ of level zero integrable representations of loop algebras and their generalizations. The category is not semisimple and so one is interested in its homological properties. We begin by looking at some approaches which are used in the study of other well--known non--semisimple categories in the representation theory of Lie algebras. This is done with a view to seeing if and how far these approaches can be made to work for $\cal I$. In the later sections we focus first on understanding the irreducible level zero modules and later on certain universal modules, the local and global Weyl modules which in many ways play a role similar to the Verma modules in the BGG--category $\cal O$. In the last section, we discuss the connections with the representation theory of finite--dimensional associative algebras and on some recent work with J. Greenstein.