Finite-dimensional representation theory of loop algebras: a survey

TL;DR

This survey reviews key results on finite-dimensional representations of loop algebras of simple complex Lie algebras, including parametrization, module descriptions, and recent advances in multiloop algebra theory.

Contribution

It provides a comprehensive overview of the classification, structure, and recent developments in the finite-dimensional representation theory of loop and multiloop algebras.

Findings

Parametrization of Weyl modules and irreducible representations

Block decomposition of the representation category

Recent progress in multiloop algebra representations

Abstract

We survey some important results concerning the finite--dimensional representations of the loop algebra of a simple complex Lie algebra, and their twisted loop subalgebras. In particular, we review the parametrization and description of the Weyl modules and of the irreducible finite--dimensional representations of such algebras, describe a block decomposition of the (non--semisimple) category of their finite--dimensional representations, and conclude with recent developments in the representation theory of multiloop algebras.