Finite-dimensional representation theory of loop algebras: a survey

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.

25 pages, updated bibliography. The author thanks N. Manning and M. Bennett for their proofread and corrections of the previous version, which are incorporated here