Irreducible finite-dimensional representations of equivariant map algebras

TL;DR

This paper classifies all irreducible finite-dimensional representations of equivariant map algebras, showing they are tensor products of evaluation and one-dimensional representations, with applications to various algebra types.

Contribution

It provides a comprehensive classification of irreducible finite-dimensional representations of equivariant map algebras, including new results for generalized Onsager algebras.

Findings

All such representations are tensor products of evaluation and one-dimensional representations.

Conditions are established under which all representations are evaluation representations.

The classification recovers known results and introduces new classifications for certain algebra types.

Abstract

Suppose a finite group acts on a scheme X and a finite-dimensional Lie algebra g. The corresponding equivariant map algebra is the Lie algebra M of equivariant regular maps from X to g. We classify the irreducible finite-dimensional representations of these algebras. In particular, we show that all such representations are tensor products of evaluation representations and one-dimensional representations, and we establish conditions ensuring that they are all evaluation representations. For example, this is always the case if M is perfect. Our results can be applied to multiloop algebras, current algebras, the Onsager algebra, and the tetrahedron algebra. Doing so, we easily recover the known classifications of irreducible finite-dimensional representations of these algebras. Moreover, we obtain previously unknown classifications of irreducible finite-dimensional representations of other types of equivariant map algebras, such as the generalized Onsager algebra.