Finite-dimensional irreducible representations of twisted loop algebras of the second kind

TL;DR

This paper classifies finite-dimensional irreducible representations of twisted loop algebras of the second kind, which are infinite-dimensional Lie algebras derived from semisimple Lie algebras with automorphisms of order at most two, using a more elementary approach.

Contribution

It provides a more elementary classification of finite-dimensional irreducible representations for these algebras, expanding on prior general classifications.

Findings

Complete classification of finite-dimensional irreducible representations.

Elementary method simplifies understanding of these representations.

Connections to equivariant map algebras established.

Abstract

Twisted loop algebras of the second kind are infinite-dimensional Lie algebras that are constructed from a semisimple Lie algebra and an automorphism on it of order at most $2$. They are examples of equivariant map algebras. The finite-dimensional irreducible representations of an arbitrary equivariant map algebra have been classified by Neher--Savage--Senesi. In this paper, we classify the finite-dimensional irreducible representations of twisted loop algebras of the second kind in a more elementary way.