A block decomposition of finite-dimensional representations of twisted loop algebras

TL;DR

This paper decomposes the category of finite-dimensional representations of twisted loop algebras into blocks using twisted spectral characters, extending the spectral character approach from untwisted to twisted cases.

Contribution

It introduces twisted spectral characters to classify blocks of finite-dimensional representations of twisted loop algebras, generalizing previous untwisted spectral character methods.

Findings

Decomposition of the representation category into indecomposable blocks.

Introduction of twisted spectral characters for classification.

Extension of spectral character techniques to twisted loop algebras.

Abstract

In this paper we consider the category of F^\sigma of finite-dimensional representations of a twisted loop algebra corresponding to a finite-dimensional Lie algebra with non-trivial diagram automorphism. Although F^\sigma is not semisimple, it can be written as a sum of indecomposable subcategories (the blocks of the category). To describe these summands, we introduce the twisted spectral characters for the twisted loop algebra. These are certain equivalence classes of the spectral characters defined by Chari and Moura for an untwisted loop algebra, which were used to provide a description of the blocks of finite--dimensional representations of the untwisted loop algebra. Here we adapt this decomposition to parametrize and describe the blocks of F^\sigma, via the twisted spectral characters.