Finite-dimensional representations of map superalgebras

TL;DR

This paper classifies all finite-dimensional irreducible modules over classical map superalgebras, provides character formulas, describes extension groups, and details the block decomposition of their module categories, with applications to affine Lie superalgebras.

Contribution

It offers a complete classification and structural description of finite-dimensional modules over map superalgebras, including formulas and block decompositions, extending to affine Lie superalgebras.

Findings

Complete classification of modules over classical map superalgebras

Formulas for supercharacters and extension groups

Block decomposition of module categories

Abstract

We obtain a complete classification of all finite-dimensional irreducible modules over classical map superalgebras, provide formulas for their (super)characters and a description of their extension groups. Furthermore, we describe the block decomposition of the category of finite-dimensional modules for such map superalgebras. As an application, we specialize our results to the case of loop superalgebras in order to obtain a classification of finite-dimensional irreducible modules and block decomposition of the category of finite-dimensional modules over affine Lie superalgebras.