Equivariant map superalgebras

TL;DR

This paper classifies all irreducible finite dimensional modules for equivariant map superalgebras, showing they are tensor products of evaluation modules or quotients of generalized Kac modules, depending on the structure of the superalgebra.

Contribution

It provides the first classification of irreducible finite dimensional modules for twisted loop superalgebras, extending understanding of superalgebra representations.

Findings

Modules are tensor products of evaluation modules parameterized by equivariant maps.

When the even part of the superalgebra is semisimple, modules are all evaluation modules.

For non-semisimple even parts, modules are quotients of generalized Kac modules.

Abstract

Suppose a group $\Gamma$ acts on a scheme $X$ and a Lie superalgebra $\mathfrak{g}$. The corresponding equivariant map superalgebra is the Lie superalgebra of equivariant regular maps from $X$ to $\mathfrak{g}$. We classify the irreducible finite dimensional modules for these superalgebras under the assumptions that the coordinate ring of $X$ is finitely generated, $\Gamma$ is finite abelian and acts freely on the rational points of $X$, and $\mathfrak{g}$ is a basic classical Lie superalgebra (or $\mathfrak{sl}(n,n)$, $n > 0$, if $\Gamma$ is trivial). We show that they are all (tensor products of) generalized evaluation modules and are parameterized by a certain set of equivariant finitely supported maps defined on $X$. Furthermore, in the case that the even part of $\mathfrak{g}$ is semisimple, we show that all such modules are in fact (tensor products of) evaluation modules. On the other hand, if the even part of $\mathfrak{g}$ is not semisimple (more generally, if $\mathfrak{g}$ is of type I), we introduce a natural generalization of Kac modules and show that all irreducible finite dimensional modules are quotients of these. As a special case, our results give the first classification of the irreducible finite dimensional modules for twisted loop superalgebras.