Equivariant map queer Lie superalgebras

TL;DR

This paper classifies all irreducible finite-dimensional representations of equivariant map queer Lie superalgebras, especially focusing on cases where the acting group is abelian and acts freely, including the twisted loop queer superalgebra.

Contribution

It provides a complete classification of irreducible finite-dimensional representations for equivariant map queer Lie superalgebras with abelian, freely acting groups, extending to the case of twisted loop superalgebras.

Findings

Representations are parameterized by $ ext{Gamma}$-equivariant finitely supported maps.

Classification includes the case of the torus, leading to twisted loop queer superalgebras.

Results generalize known classifications for queer Lie superalgebras.

Abstract

An equivariant map queer Lie superalgebra is the Lie superalgebra of regular maps from an algebraic variety (or scheme) $X$ to a queer Lie superalgebra $\mathfrak{q}$ that are equivariant with respect to the action of a finite group $\Gamma$ acting on $X$ and $\mathfrak{q}$. In this paper, we classify all irreducible finite-dimensional representations of the equivariant map queer Lie superalgebras under the assumption that $\Gamma$ is abelian and acts freely on $X$. We show that such representations are parameterized by a certain set of $\Gamma$-equivariant finitely supported maps from $X$ to the set of isomorphism classes of irreducible finite-dimensional representations of $\mathfrak{q}$. In the special case where $X$ is the torus, we obtain a classification of the irreducible finite-dimensional representations of the twisted loop queer superalgebra.