Irreducible modules for equivariant map superalgebras and their extensions

TL;DR

This paper classifies finite-dimensional irreducible modules for equivariant map superalgebras associated with periplectic Lie superalgebras, extending previous classifications to new algebraic structures and describing module extensions.

Contribution

It completes the classification of irreducible modules for equivariant map superalgebras when the Lie superalgebra is periplectic, a case not previously fully addressed.

Findings

Classified finite-dimensional irreducible modules for the specified superalgebras.

Described extensions between modules in terms of homomorphisms and extensions of finite-dimensional Lie superalgebras.

Extended the classification framework to include periplectic Lie superalgebras.

Abstract

Let $\Gamma$ be a group acting on a scheme $X$ and on a Lie superalgebra $\mathfrak{g}$, both defined over an algebraically closed field of characteristic zero $\Bbbk$. The corresponding equivariant map superalgebra $M(\mathfrak{g}, X)^\Gamma$ is the Lie superalgebra of equivariant regular maps from $X$ to $\mathfrak{g}$. In this paper we complete the classification of finite-dimensional irreducible $M(\mathfrak{g}, X)^\Gamma$-modules when $\mathfrak{g}$ is a finite-dimensional simple Lie superalgebra, $X$ is of finite type and $\Gamma$ is a finite abelian group acting freely on the rational points of $X$, by classifying these $M(\mathfrak{g},X)^\Gamma$-modules in the case where $\mathfrak{g}$ is a periplectic Lie superalgebra. We also describe extensions between irreducible modules in terms of homomorphisms and extensions between modules for certain finite-dimensional Lie superalgebras.