On the Grothendieck ring of a quasireductive Lie superalgebra

TL;DR

This paper investigates the structure of the Grothendieck ring of modules over a quasireductive Lie superalgebra, focusing on restrictions to maximal quasitoral subalgebras and providing explicit descriptions for the case of rak{q}_n.

Contribution

It generalizes character theory to quasireductive Lie superalgebras and explicitly describes the rak{h}-supercharacter ring for rak{q}_n.

Findings

Properties of module restrictions to quasitoral subalgebras are established.

Explicit realization of the rak{h}-supercharacter ring for rak{q}_n.

Special properties of restrictions in the case of rak{q}_n are proved.

Abstract

Given a Lie superalgebra $\mathfrak{g}$ and a maximal quasitoral subalgebra $\mathfrak{h}$, we consider properties of restrictions of $\mathfrak{g}$-modules to $\mathfrak{h}$. This is a natural generalization of the study of characters in the case when $\mathfrak{h}$ is an even maximal torus. We study the case of $\mathfrak{g}=\mathfrak{q}_n$ with $\mathfrak{h}$ a Cartan subalgebra, and prove several special properties of the restriction in this case, including an explicit realization of the $\mathfrak{h}$-supercharacter ring.