On associated variety for Lie superalgebras

TL;DR

This paper introduces the associated variety for modules over Lie superalgebras, linking geometric data to module properties, and establishes its relation to the atypicality degree, especially for general linear superalgebras.

Contribution

It defines the associated variety for superalgebra modules and relates it to the module's atypicality, providing a geometric perspective on representation theory.

Findings

Associated variety is a subvariety of the cone of self-commuting odd elements.

For simple superalgebras, the associated variety's orbits are classified by rank.

In the case of gl(m|n), the associated variety equals the closure of the orbit of the same rank.

Abstract

We define the associated variety $ X_{M} $ of a module $ M $ over a finite-dimensional superalgebra $ {\mathfrak g} $, and show how to extract information about $ M $ from these geometric data. $ X_{M} $ is a subvariety of the cone $ X $ of self-commuting odd elements. For finite-dimensional $ M $, $ X_{M} $ is invariant under the action of the underlying Lie group $ G_{0} $. For simple superalgebra with invariant symmetric form, $ X $ has finitely many $ G_{0} $-orbits; we associate a number (rank) to each such orbit. One can also associate a number (degree of atypicality) to an irreducible finite-dimensional representation. We prove that if $ M $ is an irreducible $ {\mathfrak g} $-module of degree of atypicality $ k $, then $ X_{M} $ lies in the closure of all orbits on $ X $ of rank $ k $. If $ {\mathfrak g}={\mathfrak g}{\mathfrak l}(m|n) $ we prove that $ X_{M} $ coincides with this closure.