Cohomology and support varieties for Lie superalgebras

TL;DR

This paper proves the finite generation of the cohomology ring for restricted Lie superalgebras over fields of characteristic p>2, enabling the development of support variety theory for their modules.

Contribution

It establishes the finite generation of cohomology rings for restricted Lie superalgebras, facilitating support variety theory for their modules.

Findings

Cohomology ring $ ext{H}^ullet(u(g), k)$ is finitely generated.

Support varieties satisfy key properties analogous to classical theory.

Framework enables new geometric methods in superalgebra representation theory.

Abstract

Let $mathfrak{g}$ be a restricted Lie superalgebra over an algebraically closed field $k$ of characteristic $p>2$. Let $\mathfrak{u}(\mathfrak{g})$ denote the restricted enveloping algebra of $\mathfrak{g}$. In this paper we prove that the cohomology ring $\HH^\bullet(\fu(\fg), k)$ is finitely generated. This allows one to define support varieties for finite dimensional $\fu(\fg)$-supermodules. We also show that support varieties for finite dimensional $\fu(\fg)$ supermodules satisfy the desirable properties of support variety theory.