On cohomology and support varieties for Lie superalgebras

TL;DR

This paper investigates the structure of support varieties for Lie superalgebras, focusing on finite generation of cohomology rings, detection subalgebras, and explicit computations for the superalgebra S(n), advancing the understanding of their cohomological properties.

Contribution

It establishes finite generation of relative cohomology rings for S(n) and introduces detection subalgebras, with explicit cohomology and support variety computations.

Findings

The relative cohomology ring of S(n) relative to S(n)_0 is finitely generated.

Support varieties of all simple modules in the category are computed.

Detection subalgebras are formulated to understand cohomology.

Abstract

Support varieties for Lie superalgebras over the complex numbers were introduced in \cite{BKN1} using the relative cohomology. In this paper we discuss finite generation of the relative cohomology rings for Lie superalgebras, we formulate a definition for subalgebras which detect the cohomology, also discuss realizability of support varieties. In the last section as an application we compute the relative cohomology ring of the Lie superalgebra $\overline{S}(n)$ relative to the graded zero component $\overline{S}(n)_0$ and show that this ring is finitely generated. We also compute support varieties of all simple modules in the category of finite dimensional $\overline{S}(n)$-modules which are completely reducible over $\overline{S}(n)_0$.