Detecting Cohomology for Lie Superalgebras

TL;DR

This paper develops a method using invariant theory to detect cohomology in Lie superalgebras, enabling the analysis of their structure through smaller subalgebras and supporting the realization of support varieties.

Contribution

It introduces a cohomological detection framework for Type I classical Lie superalgebras using invariant theory, connecting cohomology with smaller subalgebras and support varieties.

Findings

Cohomology can be detected on smaller subalgebras.

Support varieties can be realized via rank varieties for detecting subalgebras.

Answers to previously posed questions about support varieties are provided.

Abstract

In this paper we use invariant theory to develop the notion of cohomological detection for Type I classical Lie superalgebras. In particular we show that the cohomology with coefficients in an arbitrary module can be detected on smaller subalgebras. These results are used later to affirmatively answer questions, which were originally posed in \cite{BKN1} and \cite{BaKN}, about realizing support varieties for Lie superalgebras via rank varieties constructed for the smaller detecting subalgebras.