Cohomology and Support Varieties for Lie Superalgebras II

TL;DR

This paper advances the cohomological study of classical Lie superalgebras by computing support varieties for key modules, confirming conjectures, and clarifying differences between Type I and Type II superalgebras.

Contribution

It computes support varieties for Kac supermodules and simple modules in lgebra lgebras, confirming the conjecture relating support variety dimension to atypicality.

Findings

Support varieties for lgebra lgebras match conjectured dimensions

Confirmed the support variety dimension equals atypicality for lgebra lgebras

Identified differences between Type I and Type II Lie superalgebras

Abstract

In \cite{BKN} the authors initiated a study of the representation theory of classical Lie superalgebras via a cohomological approach. Detecting subalgebras were constructed and a theory of support varieties was developed. The dimension of a detecting subalgebra coincides with the defect of the Lie superalgebra and the dimension of the support variety for a simple supermodule was conjectured to equal the atypicality of the supermodule. In this paper the authors compute the support varieties for Kac supermodules for Type I Lie superalgebras and the simple supermodules for $\mathfrak{gl}(m|n)$. The latter result verifies our earlier conjecture for $\mathfrak{gl}(m|n)$. In our investigation we also delineate several of the major differences between Type I versus Type II classical Lie superalgebras. Finally, the connection between atypicality, defect and superdimension is made more precise by using the theory of support varieties and representations of Clifford superalgebras.