On support varieties for Lie superalgebras and finite supergroup schemes

TL;DR

This paper explores the structure of support varieties for Lie superalgebras and supergroup schemes, extending classical theories to new algebraic contexts and characteristic settings.

Contribution

It characterizes support varieties for finite supergroup schemes over algebraically closed fields of characteristic zero and generalizes foundational results to positive characteristic.

Findings

Support varieties have desirable properties in these settings.

Complete characterization of support varieties over algebraically closed fields of characteristic zero.

Results in positive characteristic mirror classical Lie algebra results.

Abstract

We study the spectrum of the cohomology rings of cocommutative Hopf superalgebras, restricted and non-restricted Lie superalgebras, and finite supergroup schemes. We also investigate support varieties in these settings and demonstrate that they have the desirable properties of such a theory. We completely characterize support varieties for finite supergroup schemes over algebraically closed fields of characteristic zero, while for non-restricted Lie superalgebras we obtain results in positive characteristic that are strikingly similar to results of Duflo and Serganova in characteristic zero. Our computations for restricted Lie superalgebras and infinitesimal supergroup schemes provide natural generalizations of foundational results of Friedlander and Parshall and of Bendel, Friedlander, and Suslin in the classical setting.