Equivariant cohomology and tensor categories

TL;DR

This paper introduces supercategories as a new framework for supermathematics, generalizes key results in equivariant cohomology, and extends the Duflo isomorphism to Lie superalgebras.

Contribution

It develops supercategories as an alternative to supermathematics and generalizes important theorems in equivariant cohomology and Lie superalgebra theory.

Findings

Supercategories effectively encompass supermathematical constructions.

Generalized Chern-Weil theorem for arbitrary rigid Lie algebra objects.

Proved the Duflo isomorphism for quadratic Lie superalgebras.

Abstract

We propose the notion of a supercategory as an alternative approach to supermathematics. We show that this setting is rich to carry out many of the basic constructions of supermathematics. We also prove generalizations of a number of results in equivariant cohomology, including the Chern-Weil theorem for an arbitrary rigid Lie algebra object. For a quadratic Lie algebra object we obtain a proof of the Duflo isomorphism along the lines of Alekseev-Meinrenken, thereby generalizing their result to Lie superalgebras.