The Symmetric Tensor Product on the Drinfeld Centre of a Symmetric Fusion Category

TL;DR

This paper introduces a new symmetric tensor product on the Drinfeld centre of a symmetric fusion category, exploring its structure via Tannaka duality and relating it to fiberwise tensor products of equivariant vector bundles, including super-group cases.

Contribution

It defines a symmetric tensor product on the Drinfeld centre and characterizes it through Tannaka duality, extending to super-group structures and their associated vector bundle categories.

Findings

Symmetric tensor product corresponds to fiberwise tensor product in the non-super case.

Introduces a super-version of the fiberwise tensor product for super-groups.

Establishes the equivalence between the symmetric tensor product and the super-version in the super case.

Abstract

We define a symmetric tensor product on the Drinfeld centre of a symmetric fusion category, in addition to its usual tensor product. We examine what this tensor product looks like under Tannaka duality, identifying the symmetric fusion category with the representation category of a finite (super)-group. Under this identification, the Drinfeld centre is the category of equivariant vector bundles over the finite group (underlying the super-group, in the super case). In the non-super case, we show that the symmetric tensor product corresponds to the fibrewise tensor product of these vector bundles. In the super case, we define for each super-group structure on the finite group a super-version of the fibrewise tensor product. We show that the symmetric tensor product on the Drinfeld centre of the representation category of the resulting finite super-groups corresponds to this super-version of the fibrewise tensor product on the category of equivariant vector bundles over the finite group.