Simple algebras and exact module categories

TL;DR

This paper proves a conjecture relating algebra objects' exactness to their structure as finite products of simple algebras within finite tensor categories, introducing a Jacobson radical analogue.

Contribution

It confirms a conjecture by Etingof and Ostrik and introduces a new Jacobson radical analogue for algebra objects in tensor categories.

Findings

Algebra objects are exact iff they are finite products of simple algebras.

Introduces a Jacobson radical analogue for algebra objects.

Applications to incompressible finite symmetric tensor categories.

Abstract

We verify a conjecture of Etingof and Ostrik, stating that an algebra object in a finite tensor category is exact if and only if it is a finite direct product of simple algebras. Towards that end, we introduce an analogue of the Jacobson radical of an algebra object, similar to the Jacobson radical of a finite-dimensional algebra. We give applications of our main results in the context of incompressible finite symmetric tensor categories.