Constructing non-semisimple modular categories with local modules

TL;DR

This paper develops a framework for constructing non-semisimple modular categories using local modules of rigid Frobenius algebras, extending previous semisimple results and providing new classifications.

Contribution

It generalizes the theory of local modules to non-semisimple modular categories and classifies rigid Frobenius algebras in Drinfeld centers over arbitrary characteristic.

Findings

Categories of local modules are modular in the non-semisimple setting

Provides examples of non-semisimple modular categories via local modules

Classifies rigid Frobenius algebras in Drinfeld centers of group algebra modules

Abstract

We define the class of rigid Frobenius algebras in a (non-semisimple) modular category and prove that their categories of local modules are, again, modular. This generalizes previous work of A. Kirillov, Jr. and V. Ostrik [Adv. Math. 171 (2002), no. 2] in the semisimple setup. Examples of non-semisimple modular categories via local modules, as well as connections to the authors' prior work on relative monoidal centers, are provided. In particular, we classify rigid Frobenius algebras in Drinfeld centers of module categories over group algebras, thus generalizing the classification by A. Davydov [J. Algebra 323 (2010), no. 5] to arbitrary characteristic.