Exact factorizations and extensions of finite tensor categories

TL;DR

This paper generalizes the concept of exact factorizations in finite tensor categories to nonsemisimple cases, establishing new relations with exact sequences and applying these to quasi-Hopf algebras and group scheme categories.

Contribution

It introduces the notion of exact factorizations for nonsemisimple finite tensor categories and links these to exact sequences, expanding the theoretical framework.

Findings

Extended exact factorizations to nonsemisimple categories

Connected factorizations with exact sequences of tensor categories

Provided examples illustrating the theoretical results

Abstract

We extend \cite{G} to the nonsemisimple case. We define and study exact factorizations $\B=\A\bullet \C$ of a finite tensor category $\B$ into a product of two tensor subcategories $\A,\C\subset \B$, and relate exact factorizations of finite tensor categories to exact sequences of finite tensor categories with respect to exact module categories \cite{EG}. We apply our results to study exact factorizations of quasi-Hopf algebras, and extensions of a finite group scheme theoretical tensor category \cite{G2} by another one. We also provide several examples to illustrate our results.