Exact factorizations and extensions of finite tensor categories

Abstract

We extend [13] to the nonsemisimple case. We define and study exact factorizations B=A•C of a finite tensor category B into a product of two tensor subcategories A,C, and relate exact factorizations of finite tensor categories to exact sequences of finite tensor categories with respect to exact module categories [8]. We apply our results to study exact factorizations of quasi-Hopf algebras, and extensions of a finite group scheme theoretical tensor category [14] by another one. We also provide several examples to illustrate our results.