Nondegenerate module categories

TL;DR

This paper extends the concept of nondegeneracy from braided finite tensor categories to braided module categories, establishing equivalences between nondegeneracy and factorizability, and exploring the Hopf algebra case with new examples.

Contribution

It introduces the notions of nondegeneracy and factorizability for braided module categories and proves their equivalence, along with a new monadicity result and analysis of the Hopf case.

Findings

Nondegeneracy and factorizability are equivalent for braided module categories.

Representation categories of quasitriangular comodule algebras are nondegenerate iff the algebra is factorizable.

Several explicit examples of nondegenerate and factorizable structures are provided.

Abstract

Due to the work of Shimizu (2019), various nondegeneracy conditions for braided finite tensor categories are equivalent. This theory is partially extended to braided module categories here. We introduce when a braided module category is "nondegenerate" and "factorizable", and establish that these properties are equivalent. The proof involves a new monadicity result for module categories. Lastly, we examine the Hopf case, using Kolb's (2020) notion of a quasitriangular comodule algebra to introduce "factorizable" comodule algebras. We then show that the representation category of a quasitriangular comodule algebra is nondegenerate in our sense precisely when the comodule algebra is factorizable. Several examples are provided.