Nondegenerate module categories
Chelsea Walton, Harshit Yadav
TL;DR
This work extends Shimizu’s equivalence of nondegeneracy, weak factorizability, factorizability, and trivial Müger center from braided finite tensor categories to braided module categories. It introduces precise notions of nondegeneracy and factorizability for braided module categories and proves their equivalence via a monadicity-based analysis of EM-categories and end-algebras. The authors develop a Hopf-theoretic framework, showing that for a finite-dimensional quasitriangular Hopf algebra $H$, a finite-dimensional quasitriangular left $H$-comodule algebra $B$ is factorizable if and only if the braided module category $B ext{-FdMod}$ is nondegenerate over $H ext{-FdMod}$, linking module-category nondegeneracy to factorizable comodule algebras. They further connect reflective centers to factorizable comodule algebras, providing concrete examples (including reflective algebras) and highlighting monadic and EM-structure as central tools. The results have broad implications for understanding nondegeneracy in braided module contexts and for constructing factorizable (and hence nondegenerate) examples in Hopf-algebraic settings, with potential impact on applications in quantum symmetries and categorical quantum field theory.
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.