Rigid and Separable Algebras in Fusion 2-Categories

TL;DR

This paper generalizes the concept of rigid algebras to monoidal 2-categories, exploring their properties, separability, and dimensions, with implications for fusion 2-categories and quantum algebra.

Contribution

It introduces the notion of rigid and separable algebras in monoidal 2-categories, providing criteria for adjoints, semisimplicity, and a dimension-based characterization of separability.

Findings

Rigid algebras include G-graded and G-crossed fusion categories.

Modules and bimodules over separable algebras are finite semisimple.

A connected rigid algebra is separable iff its dimension is non-zero.

Abstract

Rigid monoidal 1-categories are ubiquitous throughout quantum algebra and low-dimensional topology. We study a generalization of this notion, namely rigid algebras in an arbitrary monoidal 2-category. Examples of rigid algebras include $G$-graded fusion 1-categories, and $G$-crossed fusion 1-categories. We explore the properties of the 2-categories of modules and of bimodules over a rigid algebra, by giving a criterion for the existence of right and left adjoints. Then, we consider separable algebras, which are particularly well-behaved rigid algebras. Specifically, given a fusion 2-category, we prove that the 2-categories of modules and of bimodules over a separable algebra are finite semisimple. Finally, we define the dimension of a connected rigid algebra in a fusion 2-category, and prove that such an algebra is separable if and only if its dimension is non-zero.