Rigid and Separable Algebras in Fusion 2Categories
Abstract
Rigid monoidal 1categories are ubiquitous throughout quantum algebra and lowdimensional topology. We study a generalization of this notion, namely rigid algebras in an arbitrary monoidal 2category. Examples of rigid algebras include $G$graded fusion 1categories, and $G$crossed fusion 1categories. We explore the properties of the 2categories 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 wellbehaved rigid algebras. Specifically, given a fusion 2category, we prove that the 2categories 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 2category, and prove that such an algebra is separable if and only if its dimension is nonzero.
 Publication:

arXiv eprints
 Pub Date:
 May 2022
 DOI:
 10.48550/arXiv.2205.06453
 arXiv:
 arXiv:2205.06453
 Bibcode:
 2022arXiv220506453D
 Keywords:

 Mathematics  Quantum Algebra;
 Mathematics  Category Theory;
 18M20;
 18N25 (Primary);
 18M30;
 18N10 (Secondary)
 EPrint:
 Minor corrections