Rigid and Separable Algebras in Fusion 2-Categories
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Publication:6399049
DOI10.1016/J.AIM.2023.108967arXiv2205.06453MaRDI QIDQ6399049
Publication date: 13 May 2022
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 -graded fusion 1-categories, and -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.
2-categories, bicategories, double categories (18N10) Fusion categories, modular tensor categories, modular functors (18M20) Categorification (18N25) String diagrams and graphical calculi (18M30)
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