Subalgebras of etale algebras and fusion subcategories

TL;DR

This paper proves that in pseudo-unitary, non-degenerate braided fusion categories, any étale subalgebra of an étale algebra is also étale, and provides a new lattice description of fusion subcategories and their operations.

Contribution

It answers affirmatively the question about étale subalgebras in certain fusion categories and introduces a new lattice framework for fusion subcategories.

Findings

Étale subalgebras are also étale in pseudo-unitary, non-degenerate categories.

New lattice description of fusion subcategories and their binary operations.

Enhanced understanding of the structure of fusion subcategories.

Abstract

In \cite[Rem. 3.4]{DNO} the authors asked the question if any \'etale subalgebra of an \'etale algebra in a braided fusion category is also \'etale. We give a positive answer to this question if the braided fusion category $\cc$ is pseudo-unitary and non-degenerate. In the case of a pseudo-unitary fusion category we also give a new description of the lattice correspondence from \cite[Theorem 4.10]{DMNO}. This new description enables us to describe the two binary operations on the lattice of fusion subcategories.