Subalgebras of Étale Algebras in Braided Fusion Categories

Abstract

In (Davydov et al. Selecta Mathematica (N.S.) 19, 237–269 2013, Rem. 3.4) the authors asked the question if any étale subalgebra of an étale algebra in a braided fusion category is also étale. We give a positive answer to this question if the braided fusion category \(\mathcal {C}\) 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 (Davydov et al. J. für die reine und angewandte Mathematik. 677, 135–177 2013, Theorem 4.10). This new description enables us to describe the two binary operations on the lattice of fusion subcategories.