Non-group-theoretical semisimple Hopf algebras from group actions on fusion categories

Abstract

Given an action of a finite group G on a fusion category \({\mathcal{C}}\) we give a criterion for the category of G-equivariant objects in \({\mathcal{C}}\) to be group-theoretical, i.e., to be categorically Morita equivalent to a category of group-graded vector spaces. We use this criterion to answer affirmatively the question about existence of non-group-theoretical semisimple Hopf algebras asked by P. Etingof, V. Ostrik, and the author in [7]. Namely, we show that certain \({\mathbb{Z}}\)/2\({\mathbb{Z}}\)-equivariantizations of fusion categories constructed by D. Tambara and S. Yamagami [26] are equivalent to representation categories of non-group-theoretical semisimple Hopf algebras. We describe these Hopf algebras as extensions and show that they are upper and lower semisolvable.