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

TL;DR

This paper provides a criterion for when G-equivariant objects in a fusion category are group-theoretical, and constructs examples of non group-theoretical semisimple Hopf algebras using this criterion, answering a longstanding open question.

Contribution

It introduces a new criterion for the group-theoretical nature of G-equivariant fusion categories and constructs explicit examples of non group-theoretical semisimple Hopf algebras.

Findings

Certain Z/2Z-equivariantizations are equivalent to non group-theoretical Hopf algebras.

Constructed Hopf algebras are described as extensions and are semisolvable.

Answer affirmatively to the existence of non group-theoretical semisimple Hopf algebras.

Abstract

Given an action of a finite group G on a fusion category C we give a criterion for the category of G-equivariant objects in 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 math/0203060. Namely, we show that certain Z/2Z-equivariantizations of fusion categories constructed by D. Tambara and S. Yamagami 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.