Hopf algebra extensions of group algebras and Tambara-Yamagami categories

Abstract

We determine the structure of Hopf algebra extensions of a group algebra by the cyclic group of order 2. We study the corepresentation theory of such Hopf algebras, which provide a generalization, at the Hopf algebra level, of the so called Tambara-Yamagami fusion categories. As a byproduct, we show that every semisimple Hopf algebra of dimension $<36$ is necessarily group-theoretical; thus 36 is the smallest possible dimension where a non group-theoretical example occurs.

Main changes appear in the description in Theorem 1.1, missing case added in proof of Proposition 6.1 and reference added. To appear in Algebr. Represent. Theory