Hopf Algebra Extensions of Group Algebras and Tambara-Yamagami Categories

Abstract

We determine the structure of Hopf algebras that admit an extension of agroup algebra by the cyclic group of order 2.We study the corepresentation theory ofsuch Hopf algebras, which provide a generalization, at the Hopf algebra level, of theso called Tambara-Yamagami fusion categories. As a byproduct, we show that everysemisimple Hopf algebra of dimension < 36 is necessarily group-theoretical; thus 36is the smallest possible dimension where a non group-theoretical example occurs.