Hopf Algebra Extensions of Group Algebras and Tambara-Yamagami Categories

Abstract

Abstract. We determine the structure of Hopf algebras that admit an extension 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.

Clifford theory for tensor categories

“…The projection in (5.2) is called cocentral if π(h 1 ) ⊗ h 2 = π(h 2 ) ⊗ h 1 . This is equivalent to the weak coaction ρ being trivial (see [17,Lemma 3.3]).…”

“…Another useful construction of a tensor category starting from a G-action over a tensor category C is the G-equivariantization of C, denoted by C G . This construction has been used, for example, in [3,14,17,18,27].…”

Topological automorphism groups of compact quantum groups

“…With this in place, we have Proof This is simply a dual version of [25,Proposition 3.5], where the case of a cocentral extension of a group algebra is treated in the context of semisimple Hopf algebras and categories of modules rather than comodules are considered. The arguments can be repeated virtually verbatim.…”

On Semisimple Hopf Algebras of Dimension 2 m , II

“…An important problem in the theory of semisimple Hopf algebras is to construct examples satisfying certain properties and in the second step classify all Hopf algebras with these properties. Several such classification results were obtained in recent years, in particular in the series of papers by Masuoka (see, for example, [Ma1], [Ma2], [Ma3], [Ma4]) and by Natale (see, for example [Na1], [Na2], [Na3], [Na4], [Na5], [Na6]) as well as in a recent preprint [Kr] by Krop. In this paper we continue the project of classifying semisimple Hopf algebras of dimension 2 m over an algebraically closed field of characteristic 0 started in the papers [K1] and [K2]. In [K1] we have shown that there are exactly 16 nontrivial (that is, noncommutative and noncocommutative) semisimple Hopf algebras of dimension 16.…”