Hopf algebra extensions of group algebras and Tambara-Yamagami categories

TL;DR

This paper classifies certain Hopf algebra extensions related to group algebras and explores their corepresentation theory, revealing that all semisimple Hopf algebras below dimension 36 are group-theoretical, with 36 being the minimal dimension for exceptions.

Contribution

It provides a complete description of Hopf algebra extensions of group algebras by cyclic groups of order 2 and links them to Tambara-Yamagami categories, establishing dimension bounds for non group-theoretical Hopf algebras.

Findings

Classified Hopf algebra extensions of group algebras by cyclic groups of order 2.

Connected these extensions to Tambara-Yamagami fusion categories.

Proved all semisimple Hopf algebras of dimension less than 36 are group-theoretical.

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.