Hopf algebra deformations of binary polyhedral groups

TL;DR

This paper classifies certain semisimple Hopf algebras as deformations of binary polyhedral groups, providing new insights into their structure and representation categories, with implications for their classification.

Contribution

It demonstrates that semisimple Hopf algebras with a specific 2-dimensional comodule are deformations of binary polyhedral groups and extends existing results on cosemisimple Hopf algebras.

Findings

Semisimple Hopf algebras with a 2-dimensional comodule are abelian extensions with Z_2 quotient.

Such Hopf algebras can be viewed as deformations of binary polyhedral groups.

Strengthens results on cosemisimple Hopf algebras with 2-dimensional irreducible comodules.

Abstract

We show that semisimple Hopf algebras having a self-dual faithful irreducible comodule of dimension 2 are always obtained as abelian extensions with quotient Z_2. We prove that nontrivial Hopf algebras arising in this way can be regarded as deformations of binary polyhedral groups and describe its category of representations. We also prove a strengthening of a result of Nichols and Richmond on cosemisimple Hopf algebras with a 2-dimensional irreducible comodule in the finite dimensional context. Finally, we give some applications to the classification of certain classes of semisimple Hopf algebras.