A class of (infinite-dimensional) cosemisimple Hopf algebras constructed via abelian extensions

TL;DR

This paper investigates a class of infinite-dimensional cosemisimple Hopf algebras formed via abelian extensions, exploring their structure, simple comodules, and potential quantum group properties.

Contribution

It introduces a new family of cosemisimple Hopf algebras from abelian extensions of infinite groups, analyzing their comodules and Grothendieck rings.

Findings

Hopf algebra $Bbbk^G{}^ au ext{ extasciicircum}\#_{\sigma}Bbbk F$ is cosemisimple

Characterization of simple comodules for these Hopf algebras

Construction of new examples and analysis of quantum group structures

Abstract

In this paper, we aim to study abelian extensions for some infinite group. We show that the Hopf algebra $\Bbbk^G{}^\tau\#_{\sigma}\Bbbk F$ constructed through abelian extensions of $\Bbbk F$ by $\Bbbk^G$ for some (infinite) group $F$ and finite group $G$ is cosemisimple, and discuss when it admits a compact quantum group structure if $\Bbbk$ is the field of complex numbers $\mathbb{C}.$ We also find all the simple $\Bbbk^G{}^\tau\#_{\sigma}\Bbbk F$-comodules and attempt to determine the Grothendieck ring of the category of finite-dimensional right $\Bbbk^G{}^\tau\#_{\sigma}\Bbbk F$-comodules. Moreover, some new properties are given and some new examples are constructed.