Coquasitriangular structures on Hopf algebras constructed via abelian extensions

TL;DR

This paper characterizes when coquasitriangular structures exist on certain cosemisimple Hopf algebras formed via abelian extensions of group algebras, with specific focus on cases where G is Z2.

Contribution

It provides necessary and sufficient conditions for the existence of coquasitriangular structures on these Hopf algebras and characterizes their form.

Findings

Coquasitriangular structures exist under specific algebraic conditions.

Characterizations are provided for structures on extensions involving finite groups.

Special case analysis for G = Z2 is included.

Abstract

The aim of this paper is to study coquasitriangular structures on a class of cosemisimple Hopf algebras of the form $\Bbbk^G {}^\tau \#_{\sigma} \Bbbk F$, constructed as abelian extensions of $\Bbbk F$ by $\Bbbk^G$ for a finite group $G$ and an arbitrary group $F$. We investigate when a coquasitriangular structure exists on $\Bbbk^G{}^\tau\#_{\sigma}\Bbbk F$ and provide characterizations of its coquasitriangular structures. As an application, we study the coquasitriangular structures for the case where $G = \mathbb{Z}_2$.