Coquasitriangular structures on Hopf algebras constructed via abelian extensions

TL;DR

The paper investigates coquasitriangular structures on cosemisimple Hopf algebras arising as abelian extensions $\Bbbk^G{}^\tau\#_{\sigma}\Bbbk F$ with finite $G$ and arbitrary $F$, framing the problem via braided comodules. It derives a necessary condition expressed in terms of orbit products $O_fO_{f'}=O_{f'}O_f$ and provides a complete, explicit characterization of possible coquasitriangular structures through axioms $(CQT0)$–$(CQT3)$, including their implications for the comodule category and Grothendieck ring. The authors apply the general theory to the case $G=\mathbb{Z}_2$, obtaining a detailed description of simple comodules, the Grothendieck ring, and the allowed forms of the coquasitriangular structure when $F$ is abelian (with two canonical cases). These results clarify when such Hopf algebras admit a braiding and supply concrete classifications in the Z2-case, offering both nonexistence criteria and explicit R-matrix data in significant instances.

Abstract

The aim of this paper is to study coquasitriangular structures on a class of cosemisimple Hopf algebras of the form $\Bbbk^G {}^τ\#_σ \Bbbk F$, 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 $\Bbbk^G{}^τ\#_σ\Bbbk F$ and provide characterizations of its coquasitriangular structures. As an application, we study the coquasitriangular structures for the case where $G = \mathbb{Z}_2$.

Theorems & Definitions (64)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Remark 2.2
• Definition 2.3
• Remark 2.4
• Lemma 2.5
• Definition 2.6
• Lemma 2.7
• Lemma 2.8