Quasitriangular structures on abelian extensions of $\mathbb{Z}_{2}$

TL;DR

This paper classifies all quasitriangular structures on a specific class of semisimple Hopf algebras formed by abelian extensions of inite cyclic groups, introducing symmetry concepts and solution methods.

Contribution

It provides a complete classification of quasitriangular structures on these Hopf algebras, including conditions for special solutions and new methods for solving related equations.

Findings

All general solutions for quasitriangular structures are characterized.

Necessary and sufficient conditions for the existence of special solutions are established.

A division-like operation simplifies the classification process.

Abstract

The aim of this paper is to give all quasitriangular structures on a class of semisimple Hopf algebras constructed through abelian extensions of $\Bbbk\mathbb{Z}_{2}$ by $\Bbbk^G$ for an abelian group $G$. We first introduce the concept of symmetry of quasitriangular structures of Hopf algebras and obtain some related propositions which can be used to simplify our calculations of quasitriangular structures. Secondly, we find that quasitriangular structures of these semisimple Hopf algebras can do division-like operations. Using such operations we transform the problem of solving the quasitriangular structures into solving general solutions and giving a special solution. Then we give all general solutions and get a necessary and sufficient condition for the existence of a special solution.