On the quasitriangular structures of abelian extensions of $Z_2$

TL;DR

This paper classifies all quasitriangular structures on certain semisimple Hopf algebras formed by abelian extensions of Z2, providing a complete list of universal R-matrices for these algebraic structures.

Contribution

It identifies only two possible forms of quasitriangular structures on these Hopf algebras and fully characterizes their universal R-matrices.

Findings

Only two forms of quasitriangular structures exist for these algebras.

Complete classification of universal R-matrices for specific Hopf algebras.

Provides a framework for understanding quasitriangular structures in abelian extensions.

Abstract

The aim of this paper is to study quasitriangular structures on a class of semisimple Hopf algebras constructed through abelian extensions of $Z_2$ for an abelian group $G$. We prove that there are only two forms of them. Using such description together with some other techniques, we get a complete list of all universal $\mathcal{R}$-matrices on some Hopf algebras.