On the quasitriangular structures of abelian extensions of ℤ₂

Abstract

The aim of this paper is to study quasitriangular structures on a class of semisimple Hopf algebras ${\mathbb{k}}^G{\#}_{\sigma ,\tau }\mathbb{k}{\mathbb{Z}}_2$ constructed through abelian extensions of $\mathbb{k}{\mathbb{Z}}_2$ by ${\mathbb{k}}^G$ for an abelian group G. We prove that there are only two forms of them and we get a complete list of all universal $\mathcal{R}$-matrices of the generalized Kac-Paljutkin algebra $H_{2n^2}$ (see Section 2 for the definition).

Keywords:

• Abelian extension
• quasitriangular Hopf algebra

2020 Mathematics Subject Classification:

• 16T05 (primary)
• 16T25 (secondary)

Additional information

Funding

National Natural Science Foundation of China NSFC 11722016.