Infinitesimal $\mathcal{R}$-matrices for some families of Hopf algebras

TL;DR

This paper classifies infinitesimal R-matrices for specific families of well-known Hopf algebras, providing a deeper understanding of their quasitriangular structures and infinitesimal deformations.

Contribution

It offers a classification of infinitesimal R-matrices for several important Hopf algebra families, expanding the knowledge of their quasitriangular properties.

Findings

Classified infinitesimal R-matrices for generalized Kac-Paljutkin Hopf algebras

Identified infinitesimal R-matrices for Radford Hopf algebras

Analyzed infinitesimal R-matrices for E(n) Hopf algebras

Abstract

Given a bialgebra $H$ such that the associated trivial topological bialgebra $H[[\hbar]]$ admits a quasitriangular structure $\tilde{\mathcal{R}}=\mathcal{R}(1\otimes 1+\hbar\chi+\mathcal{O}(\hbar^2))$, one gets a distinguished element $\chi \in H \otimes H$ which is an infinitesimal $\mathcal{R}$-matrix, according to the definition given in [1]. In this paper we classify infinitesimal $\mathcal{R}$-matrices for some families of well-known Hopf algebras, among which are the generalized Kac-Paljutkin Hopf algebras $H_{2n^2}$, the Radford Hopf algebras $H_{(r,n,q)}$, and the Hopf algebras $E(n)$.