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

Abstract

Given a bialgebra $H$ such that the associated trivial topological bialgebra $H[[\hslash ]]$ admits a quasitriangular structure $\tilde{\mathcal{R}}=\mathcal{R}(1\otimes 1+\hslash \chi +\mathcal{O}({\hslash }^2))$, 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 $E(n)$.

27 pages. Added Section 3. Minor changes throughout the paper. Additional references