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

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