Symmetry phenomenon on quasitriangular Hopf algebras and its applications

Let H be a Hopf algebra. The concept of a symmetric universal ℛ-matrix of H is introduced (see Definition 2.1). Subsequently, we prove that symmetry phenomenon exists on some well-known quasitriangular Hopf algebras, including some infinite families of small quantum groups. Through an examination of symmetric universal ℛ-matrices, we present a simple method to determine universal ℛ-matrices of some Hopf algebras. Furthermore, as part of our applications, we demonstrate that the universal ℛ-matrices of K(8n,σ,τ) (refer to Sec. 2 for their definition) are symmetric, where K(8n,σ,τ) are families of abelian extensions which include the well-known eight-dimensional Kac algebra. Subsequently, we demonstrate how symmetry can be utilized to significantly simplify the determination of universal ℛ-matrices for K(8n,σ,τ), ultimately yielding the complete set of universal ℛ-matrices for this Hopf algebra.