Quasitriangular Hopf algebras whose group-like elements form an abelian group

In this paper we prove some properties of the set of group-like elements of $A$, $G(A)$, for a pointed minimal quasitriangular Hopf algebra $A$ over a field $k$ of characteristic 0, and for a pointed quasitriangular Hopf algebra which is indecomposable as a coalgebra. We first show that over a field of characteristic 0, for any pointed minimal quasitriangular Hopf algebra $A$, $G(A)$ is abelian. We show further that if $A$ is a quasitriangular Hopf algebra which is indecomposable as a coalgebra, then $G(A)$ is contained in $A_R$, the minimal quasitriangular Hopf algebra contained in $A$. As a result, one gets that over a field of characteristic 0, a pointed indecomposable quasitriangular Hopf algebra has a finite abelian group of group-like elements.