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.