A remark on almost cocommutative Hopf algebras (Q1842738)

Let \(H\) be a Hopf algebra over a commutative ground ring \(R\) with unit element. Assume that there exist elements \(g,f\in H\) such that \(\Delta(g)=\Delta(f)^{-1}\) in \(H\otimes H\) and \(s_{12}(\Delta(h))=\Delta(ghf)\) for all \(h\in H\), where \(s_{12}\) swaps the entries. Note that the algebra \(H\) with this property is quasitriangular. It is shown that \(H\) is cocommutative.