Rota-Baxter operators on cocommutative Hopf algebras

We generalize the notion of a Rota-Baxter operator on groups and weight 1 Lie algebras define study cocommutative Hopf algebra $H$. If $H=F[G]$ is group $G$ or $H=U(\mathfrak{g})$ universal enveloping $\mathfrak{g}$, then we prove that operators $H$ are in one to correspondence with corresponding algebras.