Rota-Baxter co-operators on commutative Hopf algebras

In this paper, we mainly research Rota-Baxter co-operators on commutative Hopf algebras. We get a new Hopf algebra and some Hopf co-braces via Rota-Baxter co-operators on commutative Hopf algebras. Finally, we generalize Rota-Baxter co-operators to relative Rota-Baxter co-operators of Hopf algebras, construct a Hopf cobrace by a relative Rota-Baxter co-operator of commutative Hopf algebra, and establish the relation of relative Rota-Baxter co-operators and bijective 1-cocycles.