Constructions of Rota-Baxter operators by L-R smash products

Let A and H be two cocommutative Hopf algebras such that A is an H-bimodule Hopf algebra. Suppose that R:A→ A is a linear map and B is a Rota-Baxter operator of H. In this paper we will characterize the Rota-Baxter operators on the L-R smash product A H and give the necessary and sufficient conditions to make B̄ a Rota-Baxter operator of A H. Then we will consider the dual case, and construct a Rota-Baxter co-operator on the L-R smash coproduct C H, where C and H are commutative Hopf algebras and C is an H-bicomodule Hopf algebra.