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. We characterize the Rota–Baxter operators on the L-R smash product A♮H and give necessary and sufficient conditions for B−−(a♮h)=B(h1)▹R(a)◃B(h2)♮B(h3) to be a Rota–Baxter operator of A♮H for a∈A and h∈H. Then we 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.