A construction of Rota-Baxter operators on cocommutative Hopf algebras

Let [Formula: see text] and [Formula: see text] be two cocommutative Hopf algebras over a field [Formula: see text] such that [Formula: see text] is a left [Formula: see text]-module Hopf algebra. Denote by [Formula: see text] the smash product Hopf algebra of [Formula: see text] and [Formula: see text] Let [Formula: see text] be an [Formula: see text]-linear map from [Formula: see text] to [Formula: see text], and [Formula: see text] a Rota−Baxter operator on [Formula: see text]. We prove that the linear map [Formula: see text] from [Formula: see text] to [Formula: see text] defined as [Formula: see text] for all [Formula: see text] and [Formula: see text], is a Rota−Baxter operator on [Formula: see text] if and only if [Formula: see text] is a Rota−Baxter operator on [Formula: see text] satisfying two suitable equations.