Bialgebras of Rota—Baxter (Hom-)Lie algebras of any weight

The aim of this paper is to consider the Rota—Baxterization of (Hom-) Lie bialgebra and related structures. For this, we set up the representation theory of Rota—Baxter (Hom-)Lie algebras of any weight, and then present the notion of admissible Rota—Baxter (Hom-)Lie algebras based on the dual representation. For admissible Rota—Baxter (Hom-)Lie algebras, we give the constructions of matched pair and Manin triple, and then introduce the notion of Rota—Baxter (Hom-)Lie bialgebra and also investigate the relationship among the solution of (Hom-)classical Yang—Baxter equation, $\mathcal{O}$-operator and Rota—Baxter (Hom-)Lie bialgebra. At last, we consider Rota—Baxter pre-Lie (Hom-)algebra of any weight, discuss the connections with Rota—Baxter (Hom-) Lie algebra and $\mathcal{O}$-operator.