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 a matched pair and a 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, the $\mathcal{O}$-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 the $\mathcal{O}$-operator.