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In this paper, first we introduce the notion of quadratic Rota-Baxter Liealgebras of arbitrary weight, and show that there is a one-to-onecorrespondence between factorizable Lie bialgebras and quadratic Rota-BaxterLie algebras of nonzero weight. Then we introduce the notions of matched pairs,bialgebras and Manin triples of Rota-Baxter Lie algebras of arbitrary weight,and show that Rota-Baxter Lie bialgebras, Manin triples of Rota-Baxter Liealgebras and certain matched pairs of Rota-Baxter Lie algebras are equivalent.The coadjoint representations and quadratic Rota-Baxter Lie algebras playimportant roles in the whole study. Finally we generalize some results to theLie group context. In particular, we show that there is a one-to-onecorrespondence between factorizable Poisson Lie groups and quadraticRota-Baxter Lie groups.

Factorizable Lie bialgebras, quadratic Rota-Baxter Lie algebras and Rota-Baxter Lie bialgebras