Factorizable Lie Bialgebras, Quadratic Rota–Baxter Lie Algebras and Rota–Baxter Lie Bialgebras

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