Quasitriangular and factorizable Poisson bialgebras

We introduce the notions of quasitriangular and factorizable Poisson bialgebras.A factorizable Poisson bialgebra induces a factorization of the underlying Poisson algebra.We prove that the Drinfeld classical double of a Poisson bialgebra naturally admits a factorizable Poisson bialgebra structure.Furthermore, we introduce the notion of quadratic Rota-Baxter Poisson algebras and show that a quadratic Rota-Baxter Poisson algebra of zero weight induces a triangular Poisson bialgebra.Moreover, we establish a one-to-one correspondence between factorizable Poisson bialgebras and quadratic Rota-Baxter Poisson algebras of nonzero weights.Finally, we establish the quasitriangular and factorizable theories for differential antisymmetric infinitesimal (ASI) bialgebras, and construct quasitriangular and factorizable Poisson bialgebras from quasitriangular and factorizable (commutative and cocommutative) differential ASI bialgebras respectively.1. Introduction 453 2. Quasitriangular Poisson bialgebras 457 3. Factorizable Poisson bialgebras 466 4. Quadratic Rota-Baxter Poisson algebras 468 5. Quasitriangular Poisson bialgebras via quasitriangular differential ASI bialgebras 473 References