Phase space of a Poisson algebra and the induced Pre-Poisson bialgebra

Abstract: In this paper, we first introduce the notion of a phase space of a Poisson algebra, and show that a Poisson algebra has a phase space if and only if it is sub-adjacent to a pre-Poisson algebra. Moreover, we introduce the notion of Manin triples of pre-Poisson algebras and show that there is a one-to-one correspondence between Manin triples of pre-Poisson algebras and phase spaces of Poisson algebras. Then we introduce the notion of pre-Poisson bialgebras, which is equivalent to Manin triples of pre-Poisson algebras. We study coboundary pre-Poisson bialgebras, which leads to an analogue of the classical Yang-Baxter equation. Furthermore, we introduce the notions of quasi-triangular and factorizable pre-Poisson bialgebras as special cases. A quasi-triangular pre-Poisson bialgebra gives rise to a relative Rota-Baxter operator of weight $1$. The double of a pre-Poisson bialgebra enjoys a natural factorizable pre-Poisson bialgebra structure. Finally, we introduce the notion of quadratic Rota-Baxter pre-Poisson algebras and show that there is a one-to-one correspondence between quadratic Rota-Baxter pre-Poisson algebras and factorizable pre-Poisson bialgebras. Based on this construction, we give a phase space for a Rota-Baxter symplectic Poisson algebra.