Phase space of a Poisson algebra and the induced Pre-Poisson bialgebra
You Wang, Yunhe Sheng
TL;DR
The paper introduces the phase space concept for Poisson algebras and proves a Poisson algebra has a phase space if and only if it is sub-adjacent to a pre-Poisson algebra, with a 1-1 correspondence to Manin triples of pre-Poisson algebras. It then develops pre-Poisson bialgebras as the bialgebraic counterpart of these triples, including coboundary theories and a pre-Poisson Yang-Baxter equation, Z(r)=S(r)=0, that governs coboundary constructions. Quasi-triangular and factorizable pre-Poisson bialgebras are analyzed, revealing a relative Rota-Baxter operator of weight 1 arising from quasi-triangular data and a natural double that is factorizable. A central unification is established: factorizable pre-Poisson bialgebras correspond to quadratic Rota-Baxter pre-Poisson algebras, which in turn correspond to Rota-Baxter symplectic Poisson algebras, providing a consistent framework for phase spaces via RB theory. Consequently, the work yields phase spaces for Rota-Baxter symplectic Poisson algebras and a comprehensive bridge between Poisson geometry, pre-Poisson algebra theory, and Rota-Baxter bialgebras.
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.