Quasi-triangular and factorizable Poisson bialgebras

TL;DR

This paper introduces and explores the concepts of quasi-triangular and factorizable Poisson bialgebras, establishing their structures, relationships with Rota-Baxter algebras, and connections to differential ASI bialgebras.

Contribution

It defines new classes of Poisson bialgebras, proves their properties, and establishes correspondences with Rota-Baxter Poisson algebras, extending the theory to differential ASI bialgebras.

Findings

Factorizable Poisson bialgebras induce a Poisson algebra factorization.

A quadratic Rota-Baxter Poisson algebra of zero weight yields a triangular Poisson bialgebra.

There is a one-to-one correspondence between factorizable Poisson bialgebras and quadratic Rota-Baxter Poisson algebras of nonzero weights.

Abstract

In this paper, we introduce the notions of quasi-triangular 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 quasi-triangular and factorizable theories for differential antisymmetric infinitesimal (ASI) bialgebras, and construct quasi-triangular and factorizable Poisson bialgebras from quasi-triangular and factorizable (commutative and cocommutative) differential ASI bialgebras respectively.