Noncommutative pre-Poisson bialgebras and relative Rota-Baxter operators

TL;DR

This paper develops the theory of noncommutative pre-Poisson bialgebras, linking them to solutions of the noncommutative pre-Poisson Yang-Baxter equation and introducing related Rota-Baxter operators.

Contribution

It introduces the concept of noncommutative pre-Poisson bialgebras, explores their connections to Yang-Baxter equations, and establishes a correspondence with Rota-Baxter algebras.

Findings

Equivalence among matched pairs, Manin triples, and phase spaces for noncommutative pre-Poisson algebras.

Solutions to the NPP-YBE generate noncommutative pre-Poisson bialgebras.

Quadratic Rota-Baxter noncommutative pre-Poisson algebras correspond to factorizable bialgebras.

Abstract

In this paper, we develop the bialgebra theory for coherent noncommutative pre-Poisson algebras and establish equivalences among matched pairs, Manin triples, the phase space of noncommutative Poisson algebras and noncommutative pre-Poisson bialgebras. The investigation of coboundary noncommutative pre-Poisson bialgebras naturally leads to the noncommutative pre-Poisson Yang-Baxter equation (NPP-YBE). We prove that a symmetric solution of the NPP-YBE gives rise to a (coboundary) noncommutative pre-Poisson bialgebra. Moreover, we demonstrate how solutions without the symmetry condition can also generate such bialgebras. This motivates the introduction of quasi-triangular and factorizable noncommutative pre-Poisson bialgebras.In particular, we show that a solution of the NPP-YBE with an invariant skew-symmetric part yields a quasi-triangular noncommutative pre-Poisson bialgebra.Such solutions are further interpreted as relative Rota-Baxter operators with weights. Finally, we establish a one-to-one correspondence between quadratic Rota-Baxter noncommutative pre-Poisson algebras and factorizable noncommutative pre-Poisson bialgebras.