Noncommutative Poisson bialgebras

TL;DR

This paper introduces noncommutative Poisson bialgebras, explores their structures, and connects them with solutions to the Poisson Yang-Baxter equation, expanding the algebraic framework and solution methods.

Contribution

It establishes the equivalence of various structures for noncommutative Poisson bialgebras and introduces new tools like Rota-Baxter operators and noncommutative pre-Poisson algebras.

Findings

Equivalence between matched pairs, Manin triples, and noncommutative Poisson bialgebras.

Introduction of the Poisson Yang-Baxter equation and its skew-symmetric solutions.

Construction of solutions using Rota-Baxter and O-operators.

Abstract

In this paper, we introduce the notion of a noncommutative Poisson bialgebra, and establish the equivalence between matched pairs, Manin triples and noncommutative Poisson bialgebras. Using quasi-representations and the corresponding cohomology theory of noncommutative Poisson algebras, we study coboundary noncommutative Poisson bialgebras which leads to the introduction of the Poisson Yang-Baxter equation. A skew-symmetric solution of the Poisson Yang-Baxter equation naturally gives a (coboundary) noncommutative Poisson bialgebra. Rota-Baxter operators, more generally O-operators on noncommutative Poisson algebras, and noncommutative pre-Poisson algebras are introduced, by which we construct skew-symmetric solutions of the Poisson Yang-Baxter equation in some special noncommutative Poisson algebras obtained from these structures.