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.

中文翻译：

---

非交换泊松双代数

在本文中，我们引入了非对易泊松双代数的概念，并建立了匹配对、Manin 三元组和非对易泊松双代数之间的等价关系。使用拟表示和非交换泊松代数的相应上同调理论，我们研究了共边界非交换泊松双代数，从而引入了泊松杨-巴克斯特方程。Poisson Yang-Baxter 方程的偏对称解自然会给出（共边界）非对易泊松双代数。介绍了 Rota-Baxter 算子，更一般地说是非对易泊松代数上的 O-算子，以及非对易前泊松代数，通过它们，我们在从这些结构中获得的一些特殊的非对易泊松代数中构造了 Poisson Yang-Baxter 方程的偏对称解。