标题： 双代数理论与可容许Hom-Poisson代数的$\mathcal O$-算子 显示英文标题

标题： Bialgebra theory and $\mathcal O$-operators of admissible Hom-Poisson algebras

摘要： 本文中，我们在Hom-Poisson代数的框架内引入并探讨了若干关键概念。具体而言，我们引入了可容许Hom-Poisson代数的概念，以及与此相关的这类代数的匹配对和Manin三元组的思想。随后，我们定义了纯可容许Hom-Poisson双代数的概念，并着重讨论了其与非退化对称双线性形式相关的Manin三元组结构的兼容性。这种兼容性对于理解这些代数对象之间的结构交互至关重要。此外，我们研究了作用于可容许Hom-Poisson代数上的Hom-$ \mathcal O$-算子的概念，分析了它们的性质，并建立了与可容许Hom-前Poisson代数之间的联系，揭示了这两种结构之间的关系。 显示英文摘要

摘要： In this paper, we present and explore several key concepts within the framework of Hom-Poisson algebras. Specifically, we introduce the notions of admissible Hom-Poisson algebras, along with the related ideas of matched pairs and Manin triples for such algebras. We then define the concept of a purely admissible Hom-Poisson bialgebra, placing particular emphasis on its compatibility with the Manin triple structure associated with a nondegenerate symmetric bilinear form. This compatibility is crucial for understanding the structural interplay between these algebraic objects. Additionally, we investigate the notion of Hom-$ \mathcal O$-operators acting on admissible Hom-Poisson algebras. We analyze their properties and establish a connection with admissible Hom-pre-Poisson algebras, shedding light on the relationship between these two structures.