AI 中文总结
本文推广泊松代数上的普通$\u0002mathcal{O}$-算子为扩展$\u0002mathcal{O}$-算子，提出扩展泊松杨-巴克斯特方程，建立其与后泊松代数、余边界泊松双代数等的联系，并在相关代数框架下研究三者关系。
AI 中文摘要
本文引入泊松代数上的扩展$\u0002mathcal{O}$-算子作为普通$\u0002mathcal{O}$-算子的自然推广，同时提出扩展泊松杨-巴克斯特方程。我们证明，泊松代数上权重为$λ$的$\u0002mathcal{O}$-算子可导出后泊松代数，其操作子是泊松代数操作子的三后继，且扩展$\u0002mathcal{O}$-算子可在模空间上诱导新的泊松代数结构。通过对称化子-反对称化子分解，得到了扩展$\u0002mathcal{O}$-算子的等价刻画。本文还引入了广义泊松杨-巴克斯特方程，并建立了其与余边界泊松双代数、扩展$\u0002mathcal{O}$-算子的联系。扩展$\u0002mathcal{O}$-算子的张量形式引出了扩展泊松杨-巴克斯特方程的概念，它推广了泊松杨-巴克斯特方程的概念。最后，在二次泊松代数和半直积泊松代数的框架下，研究了扩展$\u0002mathcal{O}$-算子、扩展泊松杨-巴克斯特方程与泊松杨-巴克斯特方程之间的关系。
英文摘要
This paper introduces the extended $\mathcal{O}$-operators on Poisson algebras as a natural generalization of ordinary $\mathcal{O}$-operators, together with the extended Poisson Yang-Baxter equations. We show that $\mathcal{O}$-operators of weight $λ$ on Poisson algebras give rise to post-Poisson algebras, whose operad are the trisuccessor of the operad of Poisson algebras, and that extended $\mathcal{O}$-operators induce new Poisson algebra structures on module spaces.Equivalent characterizations of extended $\mathcal{O}$-operators are obtained via the symmetrizer-antisymmetrizer decomposition. The generalized Poisson Yang-Baxter equations are also introduced, and their connections with coboundary Poisson bialgebras and extended $\mathcal{O}$-operators are established. The tensor form of extended $\mathcal{O}$-operators leads to the notion of the extended Poisson Yang-Baxter equations, which generalizes the notion of the Poisson Yang-Baxter equations. Finally, the relationships among extended $\mathcal{O}$-operators, the extended Poisson Yang-Baxter equations, and the Poisson Yang-Baxter equations are studied in the framework of quadratic Poisson algebras and semi-direct product Poisson algebras.
Comments30 pages. Comments welcome
