扩展的$\boldsymbol{\tau}$-算子、扩展的置换杨-巴克斯特方程及相关结构
Extended $\mathcal{O}$-operators, extended perm Yang-Baxter equations and related structures
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中文总结 AI 辅助
本文引入置换代数上的扩展$\boldsymbol{\tau}$-算子与扩展置换杨-巴克斯特方程，建立其与普通$\boldsymbol{\tau}$-算子、置换杨-巴克斯特方程的关系，通过二次置换代数等框架研究相关结构。
中文摘要 AI 辅助
本文引入置换代数上的扩展$\boldsymbol{\tau}$-算子概念，作为$\boldsymbol{\tau}$-算子的推广，同时引入扩展置换杨-巴克斯特方程。我们证明，置换代数上权重为$\boldsymbol{\tau}$-算子会导出后置换代数，而扩展$\boldsymbol{\tau}$-算子会产生新的置换代数结构。我们还通过$\boldsymbol{\tau}_\boldsymbol{\tau}$分解，给出扩展$\boldsymbol{\tau}$-算子关于普通$\boldsymbol{\tau}$-算子的等价刻画。引入广义置换杨-巴克斯特方程概念，并建立其与扩展$\boldsymbol{\tau}$-算子的关系。扩展$\boldsymbol{\tau}$-算子的张量形式导出扩展置换杨-巴克斯特方程（extended perm-YBE）概念，该方程推广了置换杨-巴克斯特方程。我们证明，具有$(R, \text{ad})$-不变反对称部分的extended perm-YBE的解可由扩展$\boldsymbol{\tau}$-算子刻画。此外，通过二次置换代数和半直积置换代数框架，研究了扩展$\boldsymbol{\tau}$-算子、置换杨-巴克斯特方程与extended perm-YBE之间的关系。
英文摘要
In this paper, we introduce the notions of extended $\mathcal{O}$-operators on perm algebras, as generalization of $\mathcal{O}$-operators, and the extended perm Yang-Baxter equation. We demonstrate that an $\mathcal{O}$-operator of weight $λ$ on a perm algebra gives rise to a post-perm algebra, and an extended $\mathcal{O}$-operator yields a new perm algebra structure. We also give an equivalent characterization of extended $\mathcal{O}$-operators in terms of ordinary $\mathcal{O}$-operators via the $π_\pm$ decomposition. The notion of the generalized perm Yang-Baxter equations is introduced, and their relationship with extended $\mathcal{O}$-operators is established. The tensor form of extended $\mathcal{O}$-operators leads to the notion of the extended perm Yang-Baxter equation (extended perm-YBE), which generalizes the perm Yang-Baxter equation. We establish that a solution of the extended perm-YBE with $(R, \ad)$-invariant skew-symmetric part is characterized by an extended $\mathcal{O}$-operator. Furthermore, relationships among extended $\mathcal{O}$-operators, the perm Yang-Baxter equation and the extended perm-YBE are investigated through the framework of quadratic perm algebras and semi-direct product perm algebras.
