论文标题
非均匀关联经典杨巴克斯特方程的操作员形式
Operator forms of nonhomogeneous associative classical Yang-Baxter equation
论文作者
论文摘要
This paper studies operator forms of the nonhomogeneous associative classical Yang-Baxter equation (nhacYBe), extending and generalizing such studies for the classical Yang-Baxter equation and associative Yang-Baxter equation that can be tracked back to the works of Semonov-Tian-Shansky and Kupershmidt on Rota-Baxter Lie algebras and $\mathcal{O}$-operators.通常，NHACYBE的解决方案的特征是以通用的$ \ Mathcal {O} $ - 运算符来表征。当解决方案满足不变条件时，可以通过经典$ \ Mathcal {O} $ - 运算符给出表征。当不变条件与Frobenius代数兼容时，此类溶液与Frobenius代数上的Rota-Baxter操作员有密切的关系。通常，NHACYBE的解决方案可以从Rota-baxter运算符中产生，然后是$ \ Mathcal {O} $ - 运算符时，当解决方案以半独立产品代数采用时。在另一个方向上，可以从NHACYBE的溶液中获得Rota-baxter操作员在代数的单位化中。最后，在所有尺寸的所有Unital Complecter代数中，都可以获得满足上述不变条件的NHACYBE的分类。所有这些解决方案均显示出来自Rota-Baxter操作员。
This paper studies operator forms of the nonhomogeneous associative classical Yang-Baxter equation (nhacYBe), extending and generalizing such studies for the classical Yang-Baxter equation and associative Yang-Baxter equation that can be tracked back to the works of Semonov-Tian-Shansky and Kupershmidt on Rota-Baxter Lie algebras and $\mathcal{O}$-operators. In general, solutions of the nhacYBe are characterized in terms of generalized $\mathcal{O}$-operators. The characterization can be given by the classical $\mathcal{O}$-operators precisely when the solutions satisfy an invariant condition. When the invariant condition is compatible with a Frobenius algebra, such solutions have close relationships with Rota-Baxter operators on the Frobenius algebra. In general, solutions of the nhacYBe can be produced from Rota-Baxter operators, and then from $\mathcal{O}$-operators when the solutions are taken in semi-direct product algebras. In the other direction, Rota-Baxter operators can be obtained from solutions of the nhacYBe in unitizations of algebras. Finally a classifications of solutions of the nhacYBe satisfying the mentioned invariant condition in all unital complex algebras of dimensions two and three are obtained. All these solutions are shown to come from Rota-Baxter operators.
