Abstract

In this paper, we derive pre-anti-flexible algebras structures in term of zero weight’s Rota-Baxter operators, built underlying left-symmetric algebras, view pre-anti-flexible algebras as a splitting of anti-flexible algebras, introduce pre-anti-flexible bialgebras and establish its equivalences among matched pair of anti-flexible algebras and matched pair of pre-anti-flexible algebras. Special class of pre-anti-flexible bialgebras leads to the establishment of the pre-anti-flexible Yang-Baxter equation which is the same with $\mathcal{D}$-equation. Symmetric solution of pre-anti-flexible Yang-Baxter equation gives a pre-anti-flexible bialgebra. Finally, we recall and link $\mathcal{O}$-operators of anti-flexible algebras to bimodules of pre-anti-flexible algebras and built symmetric solutions of anti-flexible Yang-Baxter equation.

摘要

在本文中，我们根据零权重的 Rota-Baxter 算子推导了预反柔代数结构，构建了底层左对称代数，将预反柔代数视为反柔代数的分裂，引入了预反-柔性双代数，并在匹配的反柔性代数对和匹配的前反柔性代数之间建立其等价。特殊类的预反柔双代数导致建立与前反柔相同的Yang-Baxter方程$\mathcal{D}$-方程。预反柔杨-巴克斯特方程的对称解给出了预反柔双代数。最后，我们回忆并链接$\mathcal{哦}$- 反柔性代数算子到前反柔性代数的双模，并建立了反柔性杨-巴克斯特方程的对称解。