论文标题
$ \ MATHCAL {O} $ - Hom-Lie代数上的运营商
$\mathcal{O}$-Operators on Hom-Lie algebras
论文作者
论文摘要
$ \ MATHCAL {O} $ - 运算符（也称为相对Rota-Baxter运算符）在Lie代数上具有多个应用程序和经典的Yang-Baxter方程式。在本文中，我们研究了hom-lie代数的$ \ Mathcal {O} $ - 运营商。我们为$ \ Mathcal {O} $ - 在Hom-Lie代数上定义了Cochain Complex，相对于表示形式。任何$ \ MATHCAL {O} $ - 操作员都会诱导Hom-Pre-Lie代数结构。我们表达了$ \ Mathcal {O} $ - 运算符的Cochain复合物，以与诱导的hom-pre-pre-lie-lie代数相关的子贴剂hom-hom-lie代数的某些hom-lie代数cochain复合物。如果hom-lie代数中的结构映射及其表示形式是可逆的，那么我们可以通过添加零心的空间来将上述Cochain复合物扩展到$ \ Mathcal {O} $ - 运算符的变形复合物。随后，我们研究了$ \ Mathcal {O} $的线性和形式变形 - 在hom-lie代数方面的运算符在变形共同体方面。最后，我们将$ S $ -Rota-baxter运算符（重量为0）和Hom-lie代数上的偏压$ r $ $ matrices的变形作为特定情况，为$ \ MATHCAL {O} $ - HOM-LIE代数的运营商。
$\mathcal{O}$-operators (also known as relative Rota-Baxter operators) on Lie algebras have several applications in integrable systems and the classical Yang-Baxter equations. In this article, we study $\mathcal{O}$-operators on hom-Lie algebras. We define cochain complex for $\mathcal{O}$-operators on hom-Lie algebras with respect to a representation. Any $\mathcal{O}$-operator induces a hom-pre-Lie algebra structure. We express the cochain complex of an $\mathcal{O}$-operator in terms of certain hom-Lie algebra cochain complex of the sub-adjacent hom-Lie algebra associated with the induced hom-pre-Lie algebra. If the structure maps in a hom-Lie algebra and its representation are invertible, then we can extend the above cochain complex to a deformation complex for $\mathcal{O}$-operators by adding the space of zero cochains. Subsequently, we study linear and formal deformations of $\mathcal{O}$-operators on hom-Lie algebras in terms of the deformation cohomology. In the end, we deduce deformations of $s$-Rota-Baxter operators (of weight 0) and skew-symmetric $r$-matrices on hom-Lie algebras as particular cases of $\mathcal{O}$-operators on hom-Lie algebras.
