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 O-operators on hom-Lie algebras. We define a cochain complex for O-operators on hom-Lie algebras with respect to a representation. Any O-operator induces a hom-pre-Lie algebra structure. We express the cochain complex of an O-operator in terms of the specific hom-Lie algebra cochain complex. 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 O-operators by adding the space of zero cochains. Subsequently, we study formal deformations of O-operators on regular 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 O-operators on hom-Lie algebras.

中文翻译：

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hom-Lie 代数上的 O 算子

Lie 代数上的 O 算子（也称为相对 Rota-Baxter 算子）在可积系统和经典的 Yang-Baxter 方程中有多种应用。在本文中，我们研究 hom-Lie 代数上的 O 算子。我们为 hom-Lie 代数上的 O-operators 定义了一个关于表示的 cochain 复合体。任何 O 算子都会引入 hom-pre-Lie 代数结构。我们根据特定的 hom-Lie 代数 cochain complex 来表达 O-operator 的 cochain complex。如果hom-Lie代数中的结构映射及其表示是可逆的，那么我们可以通过添加零个cochains的空间将上述cochain复合体扩展为O-operators的变形复合体。随后，我们根据变形上同调研究了 O 算子在规则 hom-Lie 代数上的形式变形。到底，