$\mathcal{O}$-Operators on Hom-Lie algebras

TL;DR

This paper explores $\\mathcal{O}$-operators on hom-Lie algebras, establishing their cohomology, deformation theory, and applications to Rota-Baxter operators and $r$-matrices, extending classical concepts to the hom-Lie setting.

Contribution

It introduces a cochain complex and deformation theory for $\mathcal{O}$-operators on hom-Lie algebras, and connects these to known structures like Rota-Baxter operators and $r$-matrices.

Findings

Defined cochain complex for $\mathcal{O}$-operators on hom-Lie algebras.

Established deformation theory for these operators.

Connected deformations to classical structures like Rota-Baxter operators.

Abstract

$\mathcal{O}$-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.