O -operators on hom-Lie algebras

Abstract

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.