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.

Generalized Reynolds Operators on Hom-Lie Triple Systems

“…Subsequently, the notion of a relative Rota-Baxter operator (also called an O-operator) on a Lie algebra was independently introduced by Kupershmidt [13], to better understand the classical Yang-Baxter equation and related integrable systems. Recently, relative Rota-Baxter operators have been widely studied (see [14][15][16][17][18][19]). In addition, other operators related to (relative) Rota-Baxter operators are constantly emerging.…”

Generalized Reynolds Operators on Hom-Lie Triple Systems

“…Subsequently, the notion of a relative Rota-Baxter operator (also called an O-operator) on a Lie algebra was independently introduced by Kupershmidt [13], to better understand the classical Yang-Baxter equation and related integrable systems. Recently, relative Rota-Baxter operators have been widely studied (see [14][15][16][17][18][19]). In addition, other operators related to (relative) Rota-Baxter operators are constantly emerging.…”

Cohomology of compatible BiHom-Lie algebras

“…The primary characteristic of BiHom-Lie algebras is that they are an extension of Hom-Lie algebra with one twist map α, which is defined in [4,10], where the identities characterizing BiHom-Lie algebras are twisted by two twist maps α, β. If we consider α = β then the theory of BiHom-Lie algebras deform to Hom-Lie algebras and by putting α = β = id, then we get a Lie algebra structure.…”