Twisting O-operators by (2,3)-cocycle of Hom-Lie-Yamaguti algebras with representations

Abstract

In this paper, we first introduce the notion of twisted $O$-operators on a Hom-Lie-Yamaguti algebra by a given $(2,3)$-cocycle with coefficients in a representation. We show that a twisted $O$-operator induces a Hom-Lie-Yamaguti structure. We also introduce the notion of a weighted Reynolds operator on a Hom-Lie-Yamaguti algebra, which can serve as a special case of twisted $O$-operators on Hom-Lie-Yamaguti algebras. Then, we define a cohomology of twisted $O$-operator on Hom-Lie-Yamaguiti algebras with coefficients in a representation. Furthermore, we introduce and study the Hom-NS-Lie-Yamaguti algebras as the underlying structure of the twisted $O$-operator on Hom-Lie-Yamaguti algebras. Finally, we investigate the twisted $O$-operator on Hom-Lie-Yamaguti algebras induced by the twisted $O$-operator on Hom-Lie algebras.