Twisting $\mathcal{O}$-operators by $(2,3)$-Cocycle of Hom-Lie-Yamaguti Algebras with Representations

TL;DR

This paper introduces twisted $\ ext{O}$-operators on Hom-Lie-Yamaguti algebras using $(2,3)$-cocycles, explores their properties, cohomology, and underlying structures, and connects them to twisted operators on Hom-Lie algebras.

Contribution

It defines twisted $\mathcal{O}$-operators on Hom-Lie-Yamaguti algebras, studies their cohomology, and links them to structures like Hom-NS-Lie-Yamaguti algebras and operators on Hom-Lie algebras.

Findings

Twisted $\mathcal{O}$-operators induce Hom-Lie-Yamaguti structures.

Cohomology theory for twisted $\mathcal{O}$-operators is developed.

Connections between twisted operators on different algebraic structures are established.

Abstract

In this paper, we first introduce the notion of twisted $\mathcal O$-operators on a Hom-Lie-Yamaguti algebra by a given $(2,3)$-cocycle with coefficients in a representation. We show that a twisted $\mathcal O$-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 $\mathcal O$-operators on Hom-Lie-Yamaguti algebras. Then, we define a cohomology of twisted $\mathcal O$-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 $\mathcal O$-operator on Hom-Lie-Yamaguti algebras. Finally, we investigate the twisted $\mathcal O$-operator on Hom-Lie-Yamaguti algebras induced by the twisted $\mathcal O$-operator on a Hom-Lie algebras.