Twisting $\mathcal{O}$-operators by $(2,3)$-Cocycle of Hom-Lie-Yamaguti Algebras with Representations
Sami Mabrouk, Sergei Silvestrov, Fatma Zouaidi
TL;DR
The paper develops a theory of twisted $\mathcal{O}$-operators on Hom-Lie-Yamaguti algebras driven by a $(2,3)$-cocycle and a representation, proving that such operators induce new Hom-Lie-Yamaguti structures and natural representations. It identifies the $(\lambda,\mu)$-weighted Reynolds operator as a key special case and builds a cohomology theory $\mathbf{H}_T^*(V,A)$ governing deformations of twisted operators. A central construction is the NS-Hom-Lie-Yamaguti algebra, which arises as the underlying structure associated with twisted $\mathcal{O}$-operators, and in which the operator graphs encode semidirect products. The work also connects twisted operators on Hom-Lie-Yamaguti algebras to those on Hom-Lie algebras, via induced NS-HLYA structures, offering a unifying framework for deformation and representation theory in Hom-type algebras with ternary operations.
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.