Nijenhuis Operators on 3-Hom-L-dendriform algebras

TL;DR

This paper introduces $3$-Hom-Lie-dendriform algebras, explores Nijenhuis operators on $3$-Hom-pre-Lie algebras, and discusses structures like products and complex structures, advancing the understanding of ternary Hom-Lie algebra frameworks.

Contribution

It defines $3$-Hom-Lie-dendriform algebras, introduces Nijenhuis operators in this context, and constructs new algebraic structures and conditions, extending the theory of Hom-Lie algebras.

Findings

Defined $3$-Hom-Lie-dendriform algebras.

Constructed $3$-Hom-Lie-dendriform algebras using Nijenhuis operators.

Explored product and complex structures with integrability conditions.

Abstract

The goal of this work is to introduce the notion of $3$-Hom-Lie-dendriform algebras which is the dendriform version of $3$-Hom-Lie-algebras. They can be also regarded as the ternary analogous of Hom-Lie-dendriform algebras. We give the representation of a $3$-Hom-pre-Lie algebra. Moreover, we introduce the notion of Nijenhuis operators on a $3$-Hom-pre-Lie algebra and provide some constructions of $3$-Hom-Lie-dendriform algebras in term of Nijenhuis operators. Parallelly, we introduce the notion of a product and complex structures on a $3$-Hom-Lie-dendriform algebras and there are also four types special integrability conditions.