Representations and Deformations of 3-Hom-$\rho$-Lie algebras

The aim of this paper is to introduce 3-Hom-$\rho$-Lie algebra structures generalizing the algebras of 3-Hom-Lie algebra. Also, we investigate the representations and deformations theory of this type of Hom-Lie algebras. Moreover, we introduce the definition of extensions and abelian extensions of 3-Hom-$\rho$-Lie algebras and show that associated to any abelian extension, there is a representation and a 2-cocycle.