Constructions and Cohomology of color Hom-Lie algebras

The main purpose of this paper is to define representations and a cohomology of color Hom-Lie algebras and to study some key constructions and properties. We describe Hartwig-Larsson-Silvestrov Theorem in the case of $\Gamma$-graded algebras, study one-parameter formal deformations, discuss $\alpha^{k}$-generalized derivation and provide examples.