alpha-type Chevalley-Eilenberg cohomology of Hom-Lie algebras and bialgebras

The purpose of this paper is to define an $\alpha$-type cohomology, which we call $\alpha$-type Chevalley-Eilenberg cohomology, for Hom-Lie algebras. We relate it to the known Chevalley-Eilenberg cohomology and provide explicit computations for some examples. Moreover, using this cohomology we study formal deformations of Hom-Lie algebras, where the bracket as well as the structure map $\alpha$ are deformed. Furthermore, we provide a generalization of the Grand Crochet and study, in a particular case, the $\alpha$-type cohomology for Hom-Lie bialgebras.