Abelian extensions and crossed modules of Hom-Lie algebras

In this paper we study the low dimensional cohomology groups of Hom-Lie algebras and their relation with derivations, abelian extensions and crossed modules. On one hand, we introduce the notion of $\alpha$-abelian extensions and we obtain a five term exact sequence in cohomology. On the other hand, we introduce crossed modules of Hom-Lie algebras showing their equivalence with cat$^1$-Hom-Lie algebras, and we introduce $\alpha$-crossed modules to have a better understanding of the third cohomology group.