Extensions of hom-Lie algebras in terms of cohomology

summary:We study (non-abelian) extensions of a given hom-Lie algebra and provide a geometrical interpretation of extensions, in particular, we characterize an extension of a hom-Lie algebra $\frak {g}$ by another hom-Lie algebra $\frak {h}$ and discuss the case where $\frak {h}$ has no center. We also deal with the setting of covariant exterior derivatives, Chevalley derivative, Maurer-Cartan formula, curvature and the Bianchi identity for the possible extensions in differential geometry. Moreover, we find a cohomological obstruction to the existence of extensions of hom-Lie algebras, i.e., we show that in order to have an extendible hom-Lie algebra, there should exist a trivial member of the third cohomology