Third Cohomology Group and $$\alpha $$-Crossed Extensions of Hom–Lie Algebras

Abstract

We give a proof of the fact that the third cohomology group H 3 α (L, M ) of the Hom-Lie algebra (L, αL) with coefficients in the Hom-L-module (M, αM ) is in bijection to the set Ext 2 α (L, M ) of the equivalence classes of α-crossed module extensions of (L, αL) by (M, αM ).