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Keywords:
hom-Lie algebras; cohomology of hom-Lie algebras; extensions of hom-Lie algebras
hom-Lie algebras; cohomology of hom-Lie algebras; extensions of hom-Lie algebras
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.
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