Czechoslovak Mathematical Journal, Vol. 67, No. 2, pp. 317-328, 2017

Extensions of hom-Lie algebras in terms of cohomology

Abdoreza R. Armakan, Mohammed Reza Farhangdoost

Received October 25, 2015.   First published March 1, 2017.

Abdoreza R. Armakan, Mohammad Reza Farhangdoost (corresponding author), Department of Mathematics, College of Sciences, Shiraz university, Adabiat square, Shiraz, Fars, Iran, P.O. Box 71457-44776, e-mail:,

Abstract: 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.

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

Classification (MSC 2010): 17B99, 55U15

DOI: 10.21136/CMJ.2017.0576-15

Full text available as PDF.

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