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.

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:  17B99, 55U15
DOI:  10.21136/CMJ.2017.0576-15


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Affiliations:   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: r.armakan@shirazu.ac.ir, farhang@shirazu.ac.ir

 
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