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

[1] F. Ammar, Z. Ejbehi, A. Makhlouf: Cohomology and deformations of Hom-algebras. J. Lie Theory 21 (2011), 813-836. MR 2917693 | Zbl 1237.17003
[2] F. W. Anderson, K. R. Fuller: Rings and Categories of Modules. Graduate Texts in Mathematics 13, Springer, New York (1992). DOI 10.1007/978-1-4612-4418-9 | MR 1245487 | Zbl 0765.16001
[3] S. Benayadi, A. Makhlouf: Hom-Lie algebras with symmetric invariant nondegenerate bilinear forms. J. Geom. Phys. 76 (2014), 38-60. DOI 10.1016/j.geomphys.2013.10.010 | MR 3144357 | Zbl 1331.17028
[4] J. M. Casas, M. A. Insua, N. Pacheco: On universal central extensions of Hom-Lie algebras. Hacet. J. Math. Stat. 44 (2015), 277-288. MR 3381108 | Zbl 1344.17003
[5] J. T. Hartwig, D. Larsson, S. D. Silvestrov: Deformations of Lie algebras using $\sigma$-derivations. J. Algebra 295 (2006), 314-361. DOI 10.1016/j.jalgebra.2005.07.036 | MR 2194957 | Zbl 1138.17012
[6] I. Kolář, P. W. Michor, J. Slovák: Natural Operations in Differential Geometry. Springer, Berlin (corrected electronic version) (1993). DOI 10.1007/978-3-662-02950-3 | MR 1202431 | Zbl 0782.53013
[7] A. Makhlouf, S. D. Silvestrov: Hom-algebra structures. J. Gen. Lie Theory Appl. 2 (2008), 51-64. DOI 10.4303/jglta/S070206 | MR 2399415 | Zbl 1184.17002
[8] A. Makhlouf, S. Silvestrov: Notes on 1-parameter formal deformations of Hom-associative and Hom-Lie algebras. Forum Math. 22 (2010), 715-739. DOI 10.1515/FORUM.2010.040 | MR 2661446 | Zbl 1201.17012
[9] Y. Sheng: Representations of hom-Lie algebras. Algebr. Represent. Theory 15 (2012), 1081-1098. DOI 10.1007/s10468-011-9280-8 | MR 2994017 | Zbl 1294.17001
[10] Y. Sheng, D. Chen: Hom-Lie 2-algebras. J. Algebra 376 (2013), 174-195. DOI 10.1016/j.jalgebra.2012.11.032 | MR 3003723 | Zbl 1281.17034
[11] Y. Sheng, Z. Xiong: On Hom-Lie algebras. Linear Multilinear Algebra 63 (2015), 2379-2395. DOI 10.1080/03081087.2015.1010473 | MR 3402544 | Zbl 06519840
[12] D. Yau: Enveloping algebras of Hom-Lie algebras. J. Gen. Lie Theory Appl. 2 (2008), 95-108. DOI 10.4303/jglta/S070209 | MR 2399418 | Zbl 1214.17001
[13] D. Yau: The Hom-Yang-Baxter equation, Hom-Lie algebras, and quasi-triangular bialgebras. J. Phys. A, Math. Theor. 42 (2009), Article ID 165202, 12 pages. DOI 10.1088/1751-8113/42/16/165202 | MR 2539278 | Zbl 1179.17001

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:,

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