Czechoslovak Mathematical Journal, online first, 12 pp.

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.

  [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

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