Abstract
We introduce Hom-actions, semidirect product, and establish the equivalence between split extensions and the semi-direct product extension of Hom-Leibniz algebras. We analyze the functorial properties of the universal (\(\alpha \))-central extensions of (\(\alpha \))-perfect Hom-Leibniz algebras. We establish under what conditions an automorphism or a derivation can be lifted in an \(\alpha \)-cover and we analyze the universal \(\alpha \)-central extension of the semi-direct product of two \(\alpha \)-perfect Hom-Leibniz algebras.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ammar, F., Makhlouf, A.: Hom-Lie superalgebras and Hom-Lie admissible superalgebras. J. Algebra 34(7), 1513–1528 (2010)
Ammar, F., Mabrouk, S., Makhlouf, A.: Representations and cohomology of \(n\)-ary multiplicative Hom-Nambu-Lie algebras. J. Geom. Phys. 61(10), 1898–1913 (2011)
Arnlind, J., Makhlouf, A., Silvestrov, S.: Ternary Hom-Nambu-Lie algebras induced by Hom-Lie algebras. J. Math. Phys. 51(4), (2010), 11 pp
Ataguema, H., Makhlouf, A., Silvestrov, S.: Generalization of \(n\)-ary Nambu algebras and beyond. J. Math. Phys. 50(8), (2009), 15 pp
Caenepeel, S., Goyvaerts, I.: Monoidal Hom-Hopf algebras. Comm. Algebra 39(6), 2216–2240 (2011)
Casas, J.M., Corral, N.: On universal central extensions of Leibniz algebras. Comm. Algebra 37(6), 2104–2120 (2009)
Casas, J.M., Insua, M.A., Pacheco Rego, N.: On universal central extensions of Hom-Leibniz algebras. J. Algebra Appl. 13(8), (2014), 22 pp
Casas, J.M., Khudoyberdiyev, AKh, Ladra, M., Omirov, B.A.: On the degenerations of solvable Leibniz algebras. Linear Algebra Appl. 439(2), 472–487 (2013)
Cheng, Y.S., Su, Y.C.: (Co)homology and universal central extensions of Hom-Leibniz algebras. Acta Math. Sin. 27(5), 813–830 (2011)
Cheng, Y.S., Su, Y.C.: Quantum deformations of the Heisenberg-Virasoro algebra. Algebra Colloq. 20(2), 299–308 (2013)
Hartwing, J.T., Larson, D., Silvestrov, S.D.: Deformations of Lie algebras using \(\sigma \)-derivations. J. Algebra 295, 314–361 (2006)
Issa, A.N.: Some characterization of Hom-Leibniz algebras. Int. Electron. J. Algebra 14, 1–9 (2013)
Loday, J.-L.: Une version non commutative des algèbres de Lie: les algèbres de Leibniz. L’Enseignement Mathématique 39, 269–292 (1993)
Loday, J.-L., Pirashvili, T.: Universal enveloping algebras of Leibniz algebras and (co)homology. Math. Ann. 296, 139–158 (1993)
Makhlouf, A.: Hom-alternative algebras and Hom-Jordan algebras. Int. Electron. J. Algebra 8, 177–190 (2010)
Makhlouf, A., Silvestrov, S.: Notes on 1-parameter formal deformations of Hom-associative and Hom-Lie algebras. Forum Math. 22(4), 715–739 (2010)
Makhlouf, A., Silvestrov, S.: Hom-algebra structures. J. Gen. Lie Theory Appl. 2(2), 51–64 (2008)
Rakhimov, I.S., Atan, K.A.M.: On contractions and invariants of Leibniz algebras. Bull. Malays. Math. Sci. Soc (2) 35(2A), 557–565 (2012)
Yau, D.: Hom-algebras as deformations and homology, arXiv: 0712.3515 (2007)
Yau, D.: Enveloping algebras of Hom-Lie algebras. J. Gen. Lie Theory Appl. 2(2), 95–108 (2008)
Yau, D.: Hom-Maltsev, Hom-alternative, and Hom-Jordan algebras. Int. Electron. J. Algebra 11, 177–217 (2012)
Yuan, L.: Hom-Lie color algebra structures. Comm. Algebra 40(2), 575–592 (2012)
Acknowledgments
First author was supported by Ministerio de Economía y Competitividad (Spain), grant MTM2013-43687-P (European FEDER support included).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Lee See Keong.
Rights and permissions
About this article
Cite this article
Casas, J.M., Rego, N.P. On the Universal \(\alpha \)-Central Extension of the Semi-direct Product of Hom-Leibniz Algebras. Bull. Malays. Math. Sci. Soc. 39, 1579–1602 (2016). https://doi.org/10.1007/s40840-015-0254-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40840-015-0254-6