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On the universal $ \alpha $-central extensions of the semi-direct product of Hom-preLie algebras
1. | School of Mathematics, Changchun Normal University, Changchun 130032, China |
2. | School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China |
3. | Department of Mathematics, Harbin University of Science and Technology, Harbin 150080, China |
We study Hom-actions, semidirect product and describe the relation between semi-direct product extensions and split extensions of Hom-preLie algebras. We obtain the functorial properties of the universal $ \alpha $-central extensions of $ \alpha $-perfect Hom-preLie algebras. We give that a derivation or an automorphism can be lifted in an $ \alpha $-cover with certain constraints. We provide some necessary and sufficient conditions about the universal $ \alpha $-central extension of the semi-direct product of two $ \alpha $-perfect Hom-preLie algebras.
References:
[1] |
J. M. Casas and N. Corral,
On universal central extensions of Leibniz algebras, Comm. Algebra, 37 (2009), 2104-2120.
doi: 10.1080/00927870802506234. |
[2] |
J. M. Casas and M. Ladra,
Stem extensions and stem covers of Leibniz algebras, Georgian Math. J., 9 (2002), 659-669.
|
[3] |
J. M. Casas and M. Ladra,
Computing low dimensional Leibniz homology of some perfect Leibniz algebras, Southeast Asian Bull. Math., 31 (2007), 683-690.
|
[4] |
J. M. Casas, M. A. Insua and N. P. Rego, On universal central extensions of Hom-Leibniz algebras, J. Algebra Appl., 13 (2014), 1450053, 22pp.
doi: 10.1142/S0219498814500534. |
[5] |
J. M. Casas and N. P. Rego, On the universal $\alpha$-central extension of the semi-direct product of Hom-Leibniz algebras, Bull. Malays. Math. Sci. Soc., 39 (2016), 1579-1602.
doi: 10.1007/s40840-015-0254-6. |
[6] |
J. M. Casas and A. M. Vieites,
Central extensions of perfect of Leibniz algebras, Recent Advances in Lie Theory, 25 (2002), 189-196.
|
[7] |
X. García-Martínez, E. Khmaladze and M. Ladra,
Non-abelian tensor product and homology of Lie superalgebras, J. Algebra, 440 (2015), 464-488.
doi: 10.1016/j.jalgebra.2015.05.027. |
[8] |
A. V. Gnedbaye,
Third homology groups of universal central extensions of a Lie algebra, Afrika Mat., 10 (1999), 46-63.
|
[9] |
A. V. Gnedbaye,
A non-abelian tensor product of Leibniz algebras, Ann. Inst. Fourier (Grenoble), 49 (1999), 1149-1177.
doi: 10.5802/aif.1712. |
[10] |
R. Kurdiani and T. Pirashvili,
A Leibniz algebra structure on the second tensor power, J. Lie Theory, 12 (2002), 583-596.
|
[11] |
A. Makhlouf and S. D. Silvestrov,
Hom-algebra structures, J. Gen. Lie Theory Appl., 2 (2008), 51-64.
|
[12] |
Y. Sheng,
Representations of hom-Lie algebras, Algebr. Represent. Theory, 15 (2012), 1081-1098.
doi: 10.1007/s10468-011-9280-8. |
[13] |
B. Sun, L. Y. Chen and X. Zhou, On universal $\alpha$-central extensions of Hom-preLie algebras, arXiv: 1810.09848. Google Scholar |
[14] |
D. Yau, Hom-Novikov algebras, J. Phys. A, 44 (2011), 085202, 20 pp.
doi: 10.1088/1751-8113/44/8/085202. |
show all references
References:
[1] |
J. M. Casas and N. Corral,
On universal central extensions of Leibniz algebras, Comm. Algebra, 37 (2009), 2104-2120.
doi: 10.1080/00927870802506234. |
[2] |
J. M. Casas and M. Ladra,
Stem extensions and stem covers of Leibniz algebras, Georgian Math. J., 9 (2002), 659-669.
|
[3] |
J. M. Casas and M. Ladra,
Computing low dimensional Leibniz homology of some perfect Leibniz algebras, Southeast Asian Bull. Math., 31 (2007), 683-690.
|
[4] |
J. M. Casas, M. A. Insua and N. P. Rego, On universal central extensions of Hom-Leibniz algebras, J. Algebra Appl., 13 (2014), 1450053, 22pp.
doi: 10.1142/S0219498814500534. |
[5] |
J. M. Casas and N. P. Rego, On the universal $\alpha$-central extension of the semi-direct product of Hom-Leibniz algebras, Bull. Malays. Math. Sci. Soc., 39 (2016), 1579-1602.
doi: 10.1007/s40840-015-0254-6. |
[6] |
J. M. Casas and A. M. Vieites,
Central extensions of perfect of Leibniz algebras, Recent Advances in Lie Theory, 25 (2002), 189-196.
|
[7] |
X. García-Martínez, E. Khmaladze and M. Ladra,
Non-abelian tensor product and homology of Lie superalgebras, J. Algebra, 440 (2015), 464-488.
doi: 10.1016/j.jalgebra.2015.05.027. |
[8] |
A. V. Gnedbaye,
Third homology groups of universal central extensions of a Lie algebra, Afrika Mat., 10 (1999), 46-63.
|
[9] |
A. V. Gnedbaye,
A non-abelian tensor product of Leibniz algebras, Ann. Inst. Fourier (Grenoble), 49 (1999), 1149-1177.
doi: 10.5802/aif.1712. |
[10] |
R. Kurdiani and T. Pirashvili,
A Leibniz algebra structure on the second tensor power, J. Lie Theory, 12 (2002), 583-596.
|
[11] |
A. Makhlouf and S. D. Silvestrov,
Hom-algebra structures, J. Gen. Lie Theory Appl., 2 (2008), 51-64.
|
[12] |
Y. Sheng,
Representations of hom-Lie algebras, Algebr. Represent. Theory, 15 (2012), 1081-1098.
doi: 10.1007/s10468-011-9280-8. |
[13] |
B. Sun, L. Y. Chen and X. Zhou, On universal $\alpha$-central extensions of Hom-preLie algebras, arXiv: 1810.09848. Google Scholar |
[14] |
D. Yau, Hom-Novikov algebras, J. Phys. A, 44 (2011), 085202, 20 pp.
doi: 10.1088/1751-8113/44/8/085202. |
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