On the universal $ \alpha $-central extensions of the semi-direct product of Hom-preLie algebras

Bing Sun; Liangyun Chen; Yan Cao; Bing Sun; Liangyun Chen; Yan Cao

doi:10.3934/era.2021004

Electronic Research Archive

2021, Volume 29, Issue 4: 2619-2636. doi: 10.3934/era.2021004

Previous Article Next Article

On the universal $ \alpha $-central extensions of the semi-direct product of Hom-preLie algebras

Bing Sun ^{1
,},
Liangyun Chen ^{2
,
,},
Yan Cao ^{3
,}

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

Received: 01 October 2020 Revised: 01 November 2020 Published: 01 September 2021
Primary: 17A30; Secondary: 16E40

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.
- Hom-preLie algebra,
- Hom-action,
- universal central extension,
- universal α-central extension
Citation: Bing Sun, Liangyun Chen, Yan Cao. On the universal $ \alpha $-central extensions of the semi-direct product of Hom-preLie algebras[J]. Electronic Research Archive, 2021, 29(4): 2619-2636. doi: 10.3934/era.2021004

Related Papers:

Abstract

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]	Casas J. M., Corral N. (2009) On universal central extensions of Leibniz algebras. Comm. Algebra 37: 2104-2120.
[2]	Casas J. M., Ladra M. (2002) Stem extensions and stem covers of Leibniz algebras. Georgian Math. J. 9: 659-669.
[3]	Casas J. M., Ladra M. (2007) Computing low dimensional Leibniz homology of some perfect Leibniz algebras. Southeast Asian Bull. Math. 31: 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. 10.1142/S0219498814500534 MR3225120
[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. 10.1007/s40840-015-0254-6 MR3549981
[6]	Casas J. M., Vieites A. M. (2002) Central extensions of perfect of Leibniz algebras. Recent Advances in Lie Theory 25: 189-196.
[7]	García-Martínez X., Khmaladze E., Ladra M. (2015) Non-abelian tensor product and homology of Lie superalgebras. J. Algebra 440: 464-488.
[8]	Gnedbaye A. V. (1999) Third homology groups of universal central extensions of a Lie algebra. Afrika Mat. 10: 46-63.
[9]	Gnedbaye A. V. (1999) A non-abelian tensor product of Leibniz algebras. Ann. Inst. Fourier (Grenoble) 49: 1149-1177.
[10]	Kurdiani R., Pirashvili T. (2002) A Leibniz algebra structure on the second tensor power. J. Lie Theory 12: 583-596.
[11]	Makhlouf A., Silvestrov S. D. (2008) Hom-algebra structures. J. Gen. Lie Theory Appl. 2: 51-64.
[12]	Sheng Y. (2012) Representations of hom-Lie algebras. Algebr. Represent. Theory 15: 1081-1098.
[13]	B. Sun, L. Y. Chen and X. Zhou, On universal $\alpha$-central extensions of Hom-preLie algebras, arXiv: 1810.09848.
[14]	D. Yau, Hom-Novikov algebras, J. Phys. A, 44 (2011), 085202, 20 pp. 10.1088/1751-8113/44/8/085202 MR2770370

Reader Comments

your name:*

Email:*
© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)