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September  2021, 29(4): 2619-2636. doi: 10.3934/era.2021004

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

Bing Sun 1, , Liangyun Chen 2,, and  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

* Corresponding author: chenly640@nenu.edu.cn

Received  October 2020 Revised  November 2020 Published  September 2021 Early access  January 2021

Fund Project: Supported by the National Natural Science Foundation of China (Nos. 11901057, 11771069, 12071405 and 11801121), Natural Science Foundation of Changchun Normal University, Natural Science Foundation of Heilongjiang Province of China (QC2018006) and the Fundamental Research Foundation for Universities of Heilongjiang Province(No. LGYC2018JC002)

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.

Keywords: Hom-preLie algebra, Hom-action, universal central extension, universal $ \alpha $-central extension.
Mathematics Subject Classification: Primary: 17A30; Secondary: 16E40.
Citation: Bing Sun, Liangyun Chen, Yan Cao. On the universal $ \alpha $-central extensions of the semi-direct product of Hom-preLie algebras. Electronic Research Archive, 2021, 29 (4) : 2619-2636. doi: 10.3934/era.2021004
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.  Google Scholar

[2]

J. M. Casas and M. Ladra, Stem extensions and stem covers of Leibniz algebras, Georgian Math. J., 9 (2002), 659-669.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

J. M. Casas and A. M. Vieites, Central extensions of perfect of Leibniz algebras, Recent Advances in Lie Theory, 25 (2002), 189-196.   Google Scholar

[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.  Google Scholar

[8]

A. V. Gnedbaye, Third homology groups of universal central extensions of a Lie algebra, Afrika Mat., 10 (1999), 46-63.   Google Scholar

[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.  Google Scholar

[10]

R. Kurdiani and T. Pirashvili, A Leibniz algebra structure on the second tensor power, J. Lie Theory, 12 (2002), 583-596.   Google Scholar

[11]

A. Makhlouf and S. D. Silvestrov, Hom-algebra structures, J. Gen. Lie Theory Appl., 2 (2008), 51-64.   Google Scholar

[12]

Y. Sheng, Representations of hom-Lie algebras, Algebr. Represent. Theory, 15 (2012), 1081-1098.  doi: 10.1007/s10468-011-9280-8.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

J. M. Casas and M. Ladra, Stem extensions and stem covers of Leibniz algebras, Georgian Math. J., 9 (2002), 659-669.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

J. M. Casas and A. M. Vieites, Central extensions of perfect of Leibniz algebras, Recent Advances in Lie Theory, 25 (2002), 189-196.   Google Scholar

[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.  Google Scholar

[8]

A. V. Gnedbaye, Third homology groups of universal central extensions of a Lie algebra, Afrika Mat., 10 (1999), 46-63.   Google Scholar

[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.  Google Scholar

[10]

R. Kurdiani and T. Pirashvili, A Leibniz algebra structure on the second tensor power, J. Lie Theory, 12 (2002), 583-596.   Google Scholar

[11]

A. Makhlouf and S. D. Silvestrov, Hom-algebra structures, J. Gen. Lie Theory Appl., 2 (2008), 51-64.   Google Scholar

[12]

Y. Sheng, Representations of hom-Lie algebras, Algebr. Represent. Theory, 15 (2012), 1081-1098.  doi: 10.1007/s10468-011-9280-8.  Google Scholar

[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.  Google Scholar

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