Geodesic
Leibniz algebras with absolute maximal Lie subalgebras
Algebra and discrete mathematics, Tome 29 (2020) no. 1, pp. 52-65.

Voir la notice de l'article provenant de la source Math-Net.Ru

A Lie subalgebra of a given Leibniz algebra is said to be an absolute maximal Lie subalgebra if it has codimension one. In this paper, we study some properties of non-Lie Leibniz algebras containing absolute maximal Lie subalgebras. When the dimension and codimension of their $\mathsf{Lie}$-center are greater than two, we refer to these Leibniz algebras as $s$-Leibniz algebras (strong Leibniz algebras). We provide a classification of nilpotent Leibniz $s$-algebras of dimension up to five.
Keywords: Leibniz algebras, $s$-Leibniz algebras, $\mathsf{Lie}$-center. 
@article{ADM_2020_29_1_a4,
     author = {G. R. Biyogmam and C. Tcheka},
     title = {Leibniz algebras with absolute maximal {Lie} subalgebras},
     journal = {Algebra and discrete mathematics},
     pages = {52--65},
     publisher = {mathdoc},
     volume = {29},
     number = {1},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADM_2020_29_1_a4/}
}
TY  - JOUR
AU  - G. R. Biyogmam
AU  - C. Tcheka
TI  - Leibniz algebras with absolute maximal Lie subalgebras
JO  - Algebra and discrete mathematics
PY  - 2020
SP  - 52
EP  - 65
VL  - 29
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/ADM_2020_29_1_a4/
LA  - en
ID  - ADM_2020_29_1_a4
ER  - 
%0 Journal Article
%A G. R. Biyogmam
%A C. Tcheka
%T Leibniz algebras with absolute maximal Lie subalgebras
%J Algebra and discrete mathematics
%D 2020
%P 52-65
%V 29
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/ADM_2020_29_1_a4/
%G en
%F ADM_2020_29_1_a4
G. R. Biyogmam; C. Tcheka. Leibniz algebras with absolute maximal Lie subalgebras. Algebra and discrete mathematics, Tome 29 (2020) no. 1, pp. 52-65. http://geodesic.mathdoc.fr/item/ADM_2020_29_1_a4/

[1] R. K. Amayo, “Quasi-ideals of Lie algebras II”, Proc. Lond. Math. Soc., 3:33 (1976), 37–64 | DOI | MR | Zbl

[2] Sh. A. Ayupov and B. A. Omirov, “On some classes of nilpotent Leibniz algebras”, Sibirsk. Mat. Zh., 42:1 (2001), 18–29 | MR | Zbl

[3] G. R. Biyogmam and J. M. Casas, “On Lie-isoclinic Leibniz algebras”, J. Algebra, 499 (2018), 337–357 | DOI | MR | Zbl

[4] G. R. Biyogmam and J. M. Casas, “The $c$-Nilpotent Shur $\mathrm{Lie}$-Multiplier of Leibniz Algebras”, J. Geom. Phys., 138 (2019), 55–69 | DOI | MR | Zbl

[5] D. Barnes, “On Levi's theorem for Leibniz algebras”, Bull. Aust. Math. Soc., 86 (2012), 184–185 | DOI | MR | Zbl

[6] J. M. Casas and E. Khmaladze, “On Lie-central extensions of Leibniz algebras”, RACSAM, 2016 | DOI | MR

[7] J. M. Casas and T. Van der Linden, A relative theory of universal central extensions, Preprint Number 09-2009, Pré-Publicaçoes do Departamento de Matemàtica, Universidade de Coimbra

[8] J. M. Casas and M. A. Insua, “The Schur $\mathrm{Lie}$-multiplier of Leibniz algebras”, Quaestiones Mathematicae, 41:2 (2018) | MR

[9] J. M. Casas and T. Van der Linden, “Universal central extensions in semi-abelian categories”, Appl. Categor. Struct., 22:1 (2014), 253–268 | DOI | MR | Zbl

[10] I. Demir, Classification of 5-Dimensional Complex Nilpotent Leibniz Algebras, Ph.D. Thesis, 138 pp. http://www.lib.ncsu.edu/resolver/1840.20/33418

[11] I. Demir, C. Kailash and E. Stitzinger, “On classification of four-dimensional nilpotent Leibniz algebras”, Comm. Algebra, 45:3 (2017), 1012–1018 | DOI | MR | Zbl

[12] I. Demir, C. Kailash and E. Stitzinger, “On some structure of Leibniz algebras”, Recent Advances in Representation Theory, Quantum Groups, Algebraic Geometry, and Related Topics, Contemporary Mathematics, 623, Amer. Math. Soc., Providence, RI, 2014, 41–54 | DOI | MR | Zbl

[13] V. Gorbatsevich, On some structure of Leibniz algebras, arXiv: 1302.3345v2

[14] G. Janelidze, L. Màrki and W. Tholen, “Semi-abelian categories”, J. Pure Appl. Algebra, 168 (2002), 367–386 | DOI | MR | Zbl

[15] J.-L. Loday, Cyclic homology, Grundl. Math. Wiss., 301, Springer, 1992 | MR | Zbl

[16] J.-L. Loday, “Une version non commutative des algèbres de Lie: les algèbres de Leibniz”, L'Enseignement Mathématique, 39 (1993), 269–292 | MR

[17] D. Towers, “Lie algebras all of whose maximal subalgebras have codimension one”, Proc. Edin. Math. Soc., 24 (1981), 217–219 | DOI | MR | Zbl

[18] D. Towers, “Maximal subalgebras of Lie algebras containing Engel subalgebras”, J. Pure Appl. Algebra, 216 (2012), 688–693 | DOI | MR | Zbl