zbMATH — the first resource for mathematics

Leibniz algebras with absolute maximal Lie subalgebras. (English) Zbl 07268065
Summary: 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 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.
17A32 Leibniz algebras
17B55 Homological methods in Lie (super)algebras
18B99 Special categories
Full Text: Link
[1] R.K. Amayo:Quasi-ideals of Lie algebras II, Proc. Lond. Math. Soc. ,3(33), (1976), 37-64. · Zbl 0337.17005
[2] Sh. A. Ayupov and B. A. Omirov:On some classes of nilpotent Leibniz algebras, Sibirsk. Mat. Zh.,42(1), (2001), 18-29. · Zbl 1017.17002
[3] G. R. Biyogmam and J. M. Casas:On Lie-isoclinic Leibniz algebras, J. Algebra 499(2018), 337-357. · Zbl 1397.17003
[4] G. R. Biyogmam and J. M. Casas:Thec-Nilpotent ShurLie-Multiplier of Leibniz Algebras, J. Geom. Phys.138(2019), 55-69. · Zbl 07058909
[5] D. Barnes:On Levi’s theorem for Leibniz algebras, Bull. Aust. Math. Soc.86 (2012), 184-185 · Zbl 1280.17002
[6] J. M. Casas and E. Khmaladze:On Lie-central extensions of Leibniz algebras, RACSAM (2016), DOI 10.1007/s13398-016-0274-6. · Zbl 1387.17004
[7] J. M. Casas and T. Van der Linden:A relative theory of universal central extensions, Pré-Publicaçoes do Departamento de Matemàtica, Universidade de Coimbra Preprint Number09-(2009).
[8] J. M. Casas and M. A. Insua:The SchurLie-multiplier of Leibniz algebras, Quaestiones Mathematicae,41(2) (2018). · Zbl 06994516
[9] J. M. Casas and T. Van der Linden:Universal central extensions in semi-abelian categories, Appl. Categor. Struct.22(1) (2014), 253-268. · Zbl 1358.18005
[10] I. Demir,Classification of 5-Dimensional Complex Nilpotent Leibniz Algebras., Ph.D. Thesis, http://www.lib.ncsu.edu/resolver/1840.20/33418, 138 pages. · Zbl 1439.17003
[11] I. Demir, C. Kailash and E. Stitzinger:On classification of four-dimensional nilpotent Leibniz algebras, Comm. Algebra45(3) (2017), 1012-1018. · Zbl 1418.17007
[12] I. Demir, C. Kailash and E. Stitzinger:On some structure of Leibniz algebras, in Recent Advances in Representation Theory, Quantum Groups, Algebraic Geometry, and Related Topics, Contemporary Mathematics,623, Amer. Math. Soc., Providence, RI, 41-54 (2014).
[13] V. Gorbatsevich:On some structure of Leibniz algebras,arxiv:1302.3345v2. · Zbl 1377.17002
[14] G. Janelidze, L. Màrki and W. Tholen:Semi-abelian categories, J. Pure Appl. Algebra168(2002), 367-386. · Zbl 0993.18008
[15] J.-L. Loday:Cyclic homology, Grundl. Math. Wiss. Bd.301, Springer (1992). · Zbl 0780.18009
[16] J.-L. Loday:Une version non commutative des algèbres de Lie: les algèbres de Leibniz, L’Enseignement Mathématique39(1993), 269-292.
[17] D. Towers:Lie algebras all of whose maximal subalgebras have codimension one, Proc. Edin. Math. Soc.24(1981), 217-219. · Zbl 0466.17007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.