Abstract
In this paper, we compare the abelian subalgebras and ideals of maximal dimension for finite-dimensional Leibniz algebras. We study Leibniz algebras containing abelian subalgebras of codimension 1, solvable and supersolvable Leibniz algebras for codimensions 1 and 2, nilpotent Leibniz algebras in case of codimension 2, and we also analyze the case of k-abelian p-filiform Leibniz algebras. Throughout the paper, we also give examples to clarify some results and the need for restrictions on the underlying field.
Similar content being viewed by others
Data Availability Statement
This manuscript has no associated data.
References
Albeverio, S., Omirov, B.A., Rakhimov, I.S.: Classification of 4-dimensional nilpotent complex leibniz algebras. Extracta Math. 21(3), 197–210 (2006)
Alvarez, M.A.: On rigid 2-step nilpotent Lie algebras. Algebra Colloq. 25(2), 349–360 (2018)
Ayupov, Sh.A., Omirov, B.A.: On some classes of nilpotent Leibniz algebras. Sib. Math. J. 42(1), 15–24 (2001)
Ayupov, S., Omirov, B., Rakhimov, I.: Leibniz Algebras, Structure and Classification. CRC Press, Taylor and Francis Group (2020)
Barnes, D.W.: Some theorems on Leibniz algebras. Comm. Algebra 39(7), 2463–2472 (2011)
Burde, D., Ceballos, M.: Abelian ideals of maximal dimension for solvable Lie algebras. J. Lie Theory 22(3), 741–756 (2012)
Burde, D., Steinhoff, C.: Classification of orbit closures of \(4\)-dimensional complex Lie algebras. J. Algebra 214(2), 729–739 (1999)
Cañete, E.M., Khudoyberdiyev, A.K.: The classification of \(4\)-dimensional Leibniz algebras. Linear Algebra Appl. 439(1), 273–288 (2013)
Casas, J.M., Insua, M.A., Ladra, M., Ladra, S.: An algorithm for the classification of \(3\)-dimensional complex Leibniz algebras. Linear Algebra Appl. 436(9), 3747–3756 (2012)
Casas, J.M., Khudoyberdiyev, AKh., Ladra, M., Omirov, B.A.: On the degenerations of solvable Leibniz algebras. Linear Alg. Appl. 439, 472–487 (2013)
Ceballos, M.: Abelian subalgebras and ideals of maximal dimension in Lie algebras. Ph.D. Thesis, University of Seville, (2012)
Ceballos, M., Núñez, J., Tenorio, A.F.: Algorithmic procedure to compute abelian subalgebras and ideals of maximal dimension of Leibniz algebras. Int. J. Comput. Math. 92(9), 1838–1854 (2015)
Ceballos, M., Towers, D.A.: On abelian subalgebras and ideals of maximal dimension in supersolvable Lie algebras. J. Pure Appl. Algebra 218(3), 497–503 (2014)
Demir, I.: Classification of some subclasses of \(6\)-dimensional nilpotent Leibniz algebras. Turk. J. Math. 44, 1925–1940 (2020)
Fialowski, A., Mihálka, E.: Representations of Leibniz algebras. Algebr. Represent. Theory 18, 477–490 (2015)
Gómez, J.R., Omirov, B.: On classification of filiform Leibniz algebras. Algebra Colloq. 22, 757–774 (2015)
Ismailov, N., Kaygorodov, I., Volkov, Y.: Degenerations of Leibniz and anticommutative algebras. Can. Math. Bull. 62(3), 539–549 (2022)
Jacobson, N.: Schur’s theorems on commutative matrices. Bull. Am. Math. Soc. 50, 431–436 (1944)
Khudoyberdiyev, A., Rakhimov, I.S., Said Husain, K.: On classification of 5-dimensional solvable Leibniz algebras. Linear Algebra Appl. 457, 428–454 (2014)
Khudoyberdiyev, A., Shermatova, Z.: Description of solvable Leibniz algebras with four-dimensional nilradical. Contemp. Math. 672, 217–224 (2016)
Loday, J.L.: Une version non commutative des algèbres de Lie: les algèbres de Leibniz. Enseign. Math. 2(39), 269–293 (1993)
Mohamed, N.S., Husain, S.K.S., Yunos, F.: On degenerations and invariants of low-dimensional complex nilpotent Leibniz algebras. Math. Stat. 8(2), 95–99 (2020)
Nesterenko, M., Popovych, R.: Contractions of low-dimensional Lie algebras. J. Math. Phys. 47(12), 123515 (2006)
Rakhimov, I.: On degenerations of finite-dimensional nilpotent complex Leibniz algebras. J. Math. Sci 136(3) (2006), translated from Zapiski Nauchnykh Seminarov POMI 321, 268–274 (2005)
Seegal, I.E.: A class of operator algebras which are determined by groups. Duke Math. J. 18, 221–265 (1951)
Seeley, C.: Degenerations of 6-dimensional nilpotent Lie algebras over \({\mathbb{C}}\). Commun. Algebra 18, 3493–3505 (1990)
Towers, D.A.: Abelian subalgebras and ideals of maximal dimension in supersolvable and nilpotent lie algebras. Linear Multilinear Algebra (2020). https://doi.org/10.1080/03081087.2020.1805399
Towers, D.A.: Quasi-ideals of Leibniz algebras. Commun. Algebra 48(11), 4569–4579 (2020)
Acknowledgements
The paper was partially supported by US-1262169, P20_01056, MTM2016-75024-P and FEDER.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Ceballos, M., Towers, D.A. Abelian Subalgebras and Ideals of Maximal Dimension in Solvable Leibniz Algebras. Mediterr. J. Math. 20, 97 (2023). https://doi.org/10.1007/s00009-023-02306-4
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00009-023-02306-4