Classification of n-dimensional generalized Heisenberg Lie algebras of rank (n – 4)
*Hamid DarabiCorresponding authordarabi@esfarayen.ac.irDepartment of MathematicsEsfarayen University of TechnologyEsfarayen, 9661998195, Iran0000-0001-8465-1756View full profile → , Mahdi Imanparastm.imanparast@ub.ac.irDepartment of Computer ScienceUniversity of BojnordBojnord, 9453155111, IranView full profile →
* Corresponding author · click or hover a name for details
- Received:
- 01 May 2025
- Published Online:
- 03 Oct 2026
- Article type:
- Research Article
- Language:
- EN
- Article no.:
- JDMSC-2611
- Pages:
- 1–9
Abstract
In this paper, we classify all n-dimensional generalized Heisenberg Lie algebras of rank (n – 4), and to this end, we demonstrate that for any n-dimensional generalized Heisenberg Lie algebra, if the rank is (n – 4), then such an algebra does not exist when n ⩾ 11. We also show that for a n-dimensional 2-step nilpotent Lie algebra L with the derived subalgebra of dimension r, we have r ⩽ n + [1-√8n+1/2].
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References
[1] E. Cartan, Sur la Reduction a sa Forme Canonique de la Structure dun Groupe de Transformations Finiet Continu, Amer. J. Math., vol. 18, no. 1, pp. 1-61 (1896).
[2] F. Gantmacher, On the classification of real simple Lie groups, Rec. Math. [Mat. Sbornik] N.S., vol. 5, no. 47, pp. 217-250 (1939).
[3] G. M. Mubarakzyanov, Classification of real structures of lie algebras of fifth order, Izv. Vysshikh Uchebn. Zavedeneii Mat., vol. 3, no. 34, pp. 99-106 (1963).
[4] G. M. Mubarakzyanov, Classification of solvable lie algebras of sixth order with a non-nilpotent basis element, Izv. Vysshikh Uchebn. Zavedeneii Mat., vol. 4, no. 35, pp. 104-116 (1963)
[5] K. A. Umlauf, Ueber den Zusammenhang der endlichen continuirlichen Transformation-sgruppen, insbesondere der Gruppen vom Range Null. Ph.D. Thesis, University of Leipzig, (In German) (1891).
[6] V. V. Morozov, Classification of nilpotent Lie algebras of sixth order, Izv. Vyssh. Uchebn. Zaved. Mat., vol. 4, pp. 161-171 (1958).
[7] S. Cicalo, W. A. de Graaf and C. Schneider, Six-dimensional nilpotent Lie algebras, Linear Algebra Appl., vol. 436 , no. 1, pp. 163-189 (2012).
[8] C. Seeley, 7-dimensional nilpotent Lie algebras, Trans. Amer. Math. Soc., vol. 335, no. 2, pp. 479-496 (1993).
[9] B. Ren and D. J. Meng, Some 2-step nilpotent Lie algebras I, Linear Algebra Appl., vol. 338, pp. 77-98 (2001).
[10] B. Ren and L. S. Zhu, Classification of 2-step nilpotent Lie algebras of dimension 9 with 2-dimensional center, Czechoslovak Math. J., vol. 67 , no. 3, pp. 953-965 (2017).
[11] B. Ren and L. S. Zhu, Classification of 2-step nilpotent Lie algebras of dimension 8 with 2-dimensional center, Comm. Algebra, vol. 39, no. 6, pp. 2068-2081 (2011).
[12] H. Darabi and M. Imanparast, On classification of 9-dimensional nilpotent 3-ary algebras of class two, Bull. Iranian Math. Soc., vol. 47, pp. 929-937 (2021).
[13] M. A. Alvarez, Degenerations of 8-dimensional 2-step nilpotent Lie algebras, Algebr. Represent. Theory, vol. 24, no. 5, pp. 1231-1243 (2021).
[14] D. Bensikaddour and L. Belarbi, Minimal translation surfaces in Lorentz Heisenberg space (H3, g2). J. Interdiscip. Math., vol. 24, no. 4, pp. 881-896 (2021).
[15] B. Senoussi, Helicoidal surfaces of finite type in the 3-dimensional Heisenberg group. J. Interdiscip. Math., vol. 25, no. 4, pp. 1143-1152 (2022).
[16] S. O. Bedaiwi, A computational Technique for Studying the Heisenberg Group, J. Interdiscip. Math., vol. 27, no. 3, pp. 457-466 (2023).
[17] W. A. de Graaf, Classification of 6-dimensional nilpotent Lie algebras over fields of characteristic not 2. J. Algebra, vol. 309, no. 2, pp. 640-653 (2007).
[18] M. P. Gong, Classification of Nilpotent Lie Algebras of Dimension 7 (Over Algebraically Closed Fields and R), Ph.D. Thesis, University of Waterloo, Waterloo, (1998).
[19] Z. Yan and S. Deng, The classification of two step nilpotent complex Lie algebras of dimension 8, Czechoslovak Math. J. vol. 63, no. 3, pp. 847-863 (2013).
[20] M. Eshrati, F. Saeedi and H. Darabi, On the multiplier of nilpotent n-Lie algebras, J. Algebra, vol. 450, pp. 162-172 (2016).
[21] K. Moneyhun, Isoclinisms in Lie algebras, Algebras, Groups, Geometries, vol.11, pp. 9-22 (1994).
[22] Z. Hoseini, F. Saeedi and H. Darabi, Characterization of capable nilpotent n-Lie algebras of class two by their Schur multipliers, Rend. Circ. Mat. Palermo. (2), vol. 68, pp. 541-555 (2019).
