On classification of $7$-dimensional odd-nilpotent Leibniz algebras

Year 2024, Volume: 53 Issue: 1, 121 - 129, 29.02.2024

İsmail Demir

https://doi.org/10.15672/hujms.1185538

Abstract

In this paper we extend the method of canonical form for congruence of bilinear forms to give the classification of some subclasses of $7-$dimensional nilpotent Leibniz algebras. Odd-nilpotent Leibniz algebras are defined as that its even dimensional ideals in lower central series are all zero and the classification of $7-$dimensional complex odd-nilpotent Leibniz algebras with one dimensional Leib ideal is obtained by applying the aforementioned method.

Keywords

Leibniz algebra , nilpotency , classification

References

• [1] S. Albeverio, B. A. Omirov and I. S. Rakhimov, Classification of 4-dimensional nilpotent complex Leibniz algebras, Extracta Mathematicae 21(3), 197-210, 2006.
• [2] S. A. Ayupov and B. A. Omirov, On 3-dimensional Leibniz algebras, Uzbek. Math. Zh. 1, 9-14, 1999.
• [3] A. Bloh, On a generalization of Lie algebra notion, Math. in USSR Doklady 165(3), 471-473, 1965.
• [4] J.M. Casas, M.A. Insua, M. Ladra and S. Ladra, An algorithm for the classification of 3-dimensional complex Leibniz algebras, Linear Algebra Appl. 9, 3747-3756, 2012. DOI: 10.1016/j.laa.2011.11.039
• [5] I. Demir, K. C. Misra and E. Stitzinger, On classification of four-dimensional nilpotent Leibniz Algebras, Comm. Algebra 45(3), 1012-1018, 2017. DOI: 10.1080/00927872.2016.1172626.
• [6] I. Demir, Classification of 5−dimensional complex nilpotent Leibniz algebras, in Representations of Lie Algebras, Quantum Groups and Related Topics, 95-120, Contemp. Math. 713, Amer. Math. Soc., Providence, RI, 2018.
• [7] I. Demir, Classification of some subclasses of 6-dimensional nilpotent Leibniz algebras., Turk J. Math. 44, 1925-1940, 2020.
• [8] J. L. Loday, Une version non commutative des algebres de Lie: les algebres de Leibniz, Enseign. Math. (2) 39(3-4), 269-293, 1993.
• [9] I. M. Rikhsiboev and I. S. Rakhimov, Classification of three dimensional complex Leibniz algebras, AIP Conference Proc. 1450, 358-362, 2012. DOI: 10.1063/1.4724168.
• [10] F. D. Teran, Canonical forms for congruence of matrices: a tribute to H. W. Turnbull and A. C. Aitken, 2nd meeting on Linear Algebra, Matrix Analysis and Applications, (ALAMA 2010), Valencia(Spain), 2010.