On 7-Dimensional Nilpotent Leibniz Algebras With 1-Dimensional Leib Ideal

Abstract

Leibniz algebras are nonanticommutative versions of Lie algebras. Lie algebras have many applications in many scientific areas as well as mathematical areas. Scientists from different disciplines have used specific examples of Lie algebras according to their needs. However, we mathematicians are more interested in generality than in obtaining a few examples. The classification problem for Leibniz algebras has an intrinsically wild nature as in Lie algebras. In this article, the approach of congruence classes of bilinear forms is extended to classify certain subclasses of seven-dimensional nilpotent Leibniz algebras over complex numbers. Certain cases of seven-dimensional complex nilpotent Leibniz algebras of those with one-dimensional Leib ideal and derived algebra of codimension two are classified.

Keywords

• Bilinear forms
• Classification
• Leibniz algebra
• Nilpotency

References

1. [1]. Bloh, A. 1965. On a Generalization of Lie Algebra Notion. Mathematics in USSR Doklady; 165(3): 471-473.
2. [2]. Loday, JL. 1993. Une Version Non-Commutative des Algebres de Lie: Les Algebres de Leibniz. L'Enseignement Mathematique; 39(3-4): 269-293.
3. [3]. Albeverio, S, Omirov, BA, Rakhimov, IS. 2006. Classification of 4-Dimensional Nilpotent Complex Leibniz Algebra. Extracta Mathematicae; 21(3): 197-210.
4. [4]. Rakhimov, IS, Bekbaev, UD. 2010. On Isomorphisms and Invariants of Finite Dimensional Complex Filiform Leibniz algebras. Communications in Algebra; 38: 4705-4738.
5. [5]. Casas, JM, Insua, MA, Ladra, M, Ladra, S. 2012. An Algorithm for the Classification of 3-Dimensional Complex Leibniz Algebras. Linear Algebra and its Applications; 9: 3747-3756.
6. [6]. Abdulkareem, AO, Rakhimov, IS, Husain, SK. On Seven-Dimensional Filiform Leibniz Algebras, In: Kilicman, A., Leong, W., Eshkuvatov, Z. (eds) International Conference on Mathematical Sciences and Statistics, 2014, pp 1-11.
7. [7]. Gomez, JR, Omirov, BA. 2015. On Classification of Filiform Leibniz Algebras. Algebra Colloquium; 22: 757-774.
8. [8]. Demir, I, Misra, KC, Stitzinger, E. 2017. On Classification of Four-Dimensional Nilpotent Leibniz Algebras. Communications in Algebra; 45(3): 1012-1018.

9. [9]. Rakhimov, IS, Khudoyberdiyev, AK, Omirov, BA. 2017. On Isomorphism Criterion for a Subclass of Complex Filiform Leibniz Algebras. International Journal of Algebra and Computation; 27(7): 953-972.
10. [10]. Demir, I. Classification of 5-Dimensional Complex Nilpotent Leibniz Algebras, In: N. Jing, K. C. Misra (Eds.), Representations of Lie Algebras, Quantum Groups and Related Topics, Contemporary Mathematics, Volume 713, American Mathematical Society, 2018, pp. 95-120.
11. [11]. Mohamed, NS, Husain, SK, Rakhimov, IS. 2019. Classification of a Subclass of 10-Dimensional Complex Filiform Leibniz Algebras. Malaysian Journal of Mathematical Sciences; 13(3): 465-485.
12. [12]. Demir, I. 2020. Classification of Some Subclasses of 6-Dimensional Nilpotent Leibniz Algebras. Turkish Journal of Mathematics; 44: 1925-1940.
13. [13]. Farris, L. Finite Dimensional Nilpotent Leibniz Algebras with Isomorphic Maximal Algebras, Doctoral Dissertation, North Carolina State University, 2022.
14. [14]. Demir, I. On Classification of 7-Dimensional Odd-Nilpotent Leibniz Algebras. Hacettepe Journal of Mathematics and Statistics, (in press).
15. [15]. Teran, F. 2016. Canonical Forms for Congruence of Matrices and T-palindromic Matrix Pencils: a Tribute to H. W. Turnbull and A. C. Aitken. SeMA Journal: Bulletin of the Spanish Society of Applied Mathematics; 73: 7-16.
16. [16]. Gong, MP. Classification of Nilpotent Lie Algebras of Dimension 7 (over Algebraically Closed Field 𝔽 and ℝ), Doctoral Dissertation, University of Waterloo, 1998.