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Proposed Method for Expressing Inner Derivations of Leibniz Algebras

Nadia Faiq Mohammed
https://doi.org/10.35877/454RI.asci4626

Abstract

This work is concerned with the computational perspective of a specific case of derivations, known as inner derivation mapping. In this regard, we propose an algebraic method to provide inner derivations for a finite-dimensional Leibniz algebra in matrix representation. The method is applied on four-dimensional complex Leibniz algebras obtained earlier to present comprehensive descriptions of their inner derivations. Additionally, we also exhibit the span basis of outer derivatives for these algebras.

 

Keywords

Inner derivation, Leibniz algebra, and nilpotence

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References

  1. Ayupov, S., Omirov, B., & Rakhimov, I. (2019). Leibniz algebras: structure and classification. Chapman and Hall/CRC.
    Abdelwahab, H., Calderón, A. J., & Kaygorodov, I. (2019). The algebraic and geometric classification of nilpotent binary Lie algebras. International Journal of Algebra and Computation, 29(06), 1113-1129.
    Abdou, A. Z., and Mosbahi, B. (2025). Computational Methods for Biderivations of 4-dimensional nilpotent complex leibniz algebras. arXiv preprint arXiv:2501.10887.
    Almutairi, H., & AbdGhafur, A. (2018). Derivations of some classes of Zinbiel algebras. International Journal of Pure and Applied Mathematics, 2, 12-13.
    Ayala, V., Kizil, E., & de Azevedo Tribuzy, I. (2012). On an algorithm for finding derivations of Lie algebras. Proyecciones (Antofagasta), 31(1), 81-90.
    Bermúdez, J. A., & Campoamor-Stursberg, R. (2013). On a complete rigid Leibniz non-Lie algebra in arbitrary dimension. Linear Algebra and its Applications, 438(8), 3397-3407.
    Bloh, A. (1965). On a generalization of Lie algebra notion. In Math. in USSR Doklady (Vol. 165, No. 3, pp. 471-473).
    Ismailov, N., Kaygorodov, I., & Volkov, Y. (2018). The geometric classification of Leibniz algebras. International Journal of Mathematics, 29(05), 1850035.
    Loday, J. L. (1993). Une version non commutative des algebres de Lie: les algebres de Leibniz. Les rencontres physiciens-mathématiciens de Strasbourg-RCP25, 44, 127-151.
    Mohammed, N. F., Rakhimov, I. S., & Husain, S. K. S. (2017, January). Contractions of low dimensional complex associative algebras. In AIP Conference Proceedings (Vol. 1795, No. 1, p. 020022). AIP Publishing LLC.
    Mohammed, N. F., Rakhimov, I. S., & Hussain, S. K. S. (2017, April). Cohomology spaces of low dimensional complex associative algebras. In The 4th International Conference on Mathematical Sciences: Mathematical Sciences: Championing the Way in a Problem Based and Data Driven Society (Vol. 1830, No. 1, p. 070031).
    Mohammed, N. F. , Qasim, Hanan F. & Maibed, Z. H. (2025) Outer derivations of low-dimensional diassociative algebras, Journal of Discrete Mathematical Sciences and Cryptography, 28:4-B, 1369–1373, DOI: 10.47974/JDMSC-2276
    Nesterenko, M., & Popovych, R. (2006). Contractions of low-dimensional Lie algebras. Journal of mathematical physics, 47(12).
    Rakhimov, I. S., & Al-Hossain, A. N. (2011). On derivations of low-dimensional complex Leibniz algebras. JP Journal of Algebra, Number Theory and Applications, 21(1), 69-81.
    Rakhimov, I. S., & Atan, K. A. (2012). On contractions and invariants of Leibniz algebras. Bull. Malays. Math. Sci. Soc.(2), 35(2A), 557-565.

How to Cite

Mohammed, N. F. (2026). Proposed Method for Expressing Inner Derivations of Leibniz Algebras. Journal of Applied Science, Engineering, Technology, and Education, 8(1), 130–137. https://doi.org/10.35877/454RI.asci4626

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Copyright (c) 2026 Nadia Faiq Mohammed (Author)
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