Leibniz algebras of dimension 3 over finite fields

1Yashchuk, VS
1Oles Honchar Dnipropetrovsk National University
Dopov. Nac. akad. nauk Ukr. 2018, 7:20-25
https://doi.org/10.15407/dopovidi2018.07.020
Section: Mathematics
Language: English
Abstract: 

The first thing in the study of all types of algebras is the description of algebras having small dimensions. Unlike the simpler cases of 1- and 2-dimensional Leibniz algebras, the structure of 3-dimensional Leibniz algebras is more complicated. We consider the structure of Leibniz algebras of dimension 3 over a finite field. In some cases, the structure of the algebra essentially depends on the characteristic of the field. In others, it depends on the solvability of specific equations in the field, and so on.

Keywords: factor-algebra, ideal, Leibniz algebra, nilpotent Leibniz algebra
References: 
  1. Bloh, A. M. (1965). On a generalization of the concept of Lie algebra. Dokl. AN SSSR, 165, No. 3, pp. 471-473.
  2. Bloh, A. M. (1967). Cartan—Eilenberg homology theory for a generalized class of Lie algebras. Dokl. AN SSSR, 175, No. 8, pp. 824-826.
  3. Bloh, A. M. (1971). A certain generalization of the concept of Lie algebra. Algebra and number theory. Uchenye Zapiski Moskov. Gos. Pedagog. Inst., 375, pp. 9-20.
  4. Loday, J.-L. (1993). Une version non commutative des algèbre de Lie: les algèbre de Leibniz. Enseign. Math., 39, pp. 269-293.
  5. Loday, J.-L. (1998). Cyclic homology. Grundlehren der Mathematischen Wissenschaften, Vol. 301. 2nd ed. Berlin: Springer.
  6. Kirichinko, V. V., Kurdachenko, L. A., Pypka, A. A. & Subbotin, I. Ya. (2017). The some aspects of Leibniz algebra theory. Algebra Discrete Math., 24, No. 1, pp. 1-33.
  7. Albeverio, S., Omirov, B. A. & Rakhimov, I. S. (2005). Varieties of nilpotent complex Leibniz algebras of dimension less than five. Commun. Algebra, 33, No. 5, pp. 1575-1585.
  8. Ayupov, S. A. & Omirov, B. A. (1999). On 3-dimensional Leibniz algebras. Uzbek. Math. Zh., 1, pp. 9-14.
  9. Casas, J. M., Insua, M. A., Ladra, M. & Ladra, S. (2012). An algorithm for the classification of 3-dimensional complex Leibniz algebras. Linear Algebra Appl., 436, No. 9, pp. 3747-3756.
  10. Demir, I., Misra, K. C. & Stitzinger, E. (2014). On some structures of Leibniz algebras. Recent advances in representation theory, quantum groups, algebraic geometry, and related topics. Contemporary mathematics, Vol. 623, pp. 41-54.
  11. Kurdachenko, L. A., Otal, J. & Pypka, A. A. (2016). Relationships between factors of canonical central series of Leibniz algebras. Eur. J. Math., 2, No. 2, pp. 565-577.
  12. Barnes, D. (2011). Some theorems on Leibniz algebras. Commun. Algebra, 39, No. 7, pp. 2463-2472.