1Kurdachenko, LA 2Subbotin, IYa. 3Semko, NN 1Oles Honchar Dnipropetrovsk National University 2National University, Los Angeles, USA 3State Tax Service National University of Ukraine, Irpin |
Dopov. Nac. akad. nauk Ukr. 2018, 1:10-14 |
https://doi.org/10.15407/dopovidi2018.01.010 |
Section: Mathematics |
Language: English |
Abstract: An algebra L over a field F is said to be a Leibniz algebra (more precisely a left Leibniz algebra) if it satisfies the Leibniz identity: [[a, b], c] = [a, [b, c]] – [b, [a, c]] for all a, b, c ∈ L. Leibniz algebras are generalizations of Lie algebras. We consider some classes of generalized nilpotent Leibniz algebras (hypercentral, locally nilpotent algebras, and algebras with the idealizer condition) and show their some basic properties. |
Keywords: ascendant subalgebra, center of a Leibniz algebra, hypercentral Leibniz algebra, ideal, idealizer condition, left center, Leibniz algebra, Leibniz algebra with the idealizer condition, Lie algebra, locally nilpotent Leibniz algebra, right center |
References:
- Amayo, R. K. & Stewart, I. N. (1974). Infinite-dimensional Lie algebras. Leyden: Noordhoff Intern. Publ. doi: https://doi.org/10.1007/978-94-010-2305-4
- Stewart, I. N. (1970). Infinite-dimensional Lie algebras in the spirit of infinite group theory. Compositio Matematica, 22, pp. 313-331.
- Bloh, A. M. (1965). On a generalization of the concept of Lie algebra. Dokl. AN SSSR, 165, pp. 471-473 (in Russian).
- Bloh, A. M. (1967). Cartan—Eilenberg homology theory for a generalized class of Lie algebras. Dokl. AN USSR, 175, pp. 266-268 (in Russian).
- Bloh, A. M. (1971). A certain generalization of the concept of Lie algebra. Uchenye Zapiski Moskov. Gos. Pedagog. Inst., 375, pp. 9-20 (in Russian).
- Loday, J. L. (1993). Une version non commutative des algèbres de Lie: les algèbres de Leibniz. Enseign. Math., 39, pp. 269-293.
- Kurdachenko, L. A., Otal, J. & Pypka, A. A. (2016). Relationships between factors of canonical central series of Leibniz algebras. Eur. J. Math., 2, pp. 565-577. doi: https://doi.org/10.1007/s40879-016-0093-5
- Maltsev, A. I. (1949). Nilpotent torsion-free groups. Izvestiya AN SSSR. Ser. Math.,13, pp. 201-212 (in Russian).
- Chupordya, V. A., Kurdachenko, L. A. & Subbotin, I. Ya. (2017). On some “minimal” Leibniz algebras. J. Algebra and Appl., 16, 1750082 doi: https://doi.org/10.1142/S0219498817500827
- Barnes, D. (2013). Schunck classes of soluble Leibniz algebras. Commun. Algebra, 41, pp. 4046-4065. doi: https://doi.org/10.1080/00927872.2012.700978
- Hartley, B. (1967). Locally nilpotent ideals of a Lie algebras. Proc. Cambridge Phil. Soc., 63, pp. 257-272. doi: https://doi.org/10.1017/S0305004100041177
- HiIrsch, K. A. (1955). Über local-nilpotente Gruppen. Math. Z., 63, pp. 290-291. doi: https://doi.org/10.1007/BF01187939
- Plotkin, B. I. (1955). Radical groups. Math. sbornik, 37, pp. 507-526 (in Russian).
- Plotkin, B. I. (1958). Generalized soluble and generalized nilpotent groups. Uspekhi mat. nauk,13, pp. 89-172 (in Russian).
- Plotkin, B.I. (1951). To the theory of locally nilpotent groups. Dokl. AN SSSR, 76, pp. 655-657 (in Russian).