On analogs of some group-theoretic concepts and results for Leibniz algebras

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: 
  1. 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
  2. Stewart, I. N. (1970). Infinite-dimensional Lie algebras in the spirit of infinite group theory. Compositio Matematica, 22, pp. 313-331.
  3. Bloh, A. M. (1965). On a generalization of the concept of Lie algebra. Dokl. AN SSSR, 165, pp. 471-473 (in Russian).
  4. Bloh, A. M. (1967). Cartan—Eilenberg homology theory for a generalized class of Lie algebras. Dokl. AN USSR, 175, pp. 266-268 (in Russian).
  5. 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).
  6. Loday, J. L. (1993). Une version non commutative des algèbres de Lie: les algèbres de Leibniz. Enseign. Math., 39, pp. 269-293.
  7. 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
  8. Maltsev, A. I. (1949). Nilpotent torsion-free groups. Izvestiya AN SSSR. Ser. Math.,13, pp. 201-212 (in Russian).
  9. 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
  10. Barnes, D. (2013). Schunck classes of soluble Leibniz algebras. Commun. Algebra, 41, pp. 4046-4065. doi: https://doi.org/10.1080/00927872.2012.700978
  11. 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
  12. HiIrsch, K. A. (1955). Über local-nilpotente Gruppen. Math. Z., 63, pp. 290-291. doi: https://doi.org/10.1007/BF01187939
  13. Plotkin, B. I. (1955). Radical groups. Math. sbornik, 37, pp. 507-526 (in Russian).
  14. Plotkin, B. I. (1958). Generalized soluble and generalized nilpotent groups. Uspekhi mat. nauk,13, pp. 89-172 (in Russian).
  15. Plotkin, B.I. (1951). To the theory of locally nilpotent groups. Dokl. AN SSSR, 76, pp. 655-657 (in Russian).