Abstract
We consider some the fundamental properties of the Leibniz algebras. Some results were known before, but in the paper they are proved by a single method of liezation—the transition to a Lie algebra, which gives for a number of cases greatly simplified proof. There are also some new results.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Blokh, A.M. “A Generalization of the Concept of a Lie Algebra”, Sov.Math., Dokl. 6, 1450–1452 (1965).
Blokh, A. M. “Cartan–Eilenberg Homology Theory for a Generalized Class of Lie Algebras”, Sov. Math., Dokl. 8, 824–826 (1967)
Loday, J.-L. “Une Version non Commutative des Algèbres de Lie: les Algèbres de Leibniz”, Enseign. Math., II Sér. 39, No. 3–4, 269–293 (1993).
Zinbiel, G.W. “Encyclopedia of Types of Algebras 2010”, in C. Bai, L. Guo, and J.-L. Loday. “Operads and Universal Algebra”, Proceedings of the Summer School and International Conference, Tianjin, China, July 5–9, 2010 (World Scientific, Hackensack, NJ. Nankai Series in Pure, Appl. Math. and Theor. Phys., 2012), Vol. 9, pp. 217–298.
Ayupov, Sh., Omirov, B. “On Leibniz Algebras”, in “Algebra and Operator Theory”. Proceedings of the Colloquium in Tashkent, 1997 (Kluwer Acad. Publ. Dordrecht–Boston–London, 1998), pp. 1–13.
Patsourakos, A. “On Nilpotent Properties of Leibniz Algebras”, Commun. Algebra 35, 3828–3834 (2007).
Bourbaki N. Lie Groups and Lie Algebras. Chapters 1–3 (Mir, Moscow, 1976) [Russ. transl.].
Albeverio, S., Ayupov, Sh., Omirov, B. “Cartan Subalgebras and Criterion of Solvability of Finite Dimensional Leibniz Algebras”, RevistaMatem. Complutense 19, 183–195 (2006).
Barnes, D. “On Levi’s Theorem for Leibniz Algebras”, Bull. Austral. Math. Soc. 86, 184–185 (2012).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © V.V. Gorbatsevich, 2016, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2016, No. 4, pp. 14–22.
About this article
Cite this article
Gorbatsevich, V.V. On liezation of the Leibniz algebras and its applications. Russ Math. 60, 10–16 (2016). https://doi.org/10.3103/S1066369X16040034
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1066369X16040034