Some aspects of Leibniz algebra theory
Algebra and discrete mathematics, Tome 24 (2017) no. 1, pp. 1-33.

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One of the key tendencies in the development of Leibniz algebra theory is the search for analogues of the basic results of Lie algebra theory. In this survey, we consider the reverse situation. Here the main attention is paid to the results reflecting the difference of the Leibniz algebras from the Lie algebras.
Mots-clés : Leibniz algebra, cyclic Leibniz algebra, left (right) center, lower (upper) central series, finite dimensional Leibniz algebra, nilpotent Leibniz algebra, extraspecial Leibniz algebra, bilinear form, left (right) idealizer, Frattini subalgebra, nil-radical, nil-algebra, soluble Leibniz algebra, left (right) subideal, Leibniz $T$-algebra, Baer radical.
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Vladimir V. Kirichenko; Leonid A. Kurdachenko; Aleksandr A. Pypka; Igor Ya. Subbotin. Some aspects of Leibniz algebra theory. Algebra and discrete mathematics, Tome 24 (2017) no. 1, pp. 1-33. https://geodesic-test.mathdoc.fr/item/ADM_2017_24_1_a1/

[1] S. Albeverio, B. A. Omirov, I. S. Rakhimov, “Varieties of nilpotent complex Leibniz algebras of dimension less than five”, Comm. Algebra, 33:5 (2005), 1575–1585 | DOI | MR | Zbl

[2] S. Albeverio, B. A. Omirov, I. S. Rakhimov, “Classification of 4-dimensional nilpotent complex Leibniz algebras”, Extracta Math., 21:3 (2006), 197–210 | MR | Zbl

[3] R. K. Amayo, I. Stewart, Infinite dimensional Lie algebras, Noordhoff Intern. Publ., Leyden, 1974 | MR | Zbl

[4] S. A. Ayupov, B. A. Omirov, “On Leibniz algebras”, Algebra and Operator Theory, Proceedings of the Colloquium in Tashkent, 1997, Springer Netherlands, 1998, 1–12 | MR

[5] S. A. Ayupov, B. A. Omirov, “On 3-dimensional Leibniz algebras”, Uzbek. Math. Zh., 1 (1999), 9–14 | MR

[6] R. Baer, “Situation der Untergruppen und Struktur der Gruppe”, S.-B. Heidelberg Acad. Math.-Nat. Klasse, 2 (1933), 12–17

[7] R. Baer, “Endlichkeitskriterien für Kommutatorgruppen”, Math. Ann., 124:1 (1952), 161–177 | DOI | MR | Zbl

[8] D. Barnes, “Some theorems on Leibniz algebras”, Comm. Algebra, 39:7 (2011), 2463–2472 | DOI | MR | Zbl

[9] D. Barnes, “On Engel's theorem for Leibniz algebras”, Comm. Algebra, 40:4 (2012), 1388–1389 | DOI | MR | Zbl

[10] D. Barnes, “On Levi's theorem for Leibniz algebras”, Bull. Aust. Math. Soc., 86:2 (2012), 184–185 | DOI | MR | Zbl

[11] D. Barnes, “Schunck Classes of soluble Leibniz algebras”, Comm. Algebra, 41:11 (2013), 4046–4065 | DOI | MR | Zbl

[12] C. Batten, L. Bosko-Dunbar, A. Hedges, J. T. Hird, K. Stagg, E. Stitzinger, “A Frattini theory for Leibniz algebras”, Comm. Algebra, 41:4 (2013), 1547–1557 | DOI | MR | Zbl

[13] A. M. Bloh, “On a generalization of the concept of Lie algebra”, Dokl. Akad. Nauk SSSR, 165:3 (1965), 471–473 | MR

[14] A. M. Bloh, “Cartan-Eilenberg homology theory for a generalized class of Lie algebras”, Dokl. Akad. Nauk SSSR, 175:8 (1967), 824–826 | MR

[15] A. M. Bloh, “A certain generalization of the concept of Lie algebra”, Algebra and number theory. Moskov. Gos. Ped. Inst. Uchen. Zap., 375 (1971), 9–20 | MR

[16] J. Butterfield, C. Pagonis, From Physics to Philosophy, Cambridge Univ. Press, Cambridge, 1999 | MR | Zbl

[17] E. M. Caňete, A. Kh. Khudoyberdiyev, “The Classification of 4-dimensional Leibniz algebras”, Linear Algebra Appl., 439:1 (2013), 273–288 | DOI | MR

[18] J. M. Casas, M. A. Insua, M. Ladra, S. Ladra, “An algorithm for the classification of 3-dimensional complex Leibniz algebras”, Linear Algebra Appl., 436:9 (2012), 3747–3756 | DOI | MR | Zbl

[19] V. A. Chupordya, L. A. Kurdachenko, I. Ya. Subbotin, “On some “minimal” Leibniz algebras”, J. Algebra Appl., 16:2 (2017) | MR

[20] I. Demir, K. C. Misra, E. Stitzinger, “On some structures of Leibniz algebras”, Recent Advances in Representation Theory, Quantum Groups, Algebraic Geometry, and Related Topics, Contemporary Mathematics, 623, 2014, 41–54 | DOI | MR | Zbl

[21] I. Demir, K. C. Misra, E. Stitzinger, “Classification of some solvable Leibniz algebras”, Algebr. Represent. Theor., 19:2 (2016), 405–417 | DOI | MR | Zbl

[22] I. Demir, K. C. Misra, E. Stitzinger, “On classification of four-dimensional nilpotent Leibniz algebras”, Comm. Algebra, 45:3 (2017), 1012–1018 | DOI | MR | Zbl

[23] Lie Theory and its applications in physic, IX International workshop, ed. V. Dobrev, Springer, Tokyo, 2013 | MR

[24] Noncommutative Structures in Mathematics and Physics, Proceedings of the NATO advanced research workshop, eds. S. Duplij, J. Wess, Springer, Kiev, 2001 | MR

[25] R. Farnsteiner, “On the structure of simple-semiabelian Lie algebras”, Pacific J. Math., 111:2 (1984), 287–299 | DOI | MR | Zbl

[26] A. Gejn, “Minimal noncommutative and minimal nonabelian algebras”, Comm. Algebra, 13:2 (1985), 305–328 | DOI | MR | Zbl

[27] A. Gejn, S. V. Kuznetsov, Yu. N. Mukhin, “On minimal non nilpotent Lie algebras”, Ural. Gos. Univ. Mat. Zap., 8:3 (1972), 18–27 | MR

[28] A. Gejn, Yu. N. Mukhin, “Complements to subalgebras of Lie algebras”, Ural. Gos. Univ. Mat. Zap., 12:2 (1980), 24–48 | MR | Zbl

[29] V. V. Gorbatsevich, “On liezation of the Leibniz algebras and its applications”, Russian Math., 60:4 (2016), 10–16 | DOI | MR | Zbl

[30] L. A. Kurdachenko, J. Otal, A. A. Pypka, “Relationships between factors of canonical central series of Leibniz algebras”, Eur. J. Math., 2:2 (2016), 565–577 | DOI | MR | Zbl

[31] L. A. Kurdachenko, N. N. Semko, I. Ya. Subbotin, “The Leibniz algebras whose subalgebras are ideals”, Open Math., 15:1 (2017), 92–100 | DOI | MR | Zbl

[32] L. A. Kurdachenko, I. Ya. Subbotin, V. S. Yashchuk, Leibniz Algebras Whose Subideals are Ideals, to appear | MR

[33] J.-L. Loday, “Une version non commutative des algèbres de Lie: les algèbras de Leibniz”, L'Enseignement Mathèmatique, 39 (1993), 269–293 | MR | Zbl

[34] J.-L. Loday, Cyclic homology, Grundlehren der Mathematischen Wissenschaften, 301, 2nd ed., Springer-Verlag, Berlin, 1998 | DOI | MR | Zbl

[35] J.-L. Loday, “Dialgebras”, Dialgebras and Related Operads, Lecture Notes in Math., 1763, Springer, Berlin, 2001, 7–66 | DOI | MR | Zbl

[36] B. H. Neumann, “Groups with finite classes of conjugate elements”, Proc. Lond. Math. Soc., 3:1 (1951), 178–187 | DOI | MR | Zbl

[37] A. Patsourakos, “On nilpotent properties of Leibniz algebras”, Comm. Algebra, 35:12 (2007), 3828–3834 | DOI | MR | Zbl

[38] C. B. Ray, A. Combs, N. Gin, A. Hedges, J. T. Hird, L. Zack, “Nilpotent Lie and Leibniz algebras”, Comm. Algebra, 42:6 (2014), 2404–2410 | DOI | MR | Zbl

[39] I. M. Rikhsiboev, I. S. Rakhimov, “Classification of three dimensional complex Leibniz algebras”, AIP Conference Proc., 1450 (2012), 358–362 | DOI

[40] D. Scofield, S. M. K. Sullivan, Classification of complex cyclic Leibniz algebras, 2014, arXiv: 1411.0170v2

[41] A. Shabanskaya, “Right and left solvable extensions of an associative Leibniz algebra”, Comm. Algebra, 45:6 (2017), 2633–2661 | DOI | MR | Zbl

[42] I. N. Stewart, Subideals of Lie algebras, Ph.D. Thesis, University of Warwick, 1969

[43] I. N. Stewart, “Verbal and marginal properties of non-associative algebras in the spirit of infinite group theory”, Proc. Lond. Math. Soc., 3:28 (1974), 129–140 | DOI | MR | Zbl

[44] E. Stitzinger, “Minimal non nilpotent solvable Lie algebras”, Proc. Amer. Math. Soc., 28:1 (1971), 47–49 | DOI | MR | Zbl

[45] D. Towers, “Lie algebras all whose proper subalgebras are nilpotent”, Linear Algebra Appl., 32 (1980), 61–73 | DOI | MR | Zbl

[46] S. Gómes-Vidal, A. Kh. Khudoyberdiyev, B. A. Omirov, “Some remarks on semisimple Leibniz algebras”, J. Algebra, 410 (2014), 526–540 | DOI | MR

[47] M. R. Vaughan-Lee, “Metabelian BFC $p$-groups”, J. Lond. Math. Soc., 5:4 (1972), 673–680 | DOI | MR | Zbl

[48] G. W. Zinbiel, “Encyclopedia of Types of Algebras”, Operads and Universal Algebra, Proceedings of the Summer School and International Conference, Tianjin, China, July 5–9, 2010, Nankai Series in Pure, Appl. Math. and Theor. Phys., 9, eds. C. Bai, L. Guo, and J.-L. Loday, World Scientific, Hackensack, NJ, 2012, 217–298 | MR