Classification of 4-dimensional nilpotent complex Leibniz algebras

By Sergio Albeverio, Bakhrom Omirov, Isamiddin S. Rakhimov. 2006

74 citations recorded by the source corpus.

In [8] and [9] several classes of new algebras were introduced. Some of them have two generating operations and they are called dialgebras. The first motivation to introduce such algebraic structures (related with well known Lie and associative algebras) were problems in algebraic K-theory. The categories of these algebras over their operads assemble into the commutative diagram which reflects the Koszul duality of those categories. The aim of the present paper is to study structural properties of one class of Loday’s list, namely the so called Leibniz algebras. Leibniz algebras present a “non-commutative” (to be more precise, a “nonantisymmetric”) analogue of Lie algebras and they were introduced by J.L.Loday [8], as algebras that satisfy the following identity: [x, [y, z]] = [[x, y], z]− [[x, z], y].

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/19c0b534c55fbbf60f5cda336763b62d42763f32

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. Y. Volkov
2. Ivan Kaygorodov
3. A. Pozhidaev
4. Yury Popov
5. Isamiddin S. Rakhimov
6. E. M. Cañete
7. Abror Kh. Khudoyberdiyev
8. R. Turdibaev

In Geometry and Topology

1. Y. Volkov
2. Ivan Kaygorodov
3. A. Pozhidaev
4. Yury Popov
5. E. M. Cañete
6. Abror Kh. Khudoyberdiyev
7. Isamiddin S. Rakhimov
8. Bakhrom Omirov

In Mathematical Physics

1. Y. Volkov
2. Ivan Kaygorodov
3. E. M. Cañete
4. Abror Kh. Khudoyberdiyev
5. Isamiddin S. Rakhimov
6. A. Pozhidaev
7. Bakhrom Omirov
8. Yury Popov

Citations in the corpus

• The classification of 4-dimensional Leibniz algebras (2013, background): “In [1] the law of λ5 is defined by the following products: [e1, e1] = e3, [e2, e1] = e3, [e1, e2] = e3.”
• On Description of Leibniz Algebras Corresponding to sl2 (2013, background): “, the classification problems of nilpotent part and its subclasses [1, 2, 4, 16–21], e.”
• An algorithm for the classification of 3-dimensional complex Leibniz algebras (2012, unknown): “BracketTwo[1, 3] := {1/β, 0, 0} BracketTwo[2, 3] := {0, 1, 0}”
• Degenerations of Zinbiel and nilpotent Leibniz algebras (2016, methodology): “For the right classification of nilpotent four-dimensional Leibniz algebras see [30].”
• Varieties of Nilpotent Complex Leibniz Algebras of Dimension Less than Five (2005, unknown): “As for the Leibniz algebras case, the classification can be found in paragraph 1 of the present paper and in Albeverio et al. (2003).”