Naturally Graded 2-Filiform Leibniz Algebras

Abstract

Abstract. The Leibniz algebras appear as a generalization of the Lie algebras [8]. The classification of naturally graded p-filiform Lie algebras is known [3], [4], [5], [9]. In this work we deal with the classification of 2-filiform Leibniz algebras. The study of p-filiform Leibniz non Lie algebras is solved for p = 0 (trivial) and p = 1 [1]. In this work we get the classification of naturally graded non Lie 2-filiform Leibniz algebras.

On the description of Leibniz algebras with nilindex n−3

“…• α 1 = 0: If α 4 = 0, then we easily obtain α 1 = 1. Thus, we have the algebra L 0, 3 (1,0,0,λ,−1) with λ = 0. If α 4 = 0, then choosing appropriate values of A 4 and B 4 we derive α 1 = α 4 = 1.…”

Leibniz Algebras Associated to Extensions ofsl2

“…Verifying the Leibniz identity of the above family of algebras and using the program in software Mathematica (see in [3]), we derive the condition 1 − a = 0…”

Solvable extensions of the naturally graded quasi-filiform Leibniz algebra of second type L4

“…Bosko-Dunbar, Dunbar, Hird and Stagg attempted to classify left solvable Leibniz algebras with Heisenberg nilradical [5]. Left and right solvable extensions of R 18 [1], L 1 , L 2 and L 3 [6] over the field of real numbers were found by Shabanskaya in [22,23,24].…”

“…The starting point of the present article is a naturally graded quasi-filiform non Lie Leibniz algebra of the second type L 4 , (n ≥ 4) in the notation of [6]. This algebra is left and right at the same time and an associative when n = 4.…”

“…Naturally graded quasi-filiform Leibniz algebras in any finite dimension over C were studied by Camacho, Gómez, González, Omirov [6]. They found six such algebras of the first type, where two of them depend on a parameter and eight algebras of the second type with one of them depending on a parameter.…”