On classification of four-dimensional nilpotent Leibniz algebras

Abstract

Abstract. Leibniz algebras are certain generalization of Lie algebras. In this paper we give the classification of four dimensional non-Lie nilpotent Leibniz algebras. We use the canonical forms for the congruence classes of matrices of bilinear forms and some other techniques to obtain our result.

Leibniz algebras as non-associative algebras

“…Then we obtain from identity (2.2) that L x 2 (y) = x 2 y = x(xy) − x(xy) = 0 , which shows that L x 2 = 0. Every abelian (left or right) Leibniz algebra is a Lie algebra, but there are many Leibniz algebras that are not Lie algebras (see, for example, [20,4,5,27,1,2,3,39,18,31,30,34,16,15,17,22,21,23,24,25]). We will use the following three examples to illustrate the concepts introduced in this section.…”

“…Note that in dimension 3 there are five isomorphism classes of nilpotent non-Lie Leibniz algebras of which four (including one 1-parameter family) are associative (see[22, Theorem 6.4]). In dimension 4 there are seventeen isomorphism classes of indecomposable nilpotent non-Lie Leibniz algebras of which eleven (including three 1-parameter families) are associative (see[3,, but compare this with[24] for the correct total number of isomorphism classes).…”

Minimal Nonnilpotent Leibniz Algebras

“…Recent work in Leibniz algebra often involves studying certain classes of Leibniz algebras, such as cyclic algebras [5], algebras with a certain nilradical [3,9], or algebras of a certain dimension [6,8,10]. Many of these articles involve generalizing results from Lie algebras to Leibniz algebras.…”

Leibniz algebras with absolute maximal Lie subalgebras

“…Leibniz algebras were introduced by Jean-Louis Loday (see [16]) as noncommutative versions of Lie algebras. Several authors (see [2,3,5,6,11,12] for examples) have investigated whether the results on Lie algebras can be extended to Leibniz algebras. In [5] for instance, D. Barnes proved an analogue of Levi's decomposition for Leibniz algebras, which states that every finite dimensional Leibniz algebra g over a field of characteristic zero can be written as a direct sum of its solvable radical and a semisimple Lie subalgebra.…”