The classification of 4-dimensional Leibniz algebras

Abstract

This paper is a contribution to the development of the non associative algebras theory. More precisely, this work deals with the classification of the complex 4-dimensional Leibniz algebras. Note that the classification of 4-dimensional nilpotent complex Leibniz algebras was obtained in [1]. Therefore we will only consider non nilpotent case in this work.

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.…”

(Pseudo)digraphs and Leibniz algebra isomorphisms

“…We have also introduced an algorithmic method devoted to that goal. We hope to continue with this research in the near future in order to deal with the classification of 4‐ and 5‐dimensional Leibniz algebras (see Cañete and Khudoyberdiyev and Khudoyberdiyev et al).…”

The geometric classification of Leibniz algebras

“…gives the degeneration L 1+ia 9 → N a 3 for any a ∈ C. 5 Let us consider the degeneration L 40 → L n 2 to clarify our proof. Writing nonzero products of L 40 in the basis E t i , we get…”

“…The first step in the classification of four dimensional Leibniz algebras was done in [1], where all nilpotent four dimensional Leibniz algebras were classified. Then the description of all four dimensional solvable non-nilpotent Lebniz algebras was obtained in [5]. Finally, it was proved in [21] that there is only one non-solvable indecomposable Leibniz algebra whose dimension is less or equal to four, namely, the simple Lie algebra sl 2 .…”