On the inner structure of 3-dimensional Leibniz algebras

TL;DR

This paper investigates the internal structure of three-dimensional left Leibniz algebras, providing a detailed classification and understanding of their algebraic properties.

Contribution

It offers a comprehensive classification of 3-dimensional left Leibniz algebras, revealing their internal structure and algebraic relations.

Findings

Classification of all 3-dimensional left Leibniz algebras

Description of their internal algebraic structure

Identification of key structural properties

Abstract

Let $L$ be an algebra over a field $F$ with the binary operations $+$ and $[,]$. Then $L$ is called a left Leibniz algebra if $[[a,b],c]=[a,[b,c]]-[b,[a,c]]$ for all $a,b,c\in L$. We describe the inner structure of left Leibniz algebras having dimension 3.