Classification of $5$-Dimensional Complex Nilpotent Leibniz Algebras

TL;DR

This paper classifies 5-dimensional complex non-Lie nilpotent Leibniz algebras, using canonical forms and direct methods, providing a comprehensive understanding of their structure.

Contribution

It offers the first complete classification of 5-dimensional complex non-Lie nilpotent Leibniz algebras, employing canonical forms for certain cases and direct methods for others.

Findings

Classification of all 5-dimensional complex non-Lie nilpotent Leibniz algebras

Use of canonical forms for matrices of bilinear forms in classification

Application of direct methods for remaining cases

Abstract

Leibniz algebras are certain generalization of Lie algebras. In this paper we give the classification of $5-$dimensional complex non-Lie nilpotent Leibniz algebras. We use the canonical forms for the congruence classes of matrices of bilinear forms to classify the case $\dim(A^2)=3$ and $\dim(Leib(A))=1$ which can be applied to higher dimensions. The remaining cases are classified via direct method.