Classification and (Quasi)-Centroids of Four-Dimensional Ternary Leibniz Algebras

TL;DR

This paper classifies four-dimensional ternary Leibniz algebras over algebraically closed fields of characteristic zero, detailing their internal symmetries through centroids and quasi-centroids, and extends classical binary Leibniz results.

Contribution

It provides the first complete classification of four-dimensional ternary Leibniz algebras and explicitly computes their centroids and quasi-centroids, revealing their symmetry structures.

Findings

Complete classification of four-dimensional ternary Leibniz algebras

Explicit determination of centroids and quasi-centroids

Extension of classical binary Leibniz results

Abstract

We provide a classification, up to isomorphism, of four-dimensional ternary Leibniz algebras over an algebraically closed field of characteristic zero. For each non-abelian algebra in the classification, we explicitly determine its centroid and quasi-centroid and compute their dimensions. These results offer a comprehensive description of the internal symmetries of low-dimensional ternary Leibniz algebras and extend several classical results from the binary Leibniz setting to the ternary case.