Classification of some Solvable Leibniz Algebras

TL;DR

This paper classifies certain low-dimensional non-Lie solvable Leibniz algebras with a one-dimensional derived subalgebra, using matrix canonical forms, and extends previous classifications to higher dimensions.

Contribution

It provides a new classification of non-Lie solvable Leibniz algebras of dimension up to 8 with one-dimensional derived subalgebra, employing matrix congruence forms.

Findings

Classification of non-Lie solvable Leibniz algebras of dimension ≤8

Extension of classification methods to higher dimensions

Revisit of 3-dimensional non-Lie solvable Leibniz algebras

Abstract

Leibniz algebras are certain generalization of Lie algebras. In this paper we give classification of non-Lie solvable (left) Leibniz algebras of dimension $\leq 8$ with one dimensional derived subalgebra. We use the canonical forms for the congruence classes of matrices of bilinear forms to obtain our result. Our approach can easily be extended to classify these algebras of higher dimensions. We also revisit the classification of three dimensional non-Lie solvable (left) Leibniz algebras.