Minimal linear representations of filiform Lie algebras and their application for construction of Leibniz algebras

TL;DR

This paper determines minimal faithful matrix representations of filiform Lie algebras and applies these results to classify certain Leibniz algebras derived from them in low dimensions.

Contribution

It introduces minimal faithful representations of filiform Lie algebras and uses these to classify related Leibniz algebras in low-dimensional cases.

Findings

Minimal faithful representations found for several classes of filiform Lie algebras.

Classification of Leibniz algebras associated with these Lie algebras in low dimensions.

Representation methods facilitate understanding of Leibniz algebra structures.

Abstract

In this paper we find minimal faithful representations of several classes filiform Lie algebras by means of strictly upper-triangular matrices. We investigate Leibniz algebras whose corresponding Lie algebras are filiform Lie algebras such that the action $I \times L \to I$ gives rise to a minimal faithful representation of a filiform Lie algebra. The classification up to isomorphism of such Leibniz algebras is given for low-dimensional cases.