A characterisation of nilpotent Lie algebras by invertible Leibniz-derivations

TL;DR

This paper establishes a precise characterization of nilpotent Lie algebras by demonstrating they are exactly those admitting an invertible Leibniz-derivation, extending Jacobson's classical results on derivations.

Contribution

It proves that a Lie algebra is nilpotent if and only if it admits an invertible Leibniz-derivation, clarifying the relationship between nilpotency and Leibniz-derivations.

Findings

Nilpotent Lie algebras admit invertible Leibniz-derivations.

The characterization is proven using elementary techniques.

The result extends Jacobson's classical theorem on derivations.

Abstract

Jacobson proved that if a Lie algebra admits an invertible derivation, it must be nilpotent. He also suspected, though incorrectly, that the converse might be true: that every nilpotent Lie algebra has an invertible derivation. We prove that a Lie algebra is nilpotent if and only if it admits an invertible Leibniz-derivation. The proofs are elementary in nature and are based on well-known techniques.