A characterization of nilpotent nonassociative algebras by invertible Leibniz-derivations

TL;DR

This paper extends the characterization of nilpotency via invertible Leibniz-derivations from Lie algebras to various classes of nonassociative algebras, broadening understanding of algebraic structures.

Contribution

It proves that finite-dimensional Malcev, Jordan, and other noncommutative Jordan algebras are nilpotent if they admit an invertible Leibniz-derivation, and describes Leibniz-derivations of semisimple cases.

Findings

Nilpotency characterized by invertible Leibniz-derivations in multiple algebra classes.

Complete description of Leibniz-derivations for semisimple Jordan, right alternative, and Malcev algebras.

Extension of Moens' result from Lie algebras to broader nonassociative algebra classes.

Abstract

Moens proved that a finite-dimensional Lie algebra over field of characteristic zero is nilpotent if and only if it has an invertible Leibniz-derivation. In this article we prove the analogous results for finite-dimensional Malcev, Jordan, (-1,1)-, quasiassociative, quasialternative, right alternative and Malcev-admissible noncommutative Jordan algebras over the field of characteristic zero. Also, we describe all Leibniz-derivations of semisimple Jordan, right alternative and Malcev algebras.