Classification of Complex Cyclic Leibniz Algebras

TL;DR

This paper provides a classification of cyclic Leibniz algebras, which are generated by one element and serve as a tool to explore the differences between Leibniz and Lie algebras.

Contribution

It offers the first comprehensive classification of cyclic Leibniz algebras, highlighting their unique properties and role in understanding algebraic extensions.

Findings

Cyclic Leibniz algebras have no Lie algebra counterpart.

They serve as counterexamples to extension of Lie algebra results.

The classification reveals their simple and distinct structure.

Abstract

Since Leibniz algebras were introduced by Loday as a generalization of Lie algebras, there has been a lot of interest in which results of the latter extend to the former. Cyclic algebras, those generated by one element, are a useful tool for studying these concepts. In fact, they have no Lie algebra counterpart. Their simple structure lends itself to elegant counterexamples to the extension of several important results from Lie algebras to Leibniz algebras. In this paper, we give a classification of cyclic Leibniz algebras.