Classification of algebras of level two in the variety of nilpotent algebras and Leibniz algebras

TL;DR

This paper classifies complex finite-dimensional algebras of level two within Leibniz and nilpotent algebra varieties, identifying three Leibniz algebras of this level and describing all nilpotent ones.

Contribution

It provides the first complete classification of level two algebras in the Leibniz and nilpotent algebra varieties, detailing their structure and isomorphism classes.

Findings

Three Leibniz algebras of level two identified

All nilpotent algebras of level two described

One solvable and two nilpotent Leibniz algebras of level two

Abstract

This paper is devoted to the description of complex finite-dimensional algebras of level two. We obtain the classification of algebras of level two in the variety of Leibniz algebras. It is shown that, up to isomorphism, there exist three Leibniz algebras of level two, one of which is solvable, and two of which are nilpotent. Moreover we describe all algebras of level two in the variety of nilpotent algebras.