Almost-reductive and almost-algebraic Leibniz algebras

TL;DR

This paper explores the extension of the concepts of almost-algebraic and almost-reductive structures from Lie algebras to Leibniz algebras, revealing that these classes differ significantly in the Leibniz context.

Contribution

It introduces and analyzes the notions of almost-reductive and almost-algebraic Leibniz algebras, establishing their properties and relationships to other algebraic classes.

Findings

Almost-algebraic Leibniz algebras form a strictly larger class than almost-reductive ones.

Various properties of these classes are characterized.

Connections to $\

Abstract

This paper examines whether the concept of an almost-algebraic Lie algebra developed by Auslander and Brezin in \cite{ab} can be introduced for Leibniz algebras. Two possible analogues are considered: almost-reductive and almost-algebraic Leibniz algebras. For Lie algebras these two concepts are the same, but that is not the case for Leibniz algebras, the class of almost-algebraic Leibniz algebras strictly containing that of the almost-reductive ones. Various properties of these two classes of algebras are obtained, together with some relationships to $\phi$-free, elementary, $E$-algebras and $A$-algebras.