Abelian subalgebras and ideals of maximal dimension in Leibniz algebras

TL;DR

This paper investigates the structure of Leibniz algebras by analyzing their abelian subalgebras and ideals of maximal dimension, focusing on specific classes like solvable, nilpotent, and p-filiform Leibniz algebras.

Contribution

It provides a detailed comparison of abelian subalgebras and ideals in various classes of Leibniz algebras, including new insights into codimension and nilpotency cases.

Findings

Characterization of Leibniz algebras with abelian subalgebras of codimension 1

Analysis of solvable and supersolvable Leibniz algebras for codimensions 1 and 2

Examples illustrating the necessity of field restrictions

Abstract

In this paper, we compare the abelian subalgebras and ideals of maximal dimension for finite-dimensional Leibniz algebras. We study Leibniz algebras containing abelian subalgebras of codimension 1, solvable and supersolvable Leibniz algebras for codimensions 1 and 2, nilpotent Leibniz algebras in case of codimension 2 and we also analyze the case of k-abelian p-filiform Leibniz algebras. Throughout the paper, we also give examples to clarify some results and the need for restrictions on the underlying field.