On the subalgebra lattice of a Leibniz algebra

TL;DR

This paper explores the structure of subalgebra lattices in Leibniz algebras, focusing on properties like modularity and distributivity, and highlights differences from Lie algebras due to fewer one-dimensional subalgebras.

Contribution

It initiates the study of subalgebra lattice properties in Leibniz algebras, emphasizing how their lattice conditions differ from Lie algebras.

Findings

Leibniz algebras have fewer one-dimensional subalgebras than Lie algebras.

Certain lattice properties are weaker in Leibniz algebras.

The paper characterizes Leibniz algebras with specific lattice structures.

Abstract

In this paper we begin to study the subalgebra lattice of a Leibniz algebra. In particular, we deal with Leibniz algebras whose subalgebra lattice is modular, upper semi-modular, lower semi-modular, distributive, or dually atomistic. The fact that a non-Lie Leibniz algebra has fewer one-dimensional subalgebras in general results in a number of lattice conditions being weaker than in the Lie case.