On Semi-Modular Subalgebras of Lie Algebras Over Fields of Arbitrary Characteristic

TL;DR

This paper investigates the properties of semi-modular subalgebras within Lie algebras over fields of arbitrary characteristic, establishing conditions for their equivalence with modular subalgebras and characterizing low-dimensional cases.

Contribution

It extends the understanding of semi-modular subalgebras in Lie algebras, identifying when modularity and semi-modularity coincide and characterizing low-dimensional semi-modular subalgebras.

Findings

Modularity and semi-modularity are equivalent in certain classes of Lie algebras.

Semi-modular subalgebras of dimensions one and two are characterized over specific fields.

The equivalence of modularity and semi-modularity fails in some cases over fields of characteristic three.

Abstract

This paper is a further contribution to the extensive study by a number of authors of the subalgebra lattice of a Lie algebra. It is shown that, in certain circumstances, including for all solvable algebras, for all Lie algebras over algebraically closed fields of characteristic p > 0 that have absolute toral rank less than or equal to 1 or are restricted, and for all Lie algebras having the one-and-a-half generation property, the conditions of modularity and semi-modularity are equivalent, but that the same is not true for all Lie algebras over a perfect field of characteristic three. Semi-modular subalgebras of dimensions one and two are characterised over (perfect, in the case of two-dimensional subalgebras) fields of characteristic different from 2, 3.