On the subalgebra lattice of a restricted Lie algebra]{On the subalgebra lattice of a restricted Lie algebra

TL;DR

This paper investigates the structure of the lattice of restricted subalgebras in restricted Lie algebras, focusing on properties like dual atomistic, semimodularity, and quasi-ideals, highlighting differences from the non-restricted case.

Contribution

It characterizes the lattice properties of restricted subalgebras, revealing how restrictions affect lattice conditions compared to classical Lie algebra theory.

Findings

Certain lattice properties are weaker due to non-restricted one-dimensional subalgebras.

Conditions like dual atomistic and semimodularity are analyzed in the restricted context.

The study identifies how the presence of non-restricted subalgebras influences lattice structure.

Abstract

In this paper we study the lattice of restricted subalgebras of a restricted Lie algebra. In particular, we consider those algebras in which this lattice is dually atomistic, lower or upper semimodular, or in which every restricted subalgebra is a quasi-ideal. The fact that there are one-dimensional subalgebras which are not restricted results in some of these conditions being weaker than for the corresponding conditions in the non-restricted case.