Lie 2-algebras of toral rank 3

TL;DR

This paper investigates Lie 2-algebras over algebraically closed fields of characteristic two, focusing on those with triangulable Cartan subalgebras of toral rank three, establishing properties, necessary conditions for simplicity, and non-existence results for certain dimensions.

Contribution

It provides new properties of centerless Lie 2-algebras, analyzes simple Lie 2-algebras of toral rank three, and shows non-existence of certain low-dimensional simple cases.

Findings

Derived properties of centerless Lie 2-algebras

Established a necessary condition for simplicity

Proved non-existence of simple Lie 2-algebras with certain dimensions

Abstract

In this paper we study Lie 2-algebras over an algebraically closed field of characteristic two, which have a triangulable Cartan subalgebra, and derive some general properties of centerless ones. These properties allow us to do an analysis on simple Lie 2-algebras of toral rank three and provide a necessary condition for simplicity. By means of this latter condition we also conclude that simple Lie 2-algebras with a triangulable Cartan subalgebra of toral rank three and of dimension less than or equal to 16 cannot exist.