Toral stabilizers of restricted Lie algebras

TL;DR

This paper studies the structure and properties of toral stabilizers in finite-dimensional restricted Lie algebras over algebraically closed fields of positive characteristic, providing new characterizations and detailed subalgebra classifications.

Contribution

It introduces new characterizations of solvable restricted Lie algebras and computes toral stabilizers for certain simple Lie algebras, advancing understanding of their structure.

Findings

Computed toral stabilizers for some simple Lie algebras

Provided a new characterization of solvable restricted Lie algebras

Determined subalgebra structures of exceptional simple Lie algebras

Abstract

This paper investigates the toral stabilizer of a finite-dimensional restricted Lie algebra $(\mathfrak{g},[p])$, defined over an algebraically closed field $k$ of positive characteristic. We compute the group of toral stabilizers of some simple Lie algebras and give a new characterization of the solvable restricted Lie algebras. Moreover, we determine the subalgebra structure of exceptional simple Lie algebras in good characteristic. We also provide applications, which concern the distribution of weight spaces.