A modular analogue of Morozov's theorem on maximal subalgebras of simple Lie algebras

TL;DR

This paper extends Morozov's theorem to modular Lie algebras, showing maximal subalgebras with non-zero radical are parabolic, with specific conditions on the characteristic of the field.

Contribution

It provides a modular analogue of Morozov's theorem for simple Lie algebras over fields of positive characteristic, including necessary conditions on the prime characteristic.

Findings

Maximal Lie subalgebras with non-zero radical are parabolic.

Counterexample for type E8 in characteristic 5 shows necessity of assumptions.

Uses classification theory of finite dimensional simple Lie algebras.

Abstract

Let $G$ be a simple algebraic group over an algebraically closed field of characteristic $p>0$ and suppose that $p$ is a very good prime for $G$. We prove that any maximal Lie subalgebra $M$ of $\mathfrak{g} = {\rm Lie}(G)$ with ${\rm rad}(M) \ne 0$ has the form $M = {\rm Lie}(P)$ for some maximal parabolic subgroup $P$ of $G$. We show that the assumption on $p$ is necessary by providing a counterexample for groups type ${\rm E}_8$ over fields of characteristic $5$. Our arguments rely on the main results and methods of the classification theory of finite dimensional simple Lie algebras over fields prime characteristic.