Classification of the maximal subalgebras of exceptional Lie algebras over fields of good characteristic

TL;DR

This paper classifies the maximal Lie subalgebras of exceptional simple algebraic groups over algebraically closed fields of good characteristic, identifying conditions for their structure and conjugacy classes.

Contribution

It provides a complete classification of maximal Lie subalgebras, including maximal connected subgroups, Witt subalgebras, and exotic semidirect products, in exceptional Lie algebras.

Findings

Maximal connected subgroups are classified by known conjugacy classes.

All maximal Witt subalgebras are conjugate under G.

Two conjugacy classes of exotic semidirect products exist in type E7 in characteristics 5 and 7.

Abstract

Let $G$ be an exceptional simple algebraic group over an algebraically closed field $k$ and suppose that the characteristic $p$ of $k$ is a good prime for $G$. In this paper we classify the maximal Lie subalgebras $\mathfrak{m}$ of the Lie algebra $\mathfrak{g}={\rm Lie}(G)$. Specifically, we show that one of the following holds: $\mathfrak{m}={\rm Lie}(M)$ for some maximal connected subgroup $M$ of $G$, or $\mathfrak{m}$ is a maximal Witt subalgebra of $\mathfrak{g}$, or $\mathfrak{m}$ is a maximal $\it{\mbox{exotic semidirect product}}$. The conjugacy classes of maximal connected subgroups of G are known thanks to the work of Seitz, Testerman and Liebeck--Seitz. All maximal Witt subalgebras of $\mathfrak{g}$ are $G$-conjugate and they occur when $G$ is not of type ${\rm E}_6$ and $p-1$ coincides with the Coxeter number of $G$. We show that there are two conjugacy classes of maximal exotic semidirect products in $\mathfrak{g}$, one in characteristic $5$ and one in characteristic $7$, and both occur when $G$ is a group of type ${\rm E}_7$.