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

Abstract:

Let $G$ be an exceptional simple algebraic group over an algebraically closed field $k$ and suppose that $p={\operatorname {char}}(k)$ is a good prime for $G$. In this paper we classify the maximal Lie subalgebras $\mathfrak {m}$ of the Lie algebra $\mathfrak {g}=\operatorname {Lie}(G)$. Specifically, we show that either $\mathfrak {m}=\operatorname {Lie}(M)$ 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 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 ${\mathrm {E}}_6$ 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 ${\mathrm {E}}_7$.