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

摘要

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$.