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Classification of the maximal subalgebras of exceptional Lie algebras over fields of good characteristic
Journal of the American Mathematical Society ( IF 3.5 ) Pub Date : 2019-07-19 , DOI: 10.1090/jams/926
Alexander Premet , David I. Stewart

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 ${\rm E}_7$.

中文翻译:

特殊李代数的极大子代数在良好特性域上的分类

令 $G$ 是代数闭域 $k$ 上的一个特殊的简单代数群,并假设 $k$ 的特征 $p$ 是 $G$ 的一个很好的质数。在本文中,我们对李代数 $\mathfrak{g}={\rm Lie}(G)$ 的最大李子代数 $\mathfrak{m}$ 进行分类。具体来说,我们证明以下其中一项成立: $\mathfrak{m}={\rm Lie}(M)$ 对于 $G$ 的某个最大连通子群 $M$,或 $\mathfrak{m}$ 是$\mathfrak{g}$ 或 $\mathfrak{m}$ 的极大 Witt 子代数是极大 $\it{\mbox{exotic semidirect product}}$。由于 Seitz、Testerman 和 Liebeck--Seitz 的工作,G 的最大连通子群的共轭类是已知的。$\mathfrak{g}$ 的所有最大 Witt 子代数都是 $G$-共轭的,并且当 $G$ 不是 ${\rm E}_6$ 类型并且 $p-1$ 与 $ 的 Coxeter 数重合时出现它们G$。
更新日期:2019-07-19
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