紧李群的极大子群
Maximal Subgroups of Compact Lie Groups
摘要：
本文旨在对紧李群（不一定连通）的极大子群的分类作一概述。在第一部分中，它表明，这些自然分为三种类型：（1）平凡型群，它简单地定义为相应分支群的极大子群在标准投影下的逆像，其分类构成有限群中的一个问题;（2）正规型群，它的连通单分支是正规子群;（3）正规化子型群，它们是它们自己的连通单分量的归一化子。本文还证明了如何将后两类极大子群的分类归结为：（2）无中心连通紧致单李群的有限极大$Σ$ -正规子群的分类和（3）紧致单李代数的$Σ$ -本原子代数的分类，其中$Σ$是相应的外自同构群的子群。在第二部分中，我们明确地计算了紧经典李代数的本原子代数（在相应的典型群中）的正规化子，从而得到了紧经典李群的所有（非离散）极大子群的完全分类。
英文摘要：
This report aims at giving a general overview on the classification of the maximal subgroups of compact Lie groups (not necessarily connected). In the first part, it is shown that these fall naturally into three types: (1) those of trivial type, which are simply defined as inverse images of maximal subgroups of the corresponding component group under the canonical projection and whose classification constitutes a problem in finite group theory, (2) those of normal type, whose connected one-component is a normal subgroup, and (3) those of normalizer type, which are the normalizers of their own connected one-component. It is also shown how to reduce the classification of maximal subgroups of the last two types to: (2) the classification of the finite maximal $Σ$-invariant subgroups of center-free connected compact simple Lie groups and (3) the classification of the $Σ$-primitive subalgebras of compact simple Lie algebras, where $Σ$ is a subgroup of the corresponding outer automorphism group. In the second part, we explicitly compute the normalizers of the primitive subalgebras of the compact classical Lie algebras (in the corresponding classical groups), thus arriving at the complete classification of all (non-discrete) maximal subgroups of the compact classical Lie groups.
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