A theorem of Lie groups
A theorem of Lie groups
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李群定理
DOI：
10.1090/s0002-9904-1942-07699-3
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发表时间：
1942
影响因子：
1.3
通讯作者：
L. Zippin
中科院分区：
文献类型：
--
作者：
D. Montgomery;L. Zippin
粗略地说，这个定理说，每一个足够接近G*的子群都可以被G的一个适当的元素转化成G*。这个结果可以被看作是已知事实的推广，即李群不能有任意小的子群（除了恒等），尽管我们的兴趣不是从这个观点产生的。为了使我们的意思更清楚，假设G*是不变子群，因此因子群G/G*也是李群。如果G中有一个G*附近的子群H，通过同态把G带入G/G*，它就会变成一个G/G*附近的子群。G/G*在恒等式附近的唯一子群是恒等式本身，这意味着如果H在G*附近，它实际上必须是G*的一个子群。我们看到，当G*是G的不变紧子群时，定理的结论在平凡意义上是成立的。我们证明定理1的更一般的情况是根据特定事实的G作用于陪集空间G / G *将用M .这是空间的点是G的叠合组gG * *在M G . G组的行为也可被视为一个黎曼空间和嘉当表明存在M的黎曼度量G是一组等距。这一事实在接下来的讨论中将非常重要。我们可以假设M在G下具有黎曼度规不变量，进而假设M以通常的方式被做成一个度量空间（frimachet）
Roughly, the theorem says that each subgroup near enough to G* can be transformed into G* by an appropriate element of G. This result can be regarded as a generalization of the known fact that Lie groups cannot have arbitrarily small subgroups (other than the identity), although it was not from this point of view that our interest arose. To make our meaning clear, assume that G* is an invariant subgroup so that the factor group G/G* is also a Lie group. If there were in G a subgroup H near G* it would go, by the homomorphism taking G into G/G*, into a subgroup near the identity of G/G*. The only subgroup of G/G* near the identity is the identity itself which means that if H is to be near G* it must actually be a subgroup of G*. We see that when G* is an invariant compact subgroup of G, the conclusion of the theorem is true in a trivial sense. Our proof of Theorem 1 in the more general situation is based on certain facts about the way in which G operates on the coset space G/G* which will be denoted by M. This is the space whose points are the cosets gG* of G* in G. The group G acts transitively on M which can be regarded as a Riemannian space and Cartan has shown that there exists in M a Riemannian metric for which G is a group of isometries. This fact will be of great importance in what follows. We begin, as we may, by supposing that M is endowed with a Riemannian metric invariant under G and, furthermore, we assume that M has been made into a metric space (Fréchet) in the usual way