Transformation Groups of Spheres
Transformation Groups of Spheres
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DOI：
10.2307/1968975
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发表时间：
1943-07
影响因子：
4.9
通讯作者：
D. Montgomery;H. Samelson
中科院分区：
文献类型：
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作者：
D. Montgomery;H. Samelson
1.紧李群G被称为空间TW的变换群或作用在空间W上，如果满足以下条件：a）G的每个元素g都有W到其自身上的同胚g（x）[x in TV]。B）如果g1和g2是G的元素，则g9 g2（X）] =（glg 2）（X）。c）点g（x）连续依赖于对（g，x）。条件a）和B）意味着G的单位元与单位同胚相关联。如果除了a）、B）和C）之外，还满足以下第四个条件，则称群G作用传递：d）对于TW的任何两个点x和y，G中存在一个元素g，使得g（x）= y。当d）满足时，我们说TV是G下的齐次空间。在本文中，我们取W为n维球面S'，并研究什么样的紧连通李群能传递地和有效地作用在S'上的问题（见下面的2a））。在我我们证明了一个定理的结构，这样一个群体，这表明我们的主要关注在研究这个问题是与简单的群体。在II中，我们研究的问题，简单的群体使用Killing-Cartan分类，我们发现，在一般情况下，只有那些简单的群体可以传递和有效的SL这是众所周知的。在第三章中，我们用我们的方法得出一些关于n维球面的旋转群的某些子群的结构的结论，我们用Rn表示。否则表示Rn是n + 1个真实的变量上行列式1的正交变换群。
1. The compact Lie group G is said to be a transformation group of the space TW or to act on the space W if the following conditions are satisfied: a) to every element g of G there is associated a homeomorphism g(x) [x in TV] of W onto itself. b) if gl and g2 are elements of G then g9g2(X)] = (glg2)(X). c) the point g(x) depends continuously on the pair (g, x). Conditions a) and b) imply that to the identity element of G is associated the identity homeomorphism. The group G is said to act transitively if in addition to a), b), and c) the following fourth condition is satisfied: d) for any two points x and y of TW there is an element g in G such that g(x) = y. When d) is satisfied we say that TV is a homogeneous space under G. In this paper we take for W the n-dimensional sphere S' and study the question of what compact connected Lie groups can act transitively and effectively, (see 2 a) below), on S'. In I we prove a theorem on the structure of such a group which shows us that our main concern in the study of this problem is with simple groups. In II we study the question for simple groups using the Killing-Cartan classification, and we find that in general only those simple groups can be transitive and effective on SL which are well known to be so. In III we use our methods to draw some conclusions about the structure of certain subgroups of the rotation group of the n-dimensional sphere which we denote by Rn . Otherwise expressed Rn is the group of orthogonal transformations of determinant 1 on n + 1 real variables.