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• W2332216340 abstract "In order to apply our rather deep understanding of the structure of Lie groups to the study of transformation groups it is natural to try to single out a class of transformation groups which are in some sense naturally Lie groups. In this paper we iiltroduce such a class and commence their study. In Section 1 the inotioni of a l,ie transformation group is introduced. Roughly, these are grouips II of homeomorphismlls of a space X which admit a Lie group) topology which is stronlg enough to make the evaluation mapping (ht, x) ->h (x) of II X X into X continuous, yet weak enough so that H gets all the onie-parameter subgroups it deserves by virtue of the way it acts on X (see the definiition of admissibly weak below). Such a topology is uniquely determined if it exists and our efforts are in the main concerned with the questioni of wheni it exists anid how onie may effectively put one's hands on it wlheil it does. A niatural candidate for this so-called Lie topology is of course the compact-open topology for H. However, if one considers the example of a dense one-parameter subgroup H of the torus X acting on X by translation, it appears that this is not the general answer. In this example if we modifyv the compact-open topology by adding to the open sets all their arc componlents (getting in this way what we call the modified compact-open topology), we get the Lie topology of HI. That this is a fairly general fact is onie of our maini results (Theorem 5. 14). The latter theorem moreover shows that the reason that the compact-open topology was not good enough in the above example is connected with the fact that H was not closed in the group of all homeomorphisms of X, relative to the compact-open topology. Theorem 5. 14 also states that for a large class of interestinig cases the weakness condition for a Lie topology is redundant. The remainder of the paper is concerned with developing a certain criterion for deciding when a topological group is a Lie group and applying this criterion to derive a general necessary and sufficient conldition for groups of homeomorphisms of locally compact, locally connected finite dimensional metric spaces to be Lie transformiiation grouips. The criterion is remarkable in that local compactness is niot one of the assumptions. It states in fact" @default.
• W2332216340 created "2016-06-24" @default.
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• W2332216340 date "1957-07-01" @default.
• W2332216340 modified "2023-09-23" @default.
• W2332216340 title "On a Class of Transformation Groups" @default.
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• W2332216340 doi "https://doi.org/10.2307/2372567" @default.
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