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Continuous group

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In the fundamental works on the theory of Lie groups (S. Lie, H. Poincaré, E. Cartan, H. Weyl, and others) it is a group of smooth or analytic transformations of the space $\mathbf R^n$ or $\mathbf C^n$, depending smoothly or analytically on parameters. When there are finitely many numerical parameters, a continuous group is called finite, which corresponds to the modern concept of a finite-dimensional Lie group. In the presence of parameters that are functions one speaks of an infinite continuous group, which corresponds to the modern concept of a pseudo-group of transformations. Nowadays (1988) the term "continuous group" often stands for topological group [2].

References

[1] S. Lie, G. Scheffers, "Vorlesungen über Transformationsgruppen" , Teubner (1893)
[2] L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian)
[3] N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French)
How to Cite This Entry:
Continuous group. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Continuous_group&oldid=32404
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article