Difference between revisions of "Continuous group"
From Encyclopedia of Mathematics
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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 | + | {{TEX|done}} |
| + | 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|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|pseudo-group]] of transformations. Nowadays (1988) the term "continuous group" often stands for [[Topological group|topological group]] [[#References|[2]]]. | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> S. Lie, G. Scheffers, "Vorlesungen über Transformationsgruppen" , Teubner (1893)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> S. Lie, G. Scheffers, "Vorlesungen über Transformationsgruppen" , Teubner (1893)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French)</TD></TR></table> | ||
Latest revision as of 21:21, 9 July 2014
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://encyclopediaofmath.org/index.php?title=Continuous_group&oldid=32404
Continuous group. Encyclopedia of Mathematics. URL: http://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