Namespaces
Variants
Actions

Difference between revisions of "Maximal compact subgroup"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(TeX done)
Line 1: Line 1:
''of a [[Topological group|topological group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062930/m0629301.png" />''
+
{{TEX|done}}{{MSC|22A05}}
  
A compact subgroup (cf. [[Compact group|Compact group]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062930/m0629302.png" /> which is not contained as a proper subgroup in any compact subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062930/m0629303.png" />. For example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062930/m0629304.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062930/m0629305.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062930/m0629306.png" /> for a solvable simply-connected Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062930/m0629307.png" />.
+
''of a [[topological group]] $G$''
  
In an arbitrary group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062930/m0629308.png" /> maximal compact subgroups need not exist (for example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062930/m0629309.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062930/m06293010.png" /> is an infinite-dimensional Hilbert space), and if they do exist there may be non-isomorphic ones among them.
+
A compact subgroup (cf. [[Compact group]]) $K \subset G$ which is not contained as a proper subgroup in any compact subgroup of $G$. For example, $K = \text{SO}(n)$ for $G = \text{GL}(n,\mathbf{R})$; $K=\{e\}$ for a solvable simply-connected Lie group $G$.
  
Maximal compact subgroups of Lie groups have been studied most. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062930/m06293011.png" /> is a connected Lie group, then any compact subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062930/m06293012.png" /> is contained in some maximal compact subgroup (in particular, maximal compact subgroups must exist) and all maximal compact subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062930/m06293013.png" /> are connected and conjugate to each other. The space of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062930/m06293014.png" /> is diffeomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062930/m06293015.png" />, therefore most of the topological questions about Lie groups reduce to the corresponding questions for compact Lie groups (cf. [[Lie group, compact|Lie group, compact]]).
+
In an arbitrary group $G$ maximal compact subgroups need not exist (for example, if $G = \text{GL}(V)$, where $V$ is an infinite-dimensional Hilbert space), and if they do exist there may be non-isomorphic ones among them.
 +
 
 +
Maximal compact subgroups of Lie groups have been studied most. If $G$ is a connected Lie group, then any compact subgroup of $G$ is contained in some maximal compact subgroup (in particular, maximal compact subgroups must exist) and all maximal compact subgroups of $G$ are connected and conjugate to each other. The space of the group $G$ is diffeomorphic to $K \times \mathbf{R}^n$, therefore most of the topological questions about Lie groups reduce to the corresponding questions for compact Lie groups (cf. [[Lie group, compact]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Cartan,  "La géometrie des groupes de transformations"  ''J. Math. Pures Appl.'' , '''6'''  (1927)  pp. 1–119</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Helgason,  "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press  (1978)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  E. Cartan,  "La géometrie des groupes de transformations"  ''J. Math. Pures Appl.'' , '''6'''  (1927)  pp. 1–119 {{ZBL|53.0388.01}}</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top">  S. Helgason,  "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press  (1978) {{ZBL|0451.53038}}</TD></TR>
 +
</table>

Revision as of 19:10, 12 April 2017

2020 Mathematics Subject Classification: Primary: 22A05 [MSN][ZBL]

of a topological group $G$

A compact subgroup (cf. Compact group) $K \subset G$ which is not contained as a proper subgroup in any compact subgroup of $G$. For example, $K = \text{SO}(n)$ for $G = \text{GL}(n,\mathbf{R})$; $K=\{e\}$ for a solvable simply-connected Lie group $G$.

In an arbitrary group $G$ maximal compact subgroups need not exist (for example, if $G = \text{GL}(V)$, where $V$ is an infinite-dimensional Hilbert space), and if they do exist there may be non-isomorphic ones among them.

Maximal compact subgroups of Lie groups have been studied most. If $G$ is a connected Lie group, then any compact subgroup of $G$ is contained in some maximal compact subgroup (in particular, maximal compact subgroups must exist) and all maximal compact subgroups of $G$ are connected and conjugate to each other. The space of the group $G$ is diffeomorphic to $K \times \mathbf{R}^n$, therefore most of the topological questions about Lie groups reduce to the corresponding questions for compact Lie groups (cf. Lie group, compact).

References

[1] E. Cartan, "La géometrie des groupes de transformations" J. Math. Pures Appl. , 6 (1927) pp. 1–119 Zbl 53.0388.01
[2] S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978) Zbl 0451.53038
How to Cite This Entry:
Maximal compact subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Maximal_compact_subgroup&oldid=40957
This article was adapted from an original article by V.V. Gorbatsevich (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article