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Difference between revisions of "Maximal compact subgroup"

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<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>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  E. Cartan, "La géométrie 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>
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<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>
 
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Latest revision as of 17:30, 16 July 2024

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 géométrie 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=55864
This article was adapted from an original article by V.V. Gorbatsevich (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article