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Difference between revisions of "Isotropy group"

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The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i0529701.png" /> of elements of a given group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i0529702.png" />, acting on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i0529703.png" /> as a group of transformations, that leave a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i0529704.png" /> fixed. This set turns out to be a subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i0529705.png" /> and is called the isotropy group of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i0529706.png" />. The following terminology is used with the same meaning: stationary subgroup, stabilizer, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i0529708.png" />-centralizer. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i0529709.png" /> is a Hausdorff space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i05297010.png" /> is a topological group acting continuously on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i05297011.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i05297012.png" /> is a closed subgroup. If, furthermore, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i05297013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i05297014.png" /> are locally compact, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i05297015.png" /> has a countable base and acts transitively on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i05297016.png" />, then there exists a unique homeomorphism from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i05297017.png" /> into the quotient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i05297018.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i05297019.png" /> is one of the isotropy groups; all the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i05297020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i05297021.png" />, are isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i05297022.png" />.
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The set $G_x$ of elements of a given group $G$, acting on a set $M$ as a group of transformations, that leave a point fixed. This set turns out to be a subgroup of $G$ and is called the isotropy group of the point $x$. The following terminology is used with the same meaning: stationary subgroup, stabilizer, $G$-centralizer. If $M$ is a Hausdorff space and $G$ is a topological group acting continuously on $M$, then $G_x$ is a closed subgroup. If, furthermore, $M$ and $G$ are locally compact, $G$ has a countable base and acts transitively on $M$, then there exists a unique homeomorphism from $M$ into the quotient space $G/H$, where $H$ is one of the isotropy groups; all the $G_x$, $x\in M$, are isomorphic to $H$.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i05297023.png" /> be a smooth manifold and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i05297024.png" /> a Lie group acting smoothly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i05297025.png" />. Then the isotropy group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i05297026.png" /> of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i05297027.png" /> induces a group of linear transformations of the tangent vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i05297028.png" />; the latter is called the linear isotropy group at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i05297029.png" />. On passing to tangent spaces of higher order at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052970/i05297030.png" /> one obtains natural representations of the isotropy group in the structure groups of the corresponding tangent bundles of higher order; they are called the higher-order isotropy groups (see also [[Isotropy representation|Isotropy representation]]).
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Let $M$ be a smooth manifold and $G$ a Lie group acting smoothly on $M$. Then the isotropy group $G_x$ of a point $x\in M$ induces a group of linear transformations of the tangent vector space $T_x(M)$; the latter is called the linear isotropy group at $x$. On passing to tangent spaces of higher order at the point $x$ one obtains natural representations of the isotropy group in the structure groups of the corresponding tangent bundles of higher order; they are called the higher-order isotropy groups (see also [[Isotropy representation|Isotropy representation]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.S. Pontryagin,  "Topological groups" , Princeton Univ. Press  (1958)  (Translated from Russian)</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><TR><TD valign="top">[3]</TD> <TD valign="top">  R. Sulanke,  P. Wintgen,  "Differentialgeometrie und Faserbündel" , Deutsch. Verlag Wissenschaft.  (1972)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.S. Pontryagin,  "Topological groups" , Princeton Univ. Press  (1958)  (Translated from Russian)</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><TR><TD valign="top">[3]</TD> <TD valign="top">  R. Sulanke,  P. Wintgen,  "Differentialgeometrie und Faserbündel" , Deutsch. Verlag Wissenschaft.  (1972)</TD></TR></table>

Latest revision as of 08:24, 23 August 2014

The set $G_x$ of elements of a given group $G$, acting on a set $M$ as a group of transformations, that leave a point fixed. This set turns out to be a subgroup of $G$ and is called the isotropy group of the point $x$. The following terminology is used with the same meaning: stationary subgroup, stabilizer, $G$-centralizer. If $M$ is a Hausdorff space and $G$ is a topological group acting continuously on $M$, then $G_x$ is a closed subgroup. If, furthermore, $M$ and $G$ are locally compact, $G$ has a countable base and acts transitively on $M$, then there exists a unique homeomorphism from $M$ into the quotient space $G/H$, where $H$ is one of the isotropy groups; all the $G_x$, $x\in M$, are isomorphic to $H$.

Let $M$ be a smooth manifold and $G$ a Lie group acting smoothly on $M$. Then the isotropy group $G_x$ of a point $x\in M$ induces a group of linear transformations of the tangent vector space $T_x(M)$; the latter is called the linear isotropy group at $x$. On passing to tangent spaces of higher order at the point $x$ one obtains natural representations of the isotropy group in the structure groups of the corresponding tangent bundles of higher order; they are called the higher-order isotropy groups (see also Isotropy representation).

References

[1] L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian)
[2] S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978)
[3] R. Sulanke, P. Wintgen, "Differentialgeometrie und Faserbündel" , Deutsch. Verlag Wissenschaft. (1972)
How to Cite This Entry:
Isotropy group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Isotropy_group&oldid=33104
This article was adapted from an original article by Ü. Lumiste (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article