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The natural linear representation of the [[Isotropy group|isotropy group]] of a differentiable transformation group in the tangent space to the underlying manifold. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052980/i0529801.png" /> is a group of differentiable transformations on a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052980/i0529802.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052980/i0529803.png" /> is the corresponding isotropy subgroup at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052980/i0529804.png" />, then the isotropy representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052980/i0529805.png" /> associates with each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052980/i0529806.png" /> the differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052980/i0529807.png" /> of the transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052980/i0529808.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052980/i0529809.png" />. The image of the isotropy representation, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052980/i05298010.png" />, is called the linear isotropy group at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052980/i05298011.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052980/i05298012.png" /> is a Lie group with a countable base acting smoothly and transitively on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052980/i05298013.png" />, then the tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052980/i05298014.png" /> can naturally be identified with the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052980/i05298015.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052980/i05298016.png" /> are the Lie algebras of the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052980/i05298017.png" />. Furthermore, the isotropy representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052980/i05298018.png" /> is now identified with the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052980/i05298019.png" /> induced by the restriction of the adjoint representation (cf. [[Adjoint representation of a Lie group|Adjoint representation of a Lie group]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052980/i05298020.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052980/i05298021.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052980/i05298022.png" />.
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If a [[Homogeneous space|homogeneous space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052980/i05298023.png" /> is reductive, that is, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052980/i05298024.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052980/i05298025.png" /> is an invariant subspace with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052980/i05298026.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052980/i05298027.png" /> can be identified with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052980/i05298028.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052980/i05298029.png" /> can be identified with the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052980/i05298030.png" /> (see [[#References|[3]]]). In this case, the isotropy representation is faithful (cf. [[Faithful representation|Faithful representation]]) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052980/i05298031.png" /> acts effectively.
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The isotropy representation and linear isotropy group play an important role in the study of invariant objects on homogeneous spaces (cf. [[Invariant object|Invariant object]]). The invariant tensor fields on a homogeneous space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052980/i05298032.png" /> are in one-to-one correspondence with the tensors on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052980/i05298033.png" /> that are invariant with respect to the isotropy representation. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052980/i05298034.png" /> has an invariant Riemannian metric if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052980/i05298035.png" /> has a Euclidean metric that is invariant under the linear isotropy group. There exists on the homogeneous space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052980/i05298036.png" /> a positive [[Invariant measure|invariant measure]] if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052980/i05298037.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052980/i05298038.png" />. A homogeneous space has an invariant orientation if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052980/i05298039.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052980/i05298040.png" />. The invariant linear connections on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052980/i05298041.png" /> are in one-to-one correspondence with the linear mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052980/i05298042.png" /> with the following properties:
+
The natural linear representation of the [[Isotropy group|isotropy group]] of a differentiable transformation group in the tangent space to the underlying manifold. If  $  G $
 +
is a group of differentiable transformations on a manifold  $  M $
 +
and  $  G _ {x} $
 +
is the corresponding isotropy subgroup at the point  $  x \in M $,
 +
then the isotropy representation  $  \mathop{\rm Is} _ {x} : G _ {x} \rightarrow  \mathop{\rm GL} ( T _ {x} M ) $
 +
associates with each  $  h \in G _ {x} $
 +
the differential  $  \mathop{\rm Is} _ {x} ( h) = d h _ {x} $
 +
of the transformation  $  h $
 +
at  $  x $.  
 +
The image of the isotropy representation, $  \mathop{\rm Is} _ {x} ( G _ {x} ) $,
 +
is called the linear isotropy group at  $  x $.  
 +
If  $  G $
 +
is a Lie group with a countable base acting smoothly and transitively on $  M $,
 +
then the tangent space $  T _ {x} M $
 +
can naturally be identified with the space  $  \mathfrak g / \mathfrak g _ {x} $,
 +
where  $  \mathfrak g \supset \mathfrak g _ {x} $
 +
are the Lie algebras of the groups  $  G \supset G _ {x} $.  
 +
Furthermore, the isotropy representation  $  \mathop{\rm Is} _ {x} $
 +
is now identified with the representation  $  G _ {x} \rightarrow  \mathop{\rm GL} ( \mathfrak g / \mathfrak g _ {x} ) $
 +
induced by the restriction of the adjoint representation (cf. [[Adjoint representation of a Lie group|Adjoint representation of a Lie group]])  $  \mathop{\rm Ad} _ {G} $
 +
of  $  G $
 +
to $  G _ {x} $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052980/i05298043.png" /></td> </tr></table>
+
If a [[Homogeneous space|homogeneous space]]  $  M $
 +
is reductive, that is, if  $  \mathfrak g = \mathfrak g _ {x} \dot{+} m $,
 +
where  $  m $
 +
is an invariant subspace with respect to  $  \mathop{\rm Ad} _ {G} ( G _ {x} ) $,
 +
then  $  T _ {x} M $
 +
can be identified with  $  m $,
 +
while  $  \mathop{\rm Is} _ {x} $
 +
can be identified with the representation  $  h \mapsto (  \mathop{\rm Ad} _ {G} h ) \mid  _ {m} $(
 +
see [[#References|[3]]]). In this case, the isotropy representation is faithful (cf. [[Faithful representation|Faithful representation]]) if  $  G $
 +
acts effectively.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052980/i05298044.png" /></td> </tr></table>
+
The isotropy representation and linear isotropy group play an important role in the study of invariant objects on homogeneous spaces (cf. [[Invariant object|Invariant object]]). The invariant tensor fields on a homogeneous space  $  M $
 +
are in one-to-one correspondence with the tensors on  $  T _ {x} M $
 +
that are invariant with respect to the isotropy representation. In particular,  $  M $
 +
has an invariant Riemannian metric if and only if  $  T _ {x} M $
 +
has a Euclidean metric that is invariant under the linear isotropy group. There exists on the homogeneous space  $  M $
 +
a positive [[Invariant measure|invariant measure]] if and only if  $  |  \mathop{\rm det}  A | = 1 $
 +
for all  $  A \in  \mathop{\rm Is} _ {x} ( G _ {x} ) $.
 +
A homogeneous space has an invariant orientation if and only if  $  \mathop{\rm det}  A > 0 $
 +
for all  $  A \in  \mathop{\rm Is} _ {x} ( G _ {x} ) $.
 +
The invariant linear connections on  $  M $
 +
are in one-to-one correspondence with the linear mappings  $  \Lambda : \mathfrak g \rightarrow \mathfrak g \mathfrak l ( T _ {x} M ) $
 +
with the following properties:
  
A generalization of the concept of the isotropy representation is the concept of the isotropy representation of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052980/i05298046.png" />. This is a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052980/i05298047.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052980/i05298048.png" /> into the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052980/i05298049.png" /> of invertible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052980/i05298050.png" />-jets of diffeomorphisms of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052980/i05298051.png" /> taking the zero to itself. This concept is used in the study of invariant objects of higher orders.
+
$$
 +
\left . \Lambda \right | _ {\mathfrak g _ {x}  }  = \
 +
( d  \mathop{\rm Is} _ {x} ) _ {e} ,
 +
$$
 +
 
 +
$$
 +
\Lambda ( (  \mathop{\rm Ad}  h ) X)  =  \mathop{\rm Is} _ {x} ( h) \Lambda ( X)  \mathop{\rm Is} _ {x} ( h)  ^ {-} 1 \  ( h \in G _ {x} ) .
 +
$$
 +
 
 +
A generalization of the concept of the isotropy representation is the concept of the isotropy representation of order $  r $.  
 +
This is a homomorphism $  h \rightarrow j _ {x}  ^ {r} h $
 +
of the group $  G _ {x} $
 +
into the group $  L  ^ {r} ( T _ {x} M ) $
 +
of invertible $  r $-
 +
jets of diffeomorphisms of the space $  T _ {x} M $
 +
taking the zero to itself. This concept is used in the study of invariant objects of higher orders.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Sulanke,  P. Wintgen,  "Differentialgeometrie und Faserbündel" , Deutsch. Verlag Wissenschaft.  (1972)</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">  P.K. Rashevskii,  "On the geometry of homogeneous spaces" , ''Proc. Sem. Vektor. Tenzor. Anal. i Prilozh. k Geom., Mekh. i Fiz.'' , '''9''' , Moscow-Leningrad  (1952)  pp. 49–74  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E. Cartan,  "La théorie des groupes finis et continus et l'analyse situs" , Gauthier-Villars  (1930)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  S. Kobayashi,  K. Nomizu,  "Foundations of differential geometry" , '''2''' , Interscience  (1969)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Sulanke,  P. Wintgen,  "Differentialgeometrie und Faserbündel" , Deutsch. Verlag Wissenschaft.  (1972)</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">  P.K. Rashevskii,  "On the geometry of homogeneous spaces" , ''Proc. Sem. Vektor. Tenzor. Anal. i Prilozh. k Geom., Mekh. i Fiz.'' , '''9''' , Moscow-Leningrad  (1952)  pp. 49–74  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E. Cartan,  "La théorie des groupes finis et continus et l'analyse situs" , Gauthier-Villars  (1930)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  S. Kobayashi,  K. Nomizu,  "Foundations of differential geometry" , '''2''' , Interscience  (1969)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Helgason,  "Groups and geometric analysis" , Acad. Press  (1984)  pp. Chapt. II, Sect. 4</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Helgason,  "Groups and geometric analysis" , Acad. Press  (1984)  pp. Chapt. II, Sect. 4</TD></TR></table>

Revision as of 22:13, 5 June 2020


The natural linear representation of the isotropy group of a differentiable transformation group in the tangent space to the underlying manifold. If $ G $ is a group of differentiable transformations on a manifold $ M $ and $ G _ {x} $ is the corresponding isotropy subgroup at the point $ x \in M $, then the isotropy representation $ \mathop{\rm Is} _ {x} : G _ {x} \rightarrow \mathop{\rm GL} ( T _ {x} M ) $ associates with each $ h \in G _ {x} $ the differential $ \mathop{\rm Is} _ {x} ( h) = d h _ {x} $ of the transformation $ h $ at $ x $. The image of the isotropy representation, $ \mathop{\rm Is} _ {x} ( G _ {x} ) $, is called the linear isotropy group at $ x $. If $ G $ is a Lie group with a countable base acting smoothly and transitively on $ M $, then the tangent space $ T _ {x} M $ can naturally be identified with the space $ \mathfrak g / \mathfrak g _ {x} $, where $ \mathfrak g \supset \mathfrak g _ {x} $ are the Lie algebras of the groups $ G \supset G _ {x} $. Furthermore, the isotropy representation $ \mathop{\rm Is} _ {x} $ is now identified with the representation $ G _ {x} \rightarrow \mathop{\rm GL} ( \mathfrak g / \mathfrak g _ {x} ) $ induced by the restriction of the adjoint representation (cf. Adjoint representation of a Lie group) $ \mathop{\rm Ad} _ {G} $ of $ G $ to $ G _ {x} $.

If a homogeneous space $ M $ is reductive, that is, if $ \mathfrak g = \mathfrak g _ {x} \dot{+} m $, where $ m $ is an invariant subspace with respect to $ \mathop{\rm Ad} _ {G} ( G _ {x} ) $, then $ T _ {x} M $ can be identified with $ m $, while $ \mathop{\rm Is} _ {x} $ can be identified with the representation $ h \mapsto ( \mathop{\rm Ad} _ {G} h ) \mid _ {m} $( see [3]). In this case, the isotropy representation is faithful (cf. Faithful representation) if $ G $ acts effectively.

The isotropy representation and linear isotropy group play an important role in the study of invariant objects on homogeneous spaces (cf. Invariant object). The invariant tensor fields on a homogeneous space $ M $ are in one-to-one correspondence with the tensors on $ T _ {x} M $ that are invariant with respect to the isotropy representation. In particular, $ M $ has an invariant Riemannian metric if and only if $ T _ {x} M $ has a Euclidean metric that is invariant under the linear isotropy group. There exists on the homogeneous space $ M $ a positive invariant measure if and only if $ | \mathop{\rm det} A | = 1 $ for all $ A \in \mathop{\rm Is} _ {x} ( G _ {x} ) $. A homogeneous space has an invariant orientation if and only if $ \mathop{\rm det} A > 0 $ for all $ A \in \mathop{\rm Is} _ {x} ( G _ {x} ) $. The invariant linear connections on $ M $ are in one-to-one correspondence with the linear mappings $ \Lambda : \mathfrak g \rightarrow \mathfrak g \mathfrak l ( T _ {x} M ) $ with the following properties:

$$ \left . \Lambda \right | _ {\mathfrak g _ {x} } = \ ( d \mathop{\rm Is} _ {x} ) _ {e} , $$

$$ \Lambda ( ( \mathop{\rm Ad} h ) X) = \mathop{\rm Is} _ {x} ( h) \Lambda ( X) \mathop{\rm Is} _ {x} ( h) ^ {-} 1 \ ( h \in G _ {x} ) . $$

A generalization of the concept of the isotropy representation is the concept of the isotropy representation of order $ r $. This is a homomorphism $ h \rightarrow j _ {x} ^ {r} h $ of the group $ G _ {x} $ into the group $ L ^ {r} ( T _ {x} M ) $ of invertible $ r $- jets of diffeomorphisms of the space $ T _ {x} M $ taking the zero to itself. This concept is used in the study of invariant objects of higher orders.

References

[1] R. Sulanke, P. Wintgen, "Differentialgeometrie und Faserbündel" , Deutsch. Verlag Wissenschaft. (1972)
[2] S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978)
[3] P.K. Rashevskii, "On the geometry of homogeneous spaces" , Proc. Sem. Vektor. Tenzor. Anal. i Prilozh. k Geom., Mekh. i Fiz. , 9 , Moscow-Leningrad (1952) pp. 49–74 (In Russian)
[4] E. Cartan, "La théorie des groupes finis et continus et l'analyse situs" , Gauthier-Villars (1930)
[5] S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 2 , Interscience (1969)

Comments

References

[a1] S. Helgason, "Groups and geometric analysis" , Acad. Press (1984) pp. Chapt. II, Sect. 4
How to Cite This Entry:
Isotropy representation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Isotropy_representation&oldid=47447
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article