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Difference between revisions of "Isotropy representation"

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m (tex encoded by computer)
m (fixing superscript)
 
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can be identified with  $  m $,  
 
can be identified with  $  m $,  
 
while  $  \mathop{\rm Is} _ {x} $
 
while  $  \mathop{\rm Is} _ {x} $
can be identified with the representation  $  h \mapsto (  \mathop{\rm Ad} _ {G} h ) \mid  _ {m} $(
+
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 $
see [[#References|[3]]]). In this case, the isotropy representation is faithful (cf. [[Faithful representation|Faithful representation]]) if  $  G $
 
 
acts effectively.
 
acts effectively.
  
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$$  
 
$$  
\Lambda ( (  \mathop{\rm Ad}  h ) X)  =  \mathop{\rm Is} _ {x} ( h) \Lambda ( X)  \mathop{\rm Is} _ {x} ( h)  ^ {-} 1 \  ( h \in G _ {x} ) .
+
\Lambda ( (  \mathop{\rm Ad}  h ) X)  =  \mathop{\rm Is} _ {x} ( h) \Lambda ( X)  \mathop{\rm Is} _ {x} ( h)  ^ {- 1} \  ( h \in G _ {x} ) .
 
$$
 
$$
  
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of the group  $  G _ {x} $
 
of the group  $  G _ {x} $
 
into the group  $  L  ^ {r} ( T _ {x} M ) $
 
into the group  $  L  ^ {r} ( T _ {x} M ) $
of invertible  $  r $-
+
of invertible  $  r $-jets of diffeomorphisms of the space  $  T _ {x} M $
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.
 
taking the zero to itself. This concept is used in the study of invariant objects of higher orders.
  

Latest revision as of 02:24, 16 June 2022


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=52441
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article