Isotropy representation

<h2 id="construction">Construction</h2>
Given a <a href="/facts/Lie_group_action/IIlu4h7M">Lie group action</a> 
 
 
 
 (
 G
 ,
 σ
 )
 
 
 {\displaystyle (G,\sigma )}
 
 on a manifold M, if Go is the <a href="/facts/Stabilizer_(group_theory)/Vh9VtUUw">stabilizer</a> of a point o (isotropy subgroup at o), then, for each g in Go, 
 
 
 
 
 σ
 
 g
 
 
 :
 M
 →
 M
 
 
 {\displaystyle \sigma _{g}:M\to M}
 
 fixes o and thus taking the derivative at o gives the map 
 
 
 
 (
 d
 
 σ
 
 g
 
 
 
 )
 
 o
 
 
 :
 
 T
 
 o
 
 
 M
 →
 
 T
 
 o
 
 
 M
 .
 
 
 {\displaystyle (d\sigma _{g})_{o}:T_{o}M\to T_{o}M.}
 
 By the <a href="/facts/Chain_rule/aMLYxP0x">chain rule</a>,

(
        d
        
          σ
          
            g
            h
          
        
        
          )
          
            o
          
        
        =
        d
        (
        
          σ
          
            g
          
        
        ∘
        
          σ
          
            h
          
        
        
          )
          
            o
          
        
        =
        (
        d
        
          σ
          
            g
          
        
        
          )
          
            o
          
        
        ∘
        (
        d
        
          σ
          
            h
          
        
        
          )
          
            o
          
        
      
    
    {\displaystyle (d\sigma _{gh})_{o}=d(\sigma _{g}\circ \sigma _{h})_{o}=(d\sigma _{g})_{o}\circ (d\sigma _{h})_{o}}

and thus there is a representation:

ρ
        :
        
          G
          
            o
          
        
        →
        GL
        ⁡
        (
        
          T
          
            o
          
        
        M
        )
      
    
    {\displaystyle \rho :G_{o}\to \operatorname {GL} (T_{o}M)}

given by

ρ
 (
 g
 )
 =
 (
 d
 
 σ
 
 g
 
 
 
 )
 
 o
 
 
 
 
 {\displaystyle \rho (g)=(d\sigma _{g})_{o}}
 
.
It is called the isotropy representation at o. For example, if 
 
 
 
 σ
 
 
 {\displaystyle \sigma }
 
 is a <a href="/facts/Inner_automorphism/awJPzhDa">conjugation</a> action of G on itself, then the isotropy representation 
 
 
 
 ρ
 
 
 {\displaystyle \rho }
 
 at the identity element e is the <a href="/facts/Adjoint_representation/bWG4c2su">adjoint representation</a> of 
 
 
 
 G
 =
 
 G
 
 e
 
 
 
 
 {\displaystyle G=G_{e}}
 
.

<ul><li><a href="http://www.math.toronto.edu/karshon/grad/2009-10/2010-01-11.pdf">http://www.math.toronto.edu/karshon/grad/2009-10/2010-01-11.pdf</a></li>
<li><a href="https://www.encyclopediaofmath.org/index.php/Isotropy_representation">https://www.encyclopediaofmath.org/index.php/Isotropy_representation</a></li>
<li>Kobayashi, Shoshichi; Nomizu, Katsumi (1996). <a href="/facts/Foundations_of_Differential_Geometry/lgGBjhhd">Foundations of Differential Geometry, Vol. 1</a> (New ed.). Wiley-Interscience. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-471-15733-3.</li></ul>

Isotropy representation open-in-new

Isotropy representation