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Question on group actions on vector spaces

  1. Jun 17, 2010 #1
    Hello,
    If I am given a vector space (e.g. [itex]\mathbb{R}^n[/itex]), and a group [itex]G[/itex] that acts on [itex]\mathbb{R}^n[/itex], what are the conditions that [itex]G[/itex] must satisfy so that for any given [itex]x\in\mathbb{R}^n[/itex] its orbit [itex]Gx[/itex] is a manifold ?
     
  2. jcsd
  3. Jun 17, 2010 #2
    If G is a Lie group acting smoothly on R^n, then the orbits will be immersed submanifolds.
     
  4. Jun 17, 2010 #3

    Office_Shredder

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    The orbit has the same group structure as G does by multiplication [tex]gx \cdot hx = (gh)x[/tex] and the obvious isomorphism [tex] g \mapsto gx[/tex]. So this means that your group G has to be a manifold. I'm guessing you'll want the multiplication to be continuous under the topology of the manifold or something, I'll have to think about it


    EDIT: Woops, not an isomorphism,. That's what happens when you use the word obvious.

    Ok after thinking about this for a bit, I have to ask: did you perhaps want G to act linearly on the vector space?
     
    Last edited: Jun 17, 2010
  5. Jun 22, 2010 #4
    Uhm...Sorry, could you please specify what does "to act linearly" mean?
    Thanks
     
  6. Jun 22, 2010 #5

    Office_Shredder

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    Every element of G is a linear map, so along with the normal group action rules you have

    [tex]g \cdot (\alpha x + \beta y) = \alpha g \cdot x + \beta g \cdot y[/tex]

    Otherwise the fact that we're living in a vector space is a happy coincidence and has very little bearing on the problem
     
  7. Jun 22, 2010 #6
    Ok. I got it, so yes...the group has to act linearly.
     
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