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Proving differentiability of function on a Lie group.

  1. Nov 22, 2006 #1
    On page 116 of Choquet-Bruhat, Analysis, Manifolds, and Physics, Lie groups are defined, and the first exercise after that asks you to prove that
    for a Lie group [itex]G[/itex]

    [itex]
    f:G \rightarrow G; x \mapsto x^{-1}
    [/itex]

    is differentiable. I know from the previous definitions that a function [itex] f [/itex] on a manifold is differentiable at [itex]x[/itex] if

    [itex]
    \psi \circ f \circ \phi^{-1}
    [/itex]

    is differentiable, where [itex](U, \phi)[/itex] and [itex](W, \psi)[/itex] are charts for neighborhoods of [itex] x [/itex] and [itex]y=f(x)=x^{-1}[/itex]. It's probably an indication of how weak my analysis is, but I don't see how to proceed from there. Can anyone point me in the right direction?
     
    Last edited: Nov 22, 2006
  2. jcsd
  3. Nov 22, 2006 #2
    In the books which I have used your exercise is the definition of the Lie group :smile: !?!.
     
  4. Nov 22, 2006 #3
    Ah, OK, I see it now, you just prove it from the definition of a Lie group, which is that the map
    from [itex]G \times G \rightarrow G; (x, y) \mapsto x y^{-1}[/itex] is differentiable. Just let [itex]x = e[/itex], the identity element. I was making it way too difficult. BTW, does anyone know how to keep inline Latex from showing up above the line like that.
     
    Last edited: Nov 22, 2006
  5. Nov 23, 2006 #4

    dextercioby

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    Just use [ itex ] [ /itex ].

    Daniel.
     
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