1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    User Avatar
    Science Advisor
    Homework Helper

    Just use [ itex ] [ /itex ].

    Daniel.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Proving differentiability of function on a Lie group.
Loading...