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Homework Help: Based loop groups as homogeneous spaces

  1. Nov 2, 2012 #1
    1. The problem statement, all variables and given/known data
    Let G be a compact connected Lie group define the loop group and the based loop group as [itex] LG = \{ \gamma \in C^\infty(S^1,G) \}, \Omega G = \{ \gamma \in LG : \gamma(e_{S^1}) = e_G \} [/itex] (choose whatever identification of the circle [itex] S^1 [/itex] you like ). Show that [itex] \Omega G [/itex] is a homogeneous space over [itex] LG [/itex].

    2. Relevant equations

    The loop group LG is a group under pointwise multiplication, pointwise inverse, and has as its identity the constant loop at [itex] e_G [/itex].

    3. The attempt at a solution
    The idea is supposed to be as follows: LG acts on [itex] \Omega G [/itex] by conjugation. Then the stabilizer at identity of [itex] \Omega G [/itex] (also the constant loop at identity) is supposed to be the set of all constant loops, which we may identify with G. Then [itex] \Omega G = LG/G [/itex] which turns [itex] \Omega G [/itex] into a homogeneous space.

    However, I think it's pretty clear that the stabilizer of any group action on a subgroup by conjugation is the entire group. Where do the constant functions come in?
  2. jcsd
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