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!

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

Can you offer guidance or do you also need help?



Similar Discussions: Based loop groups as homogeneous spaces
  1. Loop integral (Replies: 0)

Loading...