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Lie Group Decomposition

  1. Jul 17, 2014 #1
    Long time reader, first time poster.

    Originally, it was my contention that all Lie groups could be written as the semidirect product of a connected Lie group and a discrete Lie group. However, I no longer believe this is true.

    The next best thing I could think of was to say that a Lie group is diffeomorphic to the (smooth) product manifold of a connected Lie group and a discrete one. This is proving to be difficult!

    I'm a bit new to using general fiber bundles (I've only ever used vector bundles before!), but a friend of mine recommended me to use principal bundles. I've proven that ##\pi:G\to G/G_\mathrm{e}##, where ##G_\mathrm{e}## is the identity component of a Lie group ##G##, is a principal ##G_\mathrm{e}##-bundle. However, I really don't know what use this is to me.

    Could someone explain how this is a step in the right direction?

    Edit: Is there a particular way to get the LaTeX to render? It isn't working...
    Edit2: In another post, display math works. What about inline...?
    Edit3: Crisis averted.
     
    Last edited: Jul 17, 2014
  2. jcsd
  3. Jul 17, 2014 #2
    Clearly, I am an idiot.

    To those who might come looking for this thread later: principal bundles are nice, but it is easier to simply note that the connected components of a Lie group are diffeomorphic. :redface:
     
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