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A simple Lie Group is linear

  1. Nov 16, 2011 #1
    Suppose we have a simple Lie Group [itex]G[/itex], i.e, a Lie Group with a trivial center(the identity). Show that this group must be linear, i.e, we can map it to a Lie subgroup of [itex]GL(N)[/itex].

    So far, I have that from abstract algebra we can show a group with trivial center is isomorphic to the inner automorphisms on the group, let us say [itex]Inn(G)[/itex]. Also, [itex]Inn(G)[/itex] is clearly a group of diffeomorphisms on [itex]G[/itex]. I have a vague intuition that if we can show [itex]Inn(G)[/itex] is linear , that would show our desired result, although I am not sure how to make this more concrete.

    Does anyone have a few hints that might get me on the right track, or tools that I am missing? I am pondering if I can tie this to the Lie algebra of [itex]G[/itex] in some manner.

    Also to clarify, this is not part of a graded problem set. I saw it on an assignment some months ago, but didn't figure it out at the time and I want to think my way through it now. The forum rules say that graduate level coursework questions can be posted here, but if it's a problem I apologize for the misclassification.
    Last edited: Nov 16, 2011
  2. jcsd
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