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I have a conjecture/ possibly already a theorem

  1. Dec 20, 2008 #1
    I am not very well read so this may already exist as a theorem. If not, try to prove it, or disprove it.

    Let G be a compact group over the reals, then the maximally compact subgroup of the complexification of G is just G over the reals.

    That is the maximally compact subgroup of [tex]G_\mathbb{C}[/tex] is just [tex]
    G\left( \mathbb{R} \right)
    [/tex]


    Here's a simple example:

    SU(2) is the maximal compact subgroup of [tex]
    Sl\left( {2,\mathbb{C}} \right)
    [/tex]

    And [tex]
    SU(2)_\mathbb{C} \cong Sl\left( {2,\mathbb{C}} \right)
    [/tex]
     
  2. jcsd
  3. Dec 22, 2008 #2

    mathwonk

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    definitions would be nice. or does maximal compact mean just that? no larger compact subgroup exists? or no larger proper compact subgroup?
     
  4. Dec 22, 2008 #3

    mathwonk

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    a few minutes web search reveals, without even knowing wjhat these things mean, that any compact group is a maximal compact of its chevalley complexification.
     
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