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Regarding Lie algebras

  1. Oct 18, 2012 #1

    I thought I had a good picture of this stuff but now I suspect that I've mixed up things completely..
    For su(2) we have:
    [tex]f_{ab}\,^c = \epsilon_{ab}\,^c.[/tex]
    Which means that, for the basis of our tangentspace we get:
    [tex][\partial_a, \partial_b] = i \epsilon_{ab}\,^c\, \partial_c[/tex]
    Where we can see that the space of linear approximations is closed under the Lie bracket product.

    But what about the definition of a smooth manifold being locally euclidean?
    For R^n we know that our orthogonal partial derivatives commute,
    [tex]\partial_x\partial_y - \partial_y\partial_x = 0.[/tex]

    Shouldnt we then expect the orthogonal derivatives to commute in su(2) as well?
    This would mean that our structure constants are 0 both when a=b and when a is not equalt to b (for a orthonomal basis).
    In other words.. it looks like the Lie algebra should be quite trivial.

    This is of course not the case, but where does my mind make the mistake in this resoning?

    // Kontilera
  2. jcsd
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