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Homework Help: Compute volume of group SU(2)

  1. May 3, 2013 #1
    1. The problem statement, all variables and given/known data

    Compute the volume of the group SU(2)

    2. Relevant equations

    Possibly related: in a previous part of the problem I showed that any element

    [itex]g = cos(\theta) + i \hat{n} \cdot \vec{\sigma}sin(\theta)[/itex]

    3. The attempt at a solution

    How do I compute the infinitesimal volume element dV?
  2. jcsd
  3. May 4, 2013 #2


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    Gold Member

    There are a few ways to do this, but it depends on where you are coming from, type of course, etc. If you know that ##SU(2)=S^3##, you can use your parameterization to define coordinates on the 3-sphere as imbedded in ##\mathbb{R}^4##. If you know some more geometry, you could construct the Lie-algebra valued 1-form ##\omega = g^{-1} dg## and then use that to construct a volume form.

    In order to be more helpful (within the rules of the forum that say only give hints), you'd have to be more precise about what you already know.
  4. May 6, 2013 #3
    I have heard somewhere(I don't remember where) that
    U(n)/U(n-1)=S2n-1,where Sn is n dimensional sphere embedded in Rn+1.With U(1)=2∏ one can go on for simple manipulation to get U(n).But I can not go for any book which contain any information about this.May be someone can provide any reference.
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