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A particular SU(2) matrix

  1. Nov 15, 2015 #1
    Hello everyone,
    I have a question that will probably turn out to be trivial. I have the following matrix:

    $$
    U=\text{diag}(e^{2i\alpha},e^{-i\alpha},e^{-i\alpha}).
    $$

    This seems to me as an SU(2) matrix in the adjoint representation since it's unitary and has determinant 1. Am I right?

    If so, for a small value of [itex]\alpha[/itex] from what combination of the generators can I obtain it?


    Thanks!
     
  2. jcsd
  3. Nov 15, 2015 #2

    fresh_42

    Staff: Mentor

    SU(3). Which are your generators?
     
  4. Nov 15, 2015 #3
    Oh you're right. Then my mistake was to try to find a combination of the J=1 generators of SU(2) while I should have looked for the SU(3) generators! Then I guess my question becomes even dumber: how do I distinguish an matrix belong to the adjoint of SU(2) from one belonging to the fundamental of SU(3)?
     
  5. Nov 15, 2015 #4

    fresh_42

    Staff: Mentor

    I don't know the automorphisms ##su(2)##. They are in ##GL(3,ℂ)##, that's right.
    Have a look here. Just calculate it.

    The fundamental group of ##SU(2,ℂ)## is ##SO(3,ℝ)## - as far as I can see - generated by the following rotations

    [tex]\left[ {\begin{array}{*{20}{c}} {e^{it}}&{0}\\ {0}&{e^{-it}} \end{array}} \right] , \left[ {\begin{array}{*{20}{c}} {cos (t)}&{sin (t)}\\ {-sin (t)}&{cos (t)} \end{array}} \right] , \left[ {\begin{array}{*{20}{c}} {cos (t)}&{i * sin (t)}\\ {i *sin (t)}&{cos (t)} \end{array}} \right][/tex]

    Here are further informations.
     
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