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Left and right invariant metric on SU(2)

  1. Dec 19, 2012 #1
    1. The problem statement, all variables and given/known data

    I nedd some help to write a left-invariant and right invariant metric on SU(2)


    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Dec 19, 2012 #2

    dextercioby

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    Homework Helper

    What parametrization are you using ? And write the definitions you're working with. As per the guidelines of this particular forum, you're asked to post your work first and ask for advice/help later.

    And this looks like pure mathematics subject, why did you place it here ?
     
  4. Dec 20, 2012 #3
    I'm using the parametrization:

    \begin{array}{cc}
    x_{0} - ix_{3} & x_{1}+ix_{2} \\
    -x_{1}+iy_{2} & y_{0}+iy_{3} \end{array}

    Now I know that left invariant vector fields are obtained starting from vector tanget to the identity of the group (pauli matrices).
    by duality i can find also a basis for the left invariant forms.
    a) Is this right?

    Once I've found the left invariant forms [itex]θ^{i}[/itex]

    I can define the metric g=[itex]g_{\mu\nu}[/itex][itex]θ^{\mu}\otimesθ^{\nu}[/itex]

    b)Is this the left invariant metric i'm looking for? Are the metric coefficient totally arbitrary, a part the constraints to make the metric non degenerate and strictly positive (if i want a riemannian metric)?

    I'm sorry if this is not the the right place for my post! Thank you for helping!
     
  5. Dec 20, 2012 #4
    I'm sorry, obvyously in the matrix is x everywhere!
     
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