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The Ortogonal Representation of SU2

  1. Feb 8, 2009 #1
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Feb 8, 2009 #2
    1. The problem statement, all variables and given/known data

    The rotation group SO3 may be mapped to the 2 sphere by sending a rotation matrix to its first column. Describe the fibres of this map


    2. Relevant equations

    SO3 are the special matrices P such that PPt=I and they are 3x3 matrices with det P =1


    3. The attempt at a solution

    ok so we just wanna describe the inverse image of an element. I know the image of the element is a matrix which corresponds to some rotation is sent to a 2-sphere. But i dont really know how to describe this formally or in the other direction.


    i know we have 3x3 matrices with det =1 st. ppt

    so our map (phi):SO3 ---> S2

    but dunno where to go from here.
     
  4. Feb 8, 2009 #3

    Dick

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    Think of them as what they are, rotations. If two rotations R1 and R2 have the same first column then they map (1,0,0)=e1 to the same vector, call it v (the first column of the matrix). They both map e2 and e3 to vectors orthogonal to v. They must differ only by a rotation in the plane orthogonal to v, right?
     
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