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Math Help - Special Unitary Matrices as Manifold

  1. #1
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    Special Unitary Matrices as Manifold

    Does anybody know how to write the charts for the special 2x2 unitary matrices SU(2) as a manifold. i.e. those satisfying GG*=I where G* is the conjugate transpose and I is the identity matrix. This problem has been bugging me for ages, would be very very gratefull for any help....
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  2. #2
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    Quote Originally Posted by johnbarkwith View Post
    Does anybody know how to write the charts for the special 2x2 unitary matrices SU(2) as a manifold. i.e. those satisfying GG*=I where G* is the conjugate transpose and I is the identity matrix. This problem has been bugging me for ages, would be very very gratefull for any help....
    The general element of SU(2) can be expressed in the form \begin{bmatrix}z&w\\-\bar{w}&\bar{z}\end{bmatrix}, where z and w are complex numbers with |z|^2+|w|^2=1. If you write z=a+ib and w=c+id then you see that SU(2) is homeomorphic to the unit sphere a^2+b^2+c^2+d^2=1 in \mathbb{R}^4. So all you need to do is to find a set of charts for the sphere (using hemispheres, which are homeomorphic to the unit ball in \mathbb{R}^3).
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    I can use hyperspherical coordinates to express a,b,c,d in 3 variables as follows:

    a=cosF
    b=cosG sinH sinF
    c=sinG sinH sinF
    d=cosH sinF

    This means that the manifold has dimension 3. I have read that I will need atleast 2 charts to cover the whole manifold, however wont the chart mapping (F,G,H) to (Z=a+ib , W=c+id)
    (-W* , Z* )

    cover the whole manifold? If not can you please show me the two charts needed? Thankyou very much for your help, this problem has been bugging me for weeks now..
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  4. #4
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    has it got something to do with the fact that there is not a one to one mapping from the point (F=0,G,H) as G and H can take on any values at this point, and it will still map to a=1,b=0,c=0,d=0 ????
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  5. #5
    MHF Contributor Drexel28's Avatar
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    Re: Special Unitary Matrices as Manifold

    Quote Originally Posted by johnbarkwith View Post
    I can use hyperspherical coordinates to express a,b,c,d in 3 variables as follows:

    a=cosF
    b=cosG sinH sinF
    c=sinG sinH sinF
    d=cosH sinF

    This means that the manifold has dimension 3. I have read that I will need atleast 2 charts to cover the whole manifold, however wont the chart mapping (F,G,H) to (Z=a+ib , W=c+id)
    (-W* , Z* )

    cover the whole manifold? If not can you please show me the two charts needed? Thankyou very much for your help, this problem has been bugging me for weeks now..
    Once you have shown that SU(2)\approx \mathbb{S}^3 as Opalg pointed out you merely need to (following his advice) find a C^0-atlas \mathfrak{A} for \mathbb{S}^3 and then compose the elements of the atlas with the homeomorphism. To find that atlas for \mathbb{S}^3 just remember the words 'stereographic projection'.
    Last edited by mr fantastic; August 6th 2011 at 10:50 PM.
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