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Math Help - Bijective Map from SU(2) to U(1)

  1. #1
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    Bijective Map from SU(2) to U(1)

    how do i define a bijective map form SU(2) to U(1) where SU(2) is the special unitary 2x2 matrix and U(1) is the unitary 1x1 matrix?
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  2. #2
    Senior Member Tinyboss's Avatar
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    U(1) consists of complex numbers which are the inverse of their conjugate.

    An element of SU(2) has the form \left(\begin{array}{cc}\alpha&-\overline\beta\\ \beta&\overline\alpha\end{array}\right) such that the determinant is 1.

    Can you finish from there?
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  3. #3
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    Not really. I was thinking of the determinant map but the determinant of all elements in su(n) is 1.

    Can i instead define it as a map from the matrix ( tt you defined, which is in su(n) ) to the first element a ?

    Seems wrong to me though..
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  4. #4
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    i was thinking, can i define it from

    cos(a) -isin(a)
    isin(a) cos(a) seen as a matrix


    to

    exp(ia) which is in U(1)
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