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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- May 15th 2011, 11:25 AMalexandrabel90Bijective 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?

- May 15th 2011, 11:56 AMTinyboss
U(1) consists of complex numbers which are the inverse of their conjugate.

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

Can you finish from there? - May 15th 2011, 12:01 PMalexandrabel90
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.. - May 15th 2011, 02:56 PMalexandrabel90
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)