Thread: 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?

U(1) consists of complex numbers which are the inverse of their conjugate.

An element of SU(2) has the form such that the determinant is 1.

Can you finish from there?

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..

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)