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

• May 15th 2011, 11:25 AM
alexandrabel90
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?
• May 15th 2011, 11:56 AM
Tinyboss
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?
• May 15th 2011, 12:01 PM
alexandrabel90
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 PM
alexandrabel90
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)