Hi there,
How do I prove that SU(2) = the unit quaternion.... and show that this can be thought of the sphere S^3?
SU(2) consists of 2×2 complex matrices that are unitary and have determinant 1. Suppose that is such a matrix, where a,b,c,d are complex numbers. The condition det(U) = 1 tells you that ad–bc=1, and therefore . Also, . If U is unitary then these two matrices are the same, so comparing matrix entries you see that and . Therefore , where |a|^2 + |b|^2 = 1 (because of the condition det(U) = 1).
Now let a = x+iy, b = z+iw, where x,y,z,w are real. Then , which is the equation of the sphere S^3 in R^4. Finally, I'll leave you to check that the map gives an isomorphism from SU(2) to the group of unit quaternions.