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 $\displaystyle U = \begin{bmatrix}a&b\\c&d\end{bmatrix}$ 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 $\displaystyle U^{-1} = \begin{bmatrix}d&-b\\-c&a\end{bmatrix}$. Also, $\displaystyle U^* = \begin{bmatrix}\bar{a}&\bar{c}\\\bar{b}&\bar{d}\en d{bmatrix}$. If U is unitary then these two matrices are the same, so comparing matrix entries you see that $\displaystyle c=-\bar{b}$ and $\displaystyle d=\bar{a}$. Therefore $\displaystyle U = \begin{bmatrix}a&b\\-\bar{b}&\bar{a}\end{bmatrix}$, 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 $\displaystyle |x|^2 + |y|^2 + |z|^2 + |w|^2 = 1$, which is the equation of the sphere S^3 in R^4. Finally, I'll leave you to check that the map $\displaystyle \begin{bmatrix}x+iy&z+iw\\-z+iw&x-iy\end{bmatrix} \mapsto x+iy+jz+kw$ gives an isomorphism from SU(2) to the group of unit quaternions.