I need to understand the coset, $SU(2)/U(1)$ for the fundamental representation. How would I go about doing so?
From my understanding of coset it means that any transformation of $SU(2)$ mod a $U(1) $ are equivalent. In other words any transformation related by, $$ e^{i\phi } e ^{i \frac{1}{2}\sigma^a \alpha^a} = e ^ {i \frac{1}{2}\sigma^a \alpha^a} $$ is equivalent. But since no linear combination of Pauli matrices forms the identity (in other words $U(1) $ is not a normal subgroup of $SU(2)$) I can't just eliminate one of the generators in this way. How would I go about getting the generators for this group?
Note: I am a physicist with little formal mathematics training so I would prefer an informal explanation.