Generators of $SU(2)\times U(1)$ I'm currently reading about spontanous symmetry breaking. In particular about a Lagrangian that is invariant under $SU(2)\times U(1)$, in other words pretty standard QFT stuff. I know that the generators of $SU(2)$ are the Pauli matrices.
I was wondering how one would go about determining the generators of the group $SU(2)\times U(1) $? In particular the generator of the associated $U(1)$ symmetry. The reason for this is that I want to determine which of the generators of the symmetry $SU(2)\times U(1) $ are broken by the ground state.
A generator $T$ is said to be broken by the ground state $
    \phi = \frac{1}{\sqrt{2}}\left(\begin{array}{cc}
         0  \\
         v 
    \end{array}\right)$
if
$$Tv \neq 0.$$
I think that the generator for the corresponding $U(1)$ symmetry is $\frac{1}{2}(1+\sigma_3)$, but I dont understand how I can show this.
 A: It seems like you're fixing the representation of $\mathrm{SU}(2)$ to be $T^i=\frac{1}{2}\sigma^{i}$  (i.e., the fundamental representation).  This makes sense if you're talking about the Higgs mechanism.  
Now you want to find the generator $Y$ for the $\mathrm{U}(1)$ part of $\mathrm{SU}(2)\times\mathrm{U}(1)$.  Let's call that generator $Y$.  Now, any generator needs to satisfy the commutation relations (i.e., the structure constants) of the Lie algebra to which it belongs.  In the case of a $\mathrm{U}(1)$, it needs to commute with all the other elements of the Lie algebra, which are spanned by the $\{T^i\}$.  A suitable choice (and oftentimes, like here, your only choice) is the identity matrix times an overall constant: $Y=\tilde{Y}\,\mathbb{I}$.
For the $\mathrm{SU}(2)$ generators, the normalization is fixed by the commutation relations of the group.  However, since $Y$ commutes with everything, its normalization is unconstrained, and needs to be chosen.  We usually refer to the choice of the overall constant $\tilde{Y}$ (which we often just call $Y$, leaving the identity matrix implied) as the choice of representation.
For example, the definition of a field (like the Higgs doublet, for example) requires one to specify its representation under the gauge group of the theory we're considering.  If the gauge group is $\mathrm{SU}(2)\times\mathrm{U}(1)$, then one would specify something like the fundamental representation of $\mathrm{SU}(2)$, and $Y=\frac{1}{2}$ under $\mathrm{SU}(2)$.  (This is the representation of the Higgs doublet.)
