How to find all such non-trivial homomorphisms? At the Lie algebra level, should it just be a projection to the one-dimensional sub-space arbitrarily singled out? Could there be other Lie algebra homomorphisms? I searched on-line but failed to find anything useful. And is there a method to find them for general SU(n)? Thank you.
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There is no such homomorphism. The only proper normal subgroup of $SU(2)$ is it's center $Z=\{\pm 1\}$, and that cannot be the kernel of a homomorphism to $U(1)$ since $SU(2)/Z=SO(3)$.
Ruy
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Ah, I see. Simplicity is what I missed. Thank you! – X-Naut PhD Apr 30 '22 at 00:59