In physics people often talk about the singlet representation 1 of a Lie group, meaning the 1-dimensional trivial representation, which maps everything to the identity.
Sometimes it seems to me that people assume that any 1-dimensional representation is the trivial one. This fact is true for SU(2), and my question is:
Under what conditions does a Lie group only have one 1-dimensional representation?
My working so far:
If $G$ is Lie group and $R: G \to GL(1;\mathbb{C}) \cong U(1) $ a representation, which we will define to be a smooth group homomorphism, then the commutator subgroup of $G$ is mapped to the identity. So a relevant question is:
Under what conditions do the commutators of a Lie group generate the whole group?
Many thanks for your help