I just studied the theory of angular momentum in non-relativistic quantum mechanics, which defined angular momentum as generators of Lie group $SO(3)$. And by using the Lie construction relationship, we can deduce that the eigenvalues of angular momentum in 3D can only be integer or half integer.
Now I'm wondering what will happen if we consider 2D? Then to generalize the definition of angular momentum, we consider the 2-dim angular momentum as generator of $SO(2)$, but the Lie algebra of $SO(2)$ is "trivial". Then can we also deduce the eigenvalue information by this trivial Lie construction relationship?
Besides: I've read some material that said due to the topology of $SO(3)$, whose fundamental group is isomorphic to $Z/2Z$ (order is 2), hence there are only two types of spin can exist in 3D. But for $SO(2)$, whose fundamental group is isomorphic to $Z$ (order is infinite), then there will be infinite type of spin in 2D. I'm wondering whether these fact can be related to the eigenvalue problem that I asked above, that is: why eigenvalue properties can be related to topological property of group that closely, is there a general mathematic relationship?
