Is spin 1 described by $SO(3)$ or $SU(2)$

Question

What spin is described by which rotation group? I always only find information about spin-1/2

Answer 1

Quantum spin in nonrelativistic Quantum Mechanics is generally associated either with the projective unitary representations of the rotation group SO(3) or with the vector unitary representations of the special unitary group SU(2). To be more precise, spin comes naturally from the projective unitary representations of the full 3D-Galilei group, but only for angular momentum/rotation symmetry purposes, it is enough for one to restrict only to a subgroup of it isomorphic to SO(3).

Therefore, spin 1 we can describe in a proper (rigged) Hilbert space environment by the linear unitary representations of the SU(2) group which are in 1-1 correspondence with the representations of the su(2) Lie algebra by (essentially) self-adjoint operators.