Hello everyone, I'm going through some lecture notes and there are some things I don't understand about the whole derivation of the angular momentum multiplet. It's said that the skew-symmetric 3x3 matrices [itex]J_i[/itex] are the infinitesimal generators of the rotation group [itex]SO(3)[/itex]. Later, however, these are operators acting on a Hilbert space spanned by vectors [itex]|jm\rangle[/itex] which can take on all different kinds of dimensions depending on the spin [itex]j[/itex]. This doesn't make any sense to me! What is the relation between these operators and the matrices and how does spin 'tell' what the dimensions of the operators should be? Can anyone help?