What is the connection between complete set of commuting observables and generators of the Lie group?
I have a Hamiltonian written down in second quantized formalism and I also checked that it commutes with the generators of the Lie group - $SU(3)$. How would I construct algebraically eigenstates of this Hamiltonian? How many quantum numbers do I need?
Is there any procedure to determine full symmetry of the Hamiltonian? Like in hydrogen atom, it is not $SO(3)$ but $SO(4)$.