In quantum mechanics, ladder operators could be constructed within SU(2). The examples should be ladder operators in Quantum Harmonic Oscillator and ladder operators in angular part of Hydrogen Atom (Lx + i Ly, Lx - i Ly). In Field Theory, QED SU(2) and QCD SU(3), the creation and annihilation operators (an extended version of simple ladder operator) could also be constructed. And finally in SU(N), the infinite dimensional Quantum Harmonic Oscillator also allows for the construction of ladder operator. My question is: is that because SU(2) is the subgroup of all SU(N) group? in this way the ladder operator in SU(2) could be extended to SU(N)? Is there any good reference helps me understanding ladder operator? And the Factorial Methods solving differential equations? Is there any relation between ladder operator methods solving differential equation and Symmetry Methods solving differential equation (relying on Killing vector Field)? Thanks a lot!