I am studying elementary models of the Quantum Hall Effect. I don't have a strong background in Lie algebras so I was hoping someone could elaborate on the following observation:
For a free particle constrained to a plane with a uniform magnetic field oriented normal to the plane its Hamiltonian can be written in the same form as that of a quantum harmonic oscillator: $$H = \hbar \omega_B (a^\dagger a + 1/2).$$ There will also be another mutually commuting set of bosonic creation/annihilation operators $[b,b^\dagger]$ indexing the particle's angular momentum if you choose a symmetric gauge for the vector potential.
Solve a similar problem with a particle on a sphere surrounding a magnetic monopole, and you find instead of two mutually commuting "ladder algebras" (by the way is there an official name for this?) a representation of $SU(2) \times SU(2)$; something like
$$H \propto S^- S^+ + 1/2$$
along with another set $[L^i,L^j] = \mathrm{i} \epsilon^{ijk} L^k$ from which you can also define raising and lowering operators $L^+,L^-.$
Can someone elucidate what's happening here from a geometric/algebraic point of view? I can follow the standard calculations but do not have a good image of the "big picture."
