  The problem of quantizing a particle on a 2-sphere has been treated by
numerous approaches, including Isham's global method based on unitary
representations of a symplectic symmetry group that acts transitively on the
phase space. Here we reconsider this simple model using Isham's scheme,
enriched by a magnetic flux through the sphere via a modification of the
symplectic form. To maintain complete generality we construct the Hilbert space
directly from the symmetry algebra, which is manifestly gauge-invariant, using
ladder operators. In this way, we recover algebraically the complete
classification of quantizations, and the corresponding energy spectra for the
particle. The famous Dirac quantization condition for the monopole charge
follows from the requirement that the classical and quantum Casimir invariants
match. In an appendix we explain the relation between this approach and the
more common one that assumes from the outset a Hilbert space of wave functions
that are sections of a nontrivial line bundle over the sphere, and show how the
Casimir invariants of the algebra determine the bundle topology.
