We discuss the process to quantize systems of a charged particle on S^1 and S^2. We also add magnetic fields to allow a diversity of resulting quantum structures. The commutator algebras are shown to be the Lorentz algebras so(1, 2) and so(1, 3). However, in the case of S^2 where the magnetic field is due to a magnetic monopole at the center, we show that the constraints, expressed as operator equations, depend on the quantum number of the monopole. Therefore, the representation spaces are classified by using the monopole number.