We present two (classes of) examples of gauged Laplacian operators. The first one is a model of spin-Hall effect on a noncommutative four-sphere with isospin degrees of freedom, coming from a noncommutative instanton, and invariant under the quantum group SOθ(5). The second one, a Hall effect on a quantum 2-dimensional sphere S2q, describes ‘excitations moving on the quantum sphere’ in the field of a magnetic monopole with symmetry coming from the quantum group SUq(2). For both models, ample symmetries provide a complete diagonalization.
