Mathematics

We revise the Landau problem of an electron in a constant magnetic field. The free Hamiltonian is invariant with respect to the Zak's magnetic translations. Under the assumption of periodicity the magnetic translations  generate  a non-commutative quantum torus. For rational values of the magnetic flux duality manifests as a Morita equivalence between finite modules of the non-commutative torus. Taking the spin of the electron into account we are naturally lead to a supersymmetric extension of the magnetic translations