  Taking advantage of the two-parameter central extension of the planar Galilei
group, we construct a non relativistic particle model in the plane. Owing to
the extra structure, the coordinates do not commute. Our model can be viewed as
the non-relativistic counterpart of the relativistic anyon considered before by
Jackiw and Nair. For a particle moving in a magnetic field perpendicular to the
plane, the two parameters combine with the magnetic field to provide an
effective mass. For vanishing effective mass the phase space admits a
two-dimensional reduction, which represents the condensation to collective
``Hall'' motions and justifies the rule called ``Peierls substitution''.
Quantization yields the wave functions proposed by Laughlin to describe the
Fractional Quantum Hall Effect.
