Non-commutative structures were introduced, independently and around the same
time, in mathematical and in condensed matter physics (see Table~1). Souriau's
construction applied to the two-parameter central extension of the planar
Galilei group leads to the ``exotic'' particle, which has non-commuting
position coordinates. A Berry-phase argument applied to the Bloch electron
yields in turn a semiclassical model that has been used to explain the
anomalous/spin/optical Hall effects. The non-commutative parameter is
momentum-dependent in this case, and can take the form of a monopole in
momentum space.Comment: This is a contribution to the Proc. of the O'Raifeartaigh Symposium
on Non-Perturbative and Symmetry Methods in Field Theory (June 2006,
Budapest, Hungary), published in SIGMA (Symmetry, Integrability and Geometry:
Methods and Applications) at http://www.emis.de/journals/SIGMA
