Non-commuting coordinates in vortex dynamics and in the Hall effect, related to "exotic" Galilean symmetry

Vortex dynamics in a thin superfluid ${}^4$He film as well as in a type II superconductor is described by the classical counterpart of the model advocated by Peierls, and used for deriving the ground states of the Fractional Quantum Hall Effect. The model has non-commuting coordinates, and is obtained by reduction from a particle associated with the ``exotic'' extension of the planar Galilei group.Comment: To appear in the proceedings of the International Workshop {\it Nonlinear Physics: Theory and Experiment.}{\rm II}. Gallipoli, (Lecce, Italy), to be published by World Scientific. LaTex, 7 pages, no figure

On Schr\"odinger superalgebras

We construct, using the supersymplectic framework of Berezin, Kostant and others, two types of supersymmetric extensions of the Schr\"odinger algebra (itself a conformal extension of the Galilei algebra). An `$I$-type' extension exists in any space dimension, and for any pair of integers $N_+$ and $N_-$. It yields an $N=N_++N_-$ superalgebra, which generalizes the N=1 supersymmetry Gauntlett et al. found for a free spin-\half particle, as well as the N=2 supersymmetry of the fermionic oscillator found by Beckers et al. In two space dimensions, new, `exotic' or `$IJ$-type' extensions arise for each pair of integers $\nu_+$ and $\nu_-$, yielding an $N=2(\nu_++\nu_-)$ superalgebra of the type discovered recently by Leblanc et al. in non relativistic Chern-Simons theory. For the magnetic monopole the symmetry reduces to \o(3)\times\osp(1/1), and for the magnetic vortex it reduces to \o(2)\times\osp(1/2).Comment: On Schr\"odinger superalgebras, no figurs. Published versio