On Schrödinger superalgebras

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Abstract
Using the supersymplectic framework of Berezin, Kostant, and others, two types of supersymmetric extensions of the Schrödinger algebra (itself a conformal extension of the Galilei algebra) were constructed. 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‐1/2 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 ν+ and ν−, yielding an N=2(ν++ν−) superalgebra of the type discovered recently by Leblanc et al. in nonrelativistic Chern–Simons theory. For the magnetic monopole the symmetry reduces to o(3)×osp(1/1), and for the magnetic vortex it reduces to o(2)×osp(1/2).
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