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We study all the symmetries of the free Schr\"odinger equation in thenon-commutative plane. These symmetry transformations form aninfinite-dimensional Weyl algebra that appears naturally from a two-dimensionalHeisenberg algebra generated by Galilean boosts and momenta. These infinitehigh symmetries could be useful for constructing non-relativistic interactinghigher spin theories. A finite-dimensional subalgebra is given by theSchr\"odinger algebra which, besides the Galilei generators, contains also thedilatation and the expansion. We consider the quantization of the symmetrygenerators in both the reduced and extended phase spaces, and discuss therelation between both approaches.

Symmetries of the Free Schr\"odinger Equation in the Non-Commutative Plane