Anyons and the Landau problem in the noncommutative plane

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The Landau problem in the noncommutative plane is discussed in the context of realizations of the two-fold centrally extended planar Galilei group and the anyon theory.
6 pages, typos corrected. Based on the talks presented at the International Workshop "Supersymmetries and Quantum Symmetries", JINR, Dubna, Russia, 2005 and Summer Mini-Workshop in Theoretical Physics, CECS, Valdivia, Chile, 2006

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