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We discuss a quantum [Formula: see text] symmetry in the Landau problem, which naturally arises due to the relation between [Formula: see text] and the group of magnetic translations. The latter is connected with W ∞ and area-preserving (symplectic) diffeomorphisms which are the canonical transformations in the two-dimensional phase space. We shall discuss the hidden quantum symmetry in a 2 + 1 gauge theory with the Chern–Simons term and in a quantum Hall system, which are both connected with the Landau problem.
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