Area-preserving diffeomorphisms, W_{\infty} and $U_{q}(sl(2)) in   Chern-Simons theory and Quantum Hall system
  We discuss a quantum \qa symmetry in Landau problem, which naturally arises due to the relation between the \qa and the group of magnetic translations. The last one is connected with the \w and area-preserving (symplectic) diffeomorphisms which are the canonical transformations in the two-dimenssional phase space. We shall discuss the hidden quantum symmetry in a $2+1$ gauge theory with the Chern-Simons term and in aQuantum Hall system which are both connected with the Landau problem. 
Ian I. Kogan