AREA-PRESERVING DIFFEOMORPHISMS, W∞ AND ${\mathcal U}_q [{\rm sl}(2)]$ IN CHERN–SIMONS THEORY AND THE QUANTUM HALL SYSTEM

Abstract

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.