Area-preserving diffeomorphisms, W_{\infty} and $U_{q}(sl(2)) in Chern-Simons theory and Quantum Hall system

TL;DR

This paper explores the hidden quantum symmetries in Landau problems, Chern-Simons theory, and Quantum Hall systems, revealing connections with area-preserving diffeomorphisms and quantum algebras like W_infinity and U_q(sl(2)).

Contribution

It uncovers the role of quantum  symmetry and its relation to magnetic translations and area-preserving diffeomorphisms in 2+1 gauge theories and Quantum Hall systems.

Findings

Identification of quantum  symmetry in Landau problem

Connection between  symmetry and W_infinity algebra

Revealing the role of quantum symmetries in Chern-Simons and Quantum Hall systems

Abstract

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.