$W_{\infty}$ Algebras and Incompressibility in the Quantum Hall Effect
  We discuss how a large class of incompressible quantum Hall states can be characterized as highest weight states of different representations of the \Winf algebra. Second quantized expressions of the \Winf generators are explicitly derived in the cases of multilayer Hall states, the states proposed by Jain to explain the hierarchical filling fractions and the ones related by particle-hole conjugation. 
Dimitra Karabali