Classification of quantum Hall universality classes by ${\mathit{W}}_{1+\mathrm{\infty }}$ symmetry

We show how two-dimensional incompressible quantum fluids and their excitations can be viewed as ${\mathit{W}}_{1+\mathrm{\infty }}$ edge conformal field theories, thereby providing an algebraic characterization of incompressibility. The Kac-Radul representation theory of the ${\mathit{W}}_{1+\mathrm{\infty }}$ algebra leads then to a purely algebraic complete classification of hierarchical quantum Hall states, which encompasses all measured fractions. Spin-polarized electrons in single-layer devices can only have Abelian anyon excitations.