  Quantum Hall universality classes can be classified by $W_{1+\infty}$
symmetry. We show that this symmetry also governs the dynamics of quantum edge
excitations. The Hamiltonian of interacting electrons in the fully-filled first
Landau level is expressed in terms of $W_{1+\infty}$ generators. The spectra
for both the Coulomb and generic short-range interactions are thus found
algebraically. We prove the one-dimensional bosonization of edge excitations in
the limit of large number of particles. Moreover, the subleading corrections
are given by the higher-spin $W_{1+\infty}$ generators, which measure the
radial fluctuations of the electron density. The resulting spectrum for the
Coulomb interaction contains a logarithmic enhancement, in agreement with
experimental observations. The spectrum for generic short-range interactions is
subleading and reproduces the classical capillary frequencies. These results
are also extended to fractional filling by using symmetry arguments.
