U_q(sl(2) as Dynamical Symmetry Algebra of the Quantum Hall Effect

Quantum Hall effect wavefunctions corresponding to the filling factors 1/2p+1, 2/2p+1,..., 2p/2p+1, 1, are shown to form a basis of irreducible cyclic representation of the quantum algebra U_q(sl(2)) at q^{2p+1}=1. Thus, the wavefunctions \Psi_{P/Q} possessing filling factors P/Q<1 where Q is odd and P, Q are relatively prime integers are classified in terms of U_q(sl(2)). Adopted as dynamical symmetry this leads to non--existence of a ``universal microscopic theory" of the quantum Hall effect, defined as the eigenvalue problem of a differential operator O: O \Psi_\nu =\ell_\nu \Psi_\nu. for \nu =1,1/3,2/3,..., in the complex plane.