  We briefly review a theoretical picture of the quantum Hall effect in which
gauge symmetry of non-relativistic quantum mechanics, the representation theory
of U(1)-current algebra, and the study of odd, positive integral quadratic
forms on N-dimensional integral lattices play fundamental roles. We present a
classification of so-called minimal incompressible quantum Hall fluids for low
values of the lattice dimension N. The classification contains several new
universality classes of quantum Hall fluids which are physically interesting.
