Algebraic Aspects of the Fractional Quantum Hall Effect

Some algebraic issues of the FQHE are presented. First, it is shown that on the space of Laughlin wavefunctions describing the $\nu =1/m$ FQHE, there is an underlying $W_{\infty}$ algebra, which plays the role of a spectrum generating algebra and expresses the symmetry of the ground state. Its generators are expressed in a second quantized language in terms of fermion and vortex operators. Second, we present the naturally emerging algebraic structure once a general two-body interaction is introduced and discuss some of its properties.