HIGHER ALGEBRAIC STRUCTURES AND QUANTIZATION

We derive (quasi-)quantum groups in 2 + 1 dimensional topological field theory directly from the classical action and the path integral. Detailed computations are carried out for the Chem-Simons theory with finite gauge group. The principles behind our computations are presumably more general. We extend the classical action in a d + 1 dimensional topological theory to manifolds of dimension less than d + 1. We then ''construct'' a generalized path integral which in d + 1 dimensions reduces to the standard one and in d dimensions reproduces the quantum Hilbert space. In a 2 + 1 dimensional topological theory the path integral over the circle is the category of representations of a quasi-quantum group. In this paper we only consider finite theories, in,which the generalized path integral reduces to a finite sum. New ideas are needed to extend beyond the finite theories treated here.