HIGHER ALGEBRAIC STRUCTURES AND QUANTIZATION

We derive (quasi-)quantum groups in 2 + 1 dimensional topological fiel d theory directly from the classical action and the path integral. Det ailed computations are carried out for the Chem-Simons theory with fin ite gauge group. The principles behind our computations are presumably more general. We extend the classical action in a d + 1 dimensional t opological theory to manifolds of dimension less than d + 1. We then ' 'construct'' a generalized path integral which in d + 1 dimensions red uces to the standard one and in d dimensions reproduces the quantum Hi lbert space. In a 2 + 1 dimensional topological theory the path integr al over the circle is the category of representations of a quasi-quant um group. In this paper we only consider finite theories, in,which the generalized path integral reduces to a finite sum. New ideas are need ed to extend beyond the finite theories treated here.