Abstract

Braids on a given number of strands n can be concatenated and thereby form a group Bn. The latter possesses two different incarnations: it can be presented on a simple set of generators and relations due to E. Artin (1947), or it can be realized as the fundamental group of the space of configurations Xn of n points in the complex plane. I will explain how each of these incarnations leads to a class of representations of Bn. The topological representations arise from differential equations of Xn which are symmetric under the algebra gl(n) of nxn matrices. The algebraic representations arise instead from a deformation of this algebra known as the quantum group Uq gl(n).  Finally, I will tie the knot by relating these two classes of representations.