Braids, Q-binomials and Quantum Groups

The classical identities between the q-binomial coefficients and factorials can be generalized to a context where numbers are replaced by braids. More precisely, for every pair i, n of natural numbers, there is defined an element b (n) i of the braid group algebra kBn, and these satisfy analogs of the classical identities for the binomial coefficients. By choosing representations of the braid groups, one obtains numerical or matrix realizations of these identities, in particular one recovers the q-identities in this way. These binomial braids b (n) i play a crucial role in a simple definition of a family of quantum groups, including the quantum groups U q (C) of Drinfeld and Jimbo.