Braids, \(q\)-binomials, and quantum groups (Q1271152)

Let \(\mathcal B\) be the collection of all braid groups viewed as a category: the author considers monoidal representations of \(\mathcal B\) on a vector space \(X\). He points out that a linear operator \(R\colon X\otimes X\to X\otimes X\) defines a monoidal representation of \(\mathcal B\) if and only if it is invertible and satisfies the Yang-Baxter equation. In particular, if \(X\) is one-dimensional, one can take any invertible linear operator. If \(X\) is one-dimensional the generators of the braid group act as a multiplication by a scalar \(q\). He then shows how to reproduce well-known \(q\)-identities such as Pascal's, Vandermonde's, and Cauchy's, by translating braid identities into identities in suitable representations. He concludes by describing a way of extending this work by considering higher-dimensional representations obtained through the theory of quasitriangular Hopf algebras.