Braids, \(q\)-binomials, and quantum groups
type: Item
instance of: scholarly article
title: Braids, \(q\)-binomials, and quantum groups
zbMATH Open document ID: 0940.20043
author: Marcelo Aguiar
published in: Advances in Applied Mathematics
publication date: 1999-02-14T00:00:00Z
review text: 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.
reviewed by: Stefano Capparelli
Mathematics Subject Classification ID: 20F36
Mathematics Subject Classification ID: 05A30
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 18D10
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 05A10
Mathematics Subject Classification ID: 11B65
zbMATH DE Number: 1221689
zbMATH Keywords: braid groups
zbMATH Keywords: \(q\)-identities
zbMATH Keywords: quantum groups
zbMATH Keywords: monoidal representations
zbMATH Keywords: braid identities
zbMATH Keywords: quasi-triangular Hopf algebras
MaRDI profile type: Publication
full work available at URL: aama.1998.0585
OpenAlex ID: W2050638249
cites work: Two element lattice permutation numbers and their \(q\)-generalization
cites work: FROM LARGE CARDINALS TO BRAIDS VIA DISTRIBUTIVE ALGEBRA
cites work: Q3819193
cites work: Q3998721
cites work: Duality for base-changing morphisms of vector bundles, modules, Lie algebroids and Poisson structures
cites work: A q-difference analogue of \(U({\mathfrak g})\) and the Yang-Baxter equation
cites work: Q4317887
cites work: Introduction to quantum groups
cites work: Q4857364
cites work: Rings with several objects
cites work: Observations on group rings
cites work: A characterization of the borel-like subalgebras of quantum enveloping algebras
cites work: Bilinear form of real configuration of hyperplanes
cites work: On the foundations of combinatorial theory I. Theory of M�bius Functions
cites work: On the Foundations of Combinatorial Theory IV Finite Vector Spaces and Eulerian Generating Functions
cites work: Q5592270
DOI: 10.1006/AAMA.1998.0585
description: scientific article; zbMATH DE number 1221689