Quantum algebra structures on \(n\times n\) matrices
type: Item
instance of: scholarly article
title: Quantum algebra structures on \(n\times n\) matrices
zbMATH Open document ID: 0926.16035
author: Louis Kauffman
author: David E. Radford
published in: Journal of Algebra
publication date: 1999-10-05T00:00:00Z
review text: A quantum algebra is a triple consisting of an algebra \(A\) over a field \(k\), an invertible element \(R\in A\otimes A\), and an algebra antiisomorphism \(s\colon A\to A\) such that \(R^{-1}=(s\otimes 1_A)(R)\), \(R=(s\otimes s)(R)\), and \(R_{1,2}R_{1,3}R_{2,3}=R_{2,3}R_{1,3}R_{1,2}\). This concept can be thought of as a mechanism for computing solutions to the quantum Yang-Baxter equation or as a vehicle for generating knot invariants, 1-1 tangle invariants, or invariants of 3-manifolds. A general description of quantum algebra structures on the matrix algebra \(M_n(k)\) is developed. As a special case, the quantum algebra structures on \(M_2(k)\), \(k\) algebraically closed of characteristic 0, are determined by using the classification of the solutions to the quantum Yang-Baxter equation in dimension 2. The dual space of a finite dimensional quantum algebra is equipped with a natural structure of a quantum coalgebra, a concept defined by the authors in previous work. The 1-1 tangle invariants resulting from the quantum algebra structures of \(M_2(k)\), \(k\) algebraically closed of characteristic 0, are determined by computing in the dual quantum coalgebra.
Mathematics Subject Classification ID: 16W35
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 57M27
zbMATH DE Number: 1270859
zbMATH Keywords: quantum algebras
zbMATH Keywords: quantum coalgebras
zbMATH Keywords: quantum Yang-Baxter equation
zbMATH Keywords: 1-1 tangle invariants
zbMATH Keywords: knot invariants
zbMATH Keywords: matrix algebras
reviewed by: Sorin Dascalescu
MaRDI profile type: Publication
full work available at URL: jabr.1998.7668
OpenAlex ID: W2037757271
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DOI: 10.1006/JABR.1998.7668
description: scientific article; zbMATH DE number 1270859