Braid group representations arising from the generalized chiral Potts models
type: Item
instance of: scholarly article
title: Braid group representations arising from the generalized chiral Potts models
zbMATH Open document ID: 0770.57002
author: E. Date
author: M. Jimbo
author: Kei Miki
author: T. Miwa
published in: Pacific Journal of Mathematics
publication date: 1993-05-16T00:00:00Z
review text: A class of braid group representations are constructed for each nonsingular bilinear form \(B:(\mathbb{Z}/N\mathbb{Z})^ l\times(\mathbb{Z}/N\mathbb{Z})^ l\to\mathbb{Z}/N\mathbb{Z}\) with \(N\) odd. Let \(q\) be a primitive \(N\)-th root of unity, \(t\) a braid on \(M\) strings with \(v\) crossings and \(K=v-M+1\). Let further \(P=(p_{ij})_{1\leq i,j\leq K}\) denote a Seifert matrix for the closure of \(t\). Then associated link invariants \(\tau(t)\) are expressed as \[ \tau(t)=N^{-Kl/2}\sum_{a_ 1,\dots,a_ K}q^ Q,\quad Q=\sum_{1\leq i,j\leq K}p_{ij}B(a_ i,a_ j). \] For \(N\) an odd prime, they can be evaluated in terms of the classical Alexander polynomials if the quadratic form \(P\otimes B+{^ tP}\otimes{^ tB}\) is nonsingular. With a special choice of \(B\) these representations are Yang-Baxterized to the \({\mathfrak sl}(n)\) generalization of the chiral Potts model.
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 82B20
Mathematics Subject Classification ID: 17B37
zbMATH DE Number: 165418
zbMATH Keywords: braid group
zbMATH Keywords: representations
zbMATH Keywords: Seifert matrix
zbMATH Keywords: Alexander polynomials
MaRDI profile type: Publication
full work available at URL: pjm.1992.154.37
OpenAlex ID: W1972115019
DOI: 10.2140/PJM.1992.154.37
description: scientific article; zbMATH DE number 165418