On an article by W. Magnus on the Fricke characters of free groups
type: Item
instance of: scholarly article
title: On an article by W. Magnus on the Fricke characters of free groups
zbMATH Open document ID: 0967.16010
author: Jacques Peyrière
published in: Journal of Algebra
publication date: 2000-10-17T00:00:00Z
review text: The paper is inspired by the Fricke character of the free group \(\Gamma_{\langle a_1,\dots,a_n\rangle}\) with \(n\) generators \(a_1,\dots,a_n\) which associates to every element \(w\in\Gamma_{\langle a_1,\dots,a_n\rangle}\) a polynomial \(P_w(x_1,\dots,x_n)\) in the traces of the \(2^n-1\) products \(x_{j_1}\cdots x_{j_k}\), \(1\leq j_1<\cdots<j_k\leq n\), with the following property. For any homomorphism \(\phi\colon\Gamma_{\langle a_1,\dots,a_n\rangle}\to\text{SL}_2(\mathbb{C})\) the trace of \(\phi(w)\) is equal to \(P_w(\text{tr}(\phi(a_1)),\dots,\text{tr}(\phi(a_n)))\). \textit{W. Magnus} [Math. Z. 170, 91-103 (1980; Zbl 0433.20033)] elucidated the structure of the ideal of polynomials in \(2^n-1\) variables which vanish on any system of traces of \(\phi(a_{j_1}\cdots a_{j_n})\). Since the Fricke character is the evaluation of the trace of products of generic \(2\times 2\) matrices under the additional assumption that the determinants are equal to 1, the idea of the author of the present paper is to involve methods typical for algebras with polynomial identities. He considers the more general situation of algebras which are algebraic of degree 2 over a field and shows that these algebras are either \(2\times 2\) matrix algebras or algebras with one- or two-dimensional factor algebra modulo the nilpotent of class 3 Jacobson radical. The author makes applications to Fricke characters. He also establishes some relations between generic \(2\times 2\) matrices. Some of them are already known (maybe in another form) in the theory of PI-algebras.
reviewed by: Vesselin Drensky
Mathematics Subject Classification ID: 16R30
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 20G20
zbMATH DE Number: 1471835
zbMATH Keywords: algebras with polynomial identities
zbMATH Keywords: generic matrices
zbMATH Keywords: representations of free groups
zbMATH Keywords: Fricke characters
zbMATH Keywords: algebraic algebras of degree 2
MaRDI profile type: Publication
full work available at URL: jabr.2000.8293
OpenAlex ID: W2090437914
cites work: Minimum-dimension trace maps for substitution sequences
cites work: Trace Maps
cites work: Characters of free groups represented in the two‐dimensional special linear group
cites work: Rings of Fricke characters and automorphism groups of free groups
cites work: Q3128798
cites work: Polynomials associated with endomorphisms of free groups
cites work: The invariant theory of \(n\times n\) matrices
cites work: Finite basing of the identities of a matrix algebra of second order over a field of characteristic zero
cites work: Q4291228
cites work: On Special Linear Characters of Free Groups of Rank n ≧4
DOI: 10.1006/JABR.2000.8293
description: scientific article; zbMATH DE number 1471835