Graphs and Bergman's GK gap theorem
type: Item
instance of: scholarly article
title: Graphs and Bergman's GK gap theorem
zbMATH Open document ID: 0812.16034
author: Harold W. jun. Ellingsen
author: Daniel R. Farkas
published in: Journal of Algebra
publication date: 1995-05-09T00:00:00Z
review text: Using an intricate analysis of the combinatorics of words, Bergman proved that an algebra over a field cannot have Gelfand-Kirillov dimension strictly between 1 and 2. The paper under review clarifies and sharpens this method. Let \(K\) be a field, let \(F = K \langle x_ 1,x_ 2,\dots,x_ n\rangle\) be the free algebra on the letters \(x_ 1,\dots,x_ n\), and let \(A\) be the monomial algebra \(F/I\), where \(I\) is an ideal in \(F\) generated by a set of words in the free monoid on \(x_ 1,\dots, x_ n\). Then the set \(B\) of nonrelation words can be taken to be a basis of \(A\), and for any positive integer \(d\) a directed graph \(\Gamma\) can be constructed whose vertices are the words in \(B\) of length \(d\), and where an arrow is drawn from \(u\) to \(v\) if there exists a word of length \(d + 1\) in \(B\) having \(u\) as prefix and \(v\) as suffix. This graph is related to that of \textit{V. Ufnarovskij} [Mat. Zametki 31, 465- 472 (1982; Zbl 0491.05041)]. The authors prove that each connected component of \(\Gamma\) contains at most one directed circuit and that for each vertex in the component there is at most one directed path leading into or out of the circuit. As a consequence, if the basis \(B\) of the monomial algebra \(A\) contains at most \(d\) words of length \(d\) for some \(d\), then it has at most \((d+ 1/2)^ 2\) words of length \(m\) for any \(m \geq 2d\), improving the bound of \(d^ 3\) obtained by Bergman. An example is provided to show that the above bound is best possible. The authors acknowledge that their results have independently been obtained also by \textit{S. Kobayashi} and \textit{Y. Kobayashi} [Proc. Am. Math. Soc. 119, 1095-1104 (1993; Zbl 0807.16025)].
reviewed by: Günter Krause
Mathematics Subject Classification ID: 16P90
Mathematics Subject Classification ID: 16S10
Mathematics Subject Classification ID: 05C38
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 16S15
Mathematics Subject Classification ID: 05C20
Mathematics Subject Classification ID: 16P10
zbMATH DE Number: 562457
zbMATH Keywords: combinatorics of words
zbMATH Keywords: Gelfand-Kirillov dimension
zbMATH Keywords: free algebra
zbMATH Keywords: monomial algebra
zbMATH Keywords: free monoid
zbMATH Keywords: nonrelation words
zbMATH Keywords: connected component
zbMATH Keywords: directed circuit
zbMATH Keywords: directed path
MaRDI profile type: Publication
full work available at URL: jabr.1994.1080
OpenAlex ID: W1970085300
DOI: 10.1006/JABR.1994.1080
Recommended article: Q4276399
Recommended article: On Algebras With Gelfand-Kirillov Dimension One
Recommended article: Q3701584
Recommended article: Piecewise periodicity structure estimates in Shirshov's height theorem.
Recommended article: Length computation of matrix subalgebras of special type.
Recommended article: Q3462985
Recommended article: Sweeping words and the length of a generic vector subspace of \(M_n(\mathbb{F})\)
Recommended article: Growth alternative for Hecke-Kiselman monoids
Recommended article: Q4385026
Recommended article: Functionally recursive rings of matrices -- two examples
description: scientific article; zbMATH DE number 562457