Le support de l'algèbre de Lie libre. (Support of the free Lie algebra)
type: Item
instance of: scholarly article
title: Le support de l'algèbre de Lie libre. (Support of the free Lie algebra)
zbMATH Open document ID: 0698.17007
DOI: 10.1016/0012-365X(89)90305-1
published in: Discrete Mathematics
publication date: 1989-01-01T00:00:00Z
review text: Let \({\mathbb{Z}}<X>\) be a free associative ring with a base X. The ring \({\mathbb{Z}}<X>\) is a Lie ring relative to the multiplication \([a,b]=ab-ba\). Denote by L(X) a Lie ring generated by X in \({\mathbb{Z}}<X>\). Then L(X) is a free Lie ring with base X. There exists a unique involution * in the ring \({\mathbb{Z}}<X>\) such that \(x*=x\) for all \(x\in X.\) Theorem. Let f be a monomial in \({\mathbb{Z}}<X>\). Then the following are equivalent: (i) f does not occur in any element from L(X); (ii) either \(f=x^ n\), \(n\geq 2\), \(x\in X\), or \(f=f*\) and f has even length.
Mathematics Subject Classification ID: 17B35
Mathematics Subject Classification ID: 16S10
Mathematics Subject Classification ID: 17B60
zbMATH DE Number: 4144213
zbMATH Keywords: free associative ring
zbMATH Keywords: free Lie ring
zbMATH Keywords: involution
author: G. H. E Duchamp
author: Jean-Yves Thibon
reviewed by: Vyacheslav A. Artamonov
MaRDI profile type: Publication
cites work: Q5655596
cites work: Q3659988
description: scientific article; zbMATH DE number 4144213