The shuffle algebra on the factors of a word is free
type: Item
instance of: scholarly article
title: The shuffle algebra on the factors of a word is free
zbMATH Open document ID: 0565.68074
DOI: 10.1016/0097-3165(85)90020-2
author: Christophe Reutenauer
published in: Journal of Combinatorial Theory. Series A
publication date: 1985-01-01T00:00:00Z
review text: The author's introduction: ''Let \(w=a_ 1...a_ n\) be a word of length n on the alphabet A. A factor of w is a word of the form \(a_ i...a_ j\), \(i\leq j\). We show that the shuffle algebra over \({\mathbb{Q}}\) generated by the factors of w is isomorphic to a free commutative polynomial algebra. Acutally, this result is proved twice. Associate to the given word w the \((n+1)\) by \((n+1)\) matrix \(\sum^{n}_{i=1}a_ iE_{i,i+1}\), where the E's are elementary matrices. We define for each letter a the matrix \(\phi\) a: it is the coefficient of a in the above sum. Let \({\mathfrak M}\) be the (associative) algebra generated by the \(\phi\) a's and \({\mathcal L}\) be the Lie algebra generated by them. We show that the dimension of \({\mathfrak M}\) is equal to the number of distinct factors of w, while the dimension of \({\mathcal L}\) is equal to the transcendence degree of these factors in the shuffle algebra (cf. the theorem). In the course of the proof, we obtain that the shuffle algebra generated by the factors of w is free. Acutally, we prove a more precise result: if P is a set of words containing each left factor of any of its elements, then the shuffle algebra generated by P is free over some subset of P.''
Mathematics Subject Classification ID: 68Q45
Mathematics Subject Classification ID: 20M05
Mathematics Subject Classification ID: 16S10
Mathematics Subject Classification ID: 20M35
zbMATH DE Number: 3902067
zbMATH Keywords: shuffle algebra
zbMATH Keywords: free commutative polynomial algebra
zbMATH Keywords: Lie algebra
MaRDI profile type: Publication
full work available at URL: 0097-3165(85)90020-2
OpenAlex ID: W2007543059
cites work: Q5842421
cites work: Q4065692
cites work: Q4079524
cites work: Q3659988
cites work: Lie elements and an algebra associated with shuffles
cites work: Q3946345
description: scientific article; zbMATH DE number 3902067