Quasi-shuffle products
type: Item
instance of: scholarly article
title: Quasi-shuffle products
zbMATH Open document ID: 0959.16021
DOI: 10.1023/A:1008791603281
author: Michael E. Hoffman
published in: Journal of Algebraic Combinatorics
publication date: 2001-01-30T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/9907173
review text: Let \(k\) be a subfield of \(\mathbb{C}\), \(A\) a locally finite graded set, and let \(\mathcal A=k\langle A\rangle\) be the graded noncommutative polynomial algebra. Given a commutative associative operation on \(A\) that adds degrees, the author defines a commutative multiplication \(*\) on \(\mathcal A\), called the quasi-shuffle product. Moreover, he shows that there is a Hopf algebra structure \((\mathcal A,*,\Delta)\) and an isomorphism \(\exp\) of the shuffle Hopf algebra \((\mathcal A,\text{III},\Delta)\) onto \((\mathcal A,*,\Delta)\) such that both the set \(L\) of Lyndon words on \(A\) and their images \(\exp(L)\) freely generate the algebra \((\mathcal A,*)\). The graded dual of \((\mathcal A,*,\Delta)\) and a \(q\)-deformation \(*_q\) are also studied, and several examples are discussed.
reviewed by: Andrei Marcus
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 16S36
Mathematics Subject Classification ID: 16S80
Mathematics Subject Classification ID: 05E05
zbMATH DE Number: 1428365
zbMATH Keywords: Hopf algebras
zbMATH Keywords: shuffle algebras
zbMATH Keywords: quasi-symmetric functions
zbMATH Keywords: noncommutative symmetric functions
zbMATH Keywords: quantum shuffle products
zbMATH Keywords: graded polynomial algebras
zbMATH Keywords: quasi-shuffle products
MaRDI profile type: Publication
arXiv ID: math/9907173
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OpenAlex ID: W1561863879
description: scientific article