Mixable shuffles, quasi-shuffles and Hopf algebras.
type: Item
instance of: scholarly article
title: Mixable shuffles, quasi-shuffles and Hopf algebras.
zbMATH Open document ID: 1103.16025
author: Li Guo
author: Kurusch Ebrahimi-Fard
published in: Journal of Algebraic Combinatorics
publication date: 2006-08-16T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0506418
review text: The shuffle product, the quasi-shuffle product, the mixable shuffle product and the Rota-Baxter algebras have arisen and been studied in various contexts, but the relations between them have been mostly overlooked. This paper studies and clarifies some important relations between them. A Rota-Baxter algebra of weight \(a\) over a commutative ring \(k\) (\(a\) is in \(k\)) is a \(k\)-algebra \(R\) and a \(k\)-linear map \(P\) from \(R\) to \(R\) satisfying \(P(x)P(y)=P(xP(y))+P(P(x)y)+aP(xy)\). The shuffle product is well-known and gives a Rota-Baxter algebra of weight 0. Mixable shuffles arose in the study of Rota-Baxter algebras, especially free Rota-Baxter algebras. The quasi-shuffle product was given by \textit{M. E. Hoffman} [J. Algebr. Comb. 11, No. 1, 49-68 (2000; Zbl 0959.16021)] in a recursive manner. We do not give the technical definitions of these products here. The authors of the paper under review show first that the recursive definition of the quasi-shuffle product has an explicit form in terms of the mixable shuffle product, and that both can be derived from a Rota-Baxter algebra of weight 1. They are both generalizations of the shuffle product. They show that the quasi-shuffle product on a locally finite set is a subalgebra of a mixable shuffle algebra. Finally, they use the Hopf algebra structures on quasi-shuffle algebras to obtain Hopf algebra structures on a free Rota-Baxter algebra, which generalizes previous results of \textit{G. E. Andrews, L. Guo, W. Keigher} and \textit{K. Ono} [Trans. Am. Math. Soc. 355, No. 11, 4639-4656 (2003; Zbl 1056.16025)].
Mathematics Subject Classification ID: 16W30
zbMATH DE Number: 5046870
zbMATH Keywords: shuffle products
zbMATH Keywords: Hopf algebras
zbMATH Keywords: quasi-shuffle products
zbMATH Keywords: free Rota-Baxter algebras
reviewed by: Earl J. Taft
MaRDI profile type: Publication
OpenAlex ID: W1971645711
arXiv ID: math/0506418
cites work: Quadri-algebras
cites work: Baxter algebras and Hopf algebras
cites work: An analytic problem whose solution follows from a simple algebraic identity
cites work: The algebra and combinatorics of shuffles and multiple zeta values
cites work: Multiple \(q\)-zeta values
cites work: On the structure of free Baxter algebras
cites work: Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula
cites work: Renormalization in quantum field theory and the Riemann-Hilbert problem. I: The Hopf algebra structure of graphs and the main theorem
cites work: Renormalization in quantum field theory and the Riemann-Hilbert problem. II: The \(\beta\)-function, diffeomorphisms and the renormalization group.
cites work: Loday-type algebras and the Rota-Baxter relation
cites work: On the associative Nijenhuis relation
cites work: On products and duality of binary, quadratic, regular operads
cites work: Rota-Baxter algebras and dendriform algebras.
cites work: Q3626820
cites work: Integrable renormalization I: The ladder case
cites work: Integrable renormalization. II: The general case
cites work: Spitzer's identity and the algebraic Birkhoff decomposition in pQFT
cites work: Q3153360
cites work: Baxter algebras and the umbral calculus
cites work: BAXTER ALGEBRAS, STIRLING NUMBERS AND PARTITIONS
cites work: Baxter algebras and shuffle products
cites work: On free Baxter algebras: Completions and the internal construction
cites work: The algebra of multiple harmonic series
cites work: Quasi-shuffle products
cites work: Relations of multiple zeta values and their algebraic expression.
cites work: On the Hopf algebra strucutre of perturbative quantum field theories
cites work: Chen's iterated integral represents the operator product expansion
cites work: Ennea-algebras
cites work: Representations of Hopf algebras arising from Lie algebras of Cartan type
cites work: Q4656995
cites work: Q4274976
cites work: Q3134850
cites work: Baxter algebras and combinatorial identities. I
cites work: Baxter algebras and combinatorial identities. II
cites work: Q5668520
cites work: Ten mathematics problems I will never solve
cites work: A Combinatorial Lemma and Its Application to Probability Theory
cites work: Q3818510
cites work: Q5590089
DOI: 10.1007/S10801-006-9103-X
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description: scientific article