Natural endomorphisms of quasi-shuffle Hopf algebras

Jean-Christophe Novelli; Frédéric Patras; Jean-Yves Thibon

Bulletin de la Société Mathématique de France (2013)

  • Volume: 141, Issue: 1, page 107-130
  • ISSN: 0037-9484

Abstract

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The Hopf algebra of word-quasi-symmetric functions ( 𝐖𝐐𝐒𝐲𝐦 ), a noncommutative generalization of the Hopf algebra of quasi-symmetric functions, can be endowed with an internal product that has several compatibility properties with the other operations on 𝐖𝐐𝐒𝐲𝐦 . This extends constructions familiar and central in the theory of free Lie algebras, noncommutative symmetric functions and their various applications fields, and allows to interpret 𝐖𝐐𝐒𝐲𝐦 as a convolution algebra of linear endomorphisms of quasi-shuffle algebras. We then use this interpretation to study the fine structure of quasi-shuffle algebras (MZVs, free Rota-Baxter algebras...). In particular, we compute their Adams operations and prove the existence of generalized Eulerian idempotents, that is, of a canonical left-inverse to the natural surjection map to their indecomposables, allowing for the combinatorial construction of free polynomial generators for these algebras.

How to cite

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Novelli, Jean-Christophe, Patras, Frédéric, and Thibon, Jean-Yves. "Natural endomorphisms of quasi-shuffle Hopf algebras." Bulletin de la Société Mathématique de France 141.1 (2013): 107-130. <http://eudml.org/doc/272674>.

@article{Novelli2013,
abstract = {The Hopf algebra of word-quasi-symmetric functions ($\{\bf WQSym\}$), a noncommutative generalization of the Hopf algebra of quasi-symmetric functions, can be endowed with an internal product that has several compatibility properties with the other operations on $\{\bf WQSym\}$. This extends constructions familiar and central in the theory of free Lie algebras, noncommutative symmetric functions and their various applications fields, and allows to interpret $\{\bf WQSym\}$ as a convolution algebra of linear endomorphisms of quasi-shuffle algebras. We then use this interpretation to study the fine structure of quasi-shuffle algebras (MZVs, free Rota-Baxter algebras...). In particular, we compute their Adams operations and prove the existence of generalized Eulerian idempotents, that is, of a canonical left-inverse to the natural surjection map to their indecomposables, allowing for the combinatorial construction of free polynomial generators for these algebras.},
author = {Novelli, Jean-Christophe, Patras, Frédéric, Thibon, Jean-Yves},
journal = {Bulletin de la Société Mathématique de France},
keywords = {quasi-shuffle; word quasi-symmetric function; convolution; Hopf algebra; surjection; Adams operation; eulerian idempotent; multiple zeta values},
language = {eng},
number = {1},
pages = {107-130},
publisher = {Société mathématique de France},
title = {Natural endomorphisms of quasi-shuffle Hopf algebras},
url = {http://eudml.org/doc/272674},
volume = {141},
year = {2013},
}

TY - JOUR
AU - Novelli, Jean-Christophe
AU - Patras, Frédéric
AU - Thibon, Jean-Yves
TI - Natural endomorphisms of quasi-shuffle Hopf algebras
JO - Bulletin de la Société Mathématique de France
PY - 2013
PB - Société mathématique de France
VL - 141
IS - 1
SP - 107
EP - 130
AB - The Hopf algebra of word-quasi-symmetric functions (${\bf WQSym}$), a noncommutative generalization of the Hopf algebra of quasi-symmetric functions, can be endowed with an internal product that has several compatibility properties with the other operations on ${\bf WQSym}$. This extends constructions familiar and central in the theory of free Lie algebras, noncommutative symmetric functions and their various applications fields, and allows to interpret ${\bf WQSym}$ as a convolution algebra of linear endomorphisms of quasi-shuffle algebras. We then use this interpretation to study the fine structure of quasi-shuffle algebras (MZVs, free Rota-Baxter algebras...). In particular, we compute their Adams operations and prove the existence of generalized Eulerian idempotents, that is, of a canonical left-inverse to the natural surjection map to their indecomposables, allowing for the combinatorial construction of free polynomial generators for these algebras.
LA - eng
KW - quasi-shuffle; word quasi-symmetric function; convolution; Hopf algebra; surjection; Adams operation; eulerian idempotent; multiple zeta values
UR - http://eudml.org/doc/272674
ER -

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