Natural endomorphisms of quasi-shuffle Hopf algebras
Jean-Christophe Novelli
- Fonction : Auteur
- PersonId : 1286032
- IdRef : 050202596
F. Patras
- Fonction : Auteur
- PersonId : 3624
- IdHAL : patras-frederic
- ORCID : 0000-0002-8098-8279
- IdRef : 058707972
Jean-Yves Thibon
- Fonction : Auteur
- PersonId : 4928
- IdHAL : jean-yves-thibon
- ORCID : 0000-0002-8976-4044
- IdRef : 033276498
Résumé
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.
Format du dépôt | Notice |
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Type de dépôt | Article dans une revue |
Titre |
en
Natural endomorphisms of quasi-shuffle Hopf algebras
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Résumé |
en
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.
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Auteur(s) |
Jean-Christophe Novelli
1
, F. Patras
2
, Jean-Yves Thibon
1
1
LIGM -
Laboratoire d'Informatique Gaspard-Monge
( 3210 )
- Université de Paris-Est Marne-la-Vallée, Cité Descartes, Bâtiment Copernic, 5 bd Descartes, 77454 Marne-la-Vallée Cedex 2
- France
2
JAD -
Laboratoire Jean Alexandre Dieudonné
( 26 )
- Université de Nice - Sophia Antipolis U.M.R. no 6621 du C.N.R.S. Parc Valrose 06108 Nice Cedex 02 France
- France
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Comité de lecture |
Oui
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Vulgarisation |
Non
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Langue du document |
Anglais
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Nom de la revue |
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Date de production/écriture |
2011-01-04
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Audience |
Internationale
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Date de publication |
2013
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Volume |
141
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Page/Identifiant |
107-130
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Commentaire |
18 pages
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Domaine(s) |
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arXiv Id | 1101.0725 |
DOI | 10.24033/bsmf.2644 |
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