# Natural endomorphisms of quasi-shuffle Hopf algebras

Abstract : 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.
Type de document :
Article dans une revue
Bulletin de la société mathématique de France, 2013, 141, pp.107-130
Domaine :

https://hal.archives-ouvertes.fr/hal-00823090
Contributeur : Jean-Yves Thibon <>
Soumis le : jeudi 16 mai 2013 - 10:41:37
Dernière modification le : mercredi 11 avril 2018 - 12:12:02

### Identifiants

• HAL Id : hal-00823090, version 1
• ARXIV : 1101.0725

### Citation

Jean-Christophe Novelli, F. Patras, Jean-Yves Thibon. Natural endomorphisms of quasi-shuffle Hopf algebras. Bulletin de la société mathématique de France, 2013, 141, pp.107-130. 〈hal-00823090〉

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