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.
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Contributor : Jean-Yves Thibon <>
Submitted on : Thursday, May 16, 2013 - 10:41:37 AM
Last modification on : Wednesday, April 11, 2018 - 12:12:02 PM

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


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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