Natural endomorphisms of shuffle algebras

Abstract : We focus in this text on the adaptation to the study of shuffles of the main combinatorial tool in the theory of free Lie algebras, namely the existence of a universal algebra of endomorphisms for tensor and other cocommutative Hopf algebras: the family of Solomon's descent algebras of type A. We show that there exists similarly a natural endomorphism algebra for commutative shuffle algebras, which is a natural extension of the Malvenuto-Reutenauer Hopf algebra of permutations, or algebra of free quasi-symmetric functions. We study this new algebra for its own, establish freeness properties, study its generators, bases, and also feature its relations to the internal structure of shuffle algebras.
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Submitted on : Monday, May 14, 2012 - 11:36:07 AM
Last modification on : Monday, May 28, 2018 - 4:36:02 PM
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  • HAL Id : hal-00696965, version 1
  • ARXIV : 1205.2986

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Loïc Foissy, Frédéric Patras. Natural endomorphisms of shuffle algebras. 2012. ⟨hal-00696965⟩

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