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Combinatorial Hopf algebras from PROs

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Abstract

We introduce a general construction that takes as input a so-called stiff PRO and that outputs a Hopf algebra. Stiff PROs are particular PROs that can be described by generators and relations with precise conditions. Our construction generalizes the classical construction from operads to Hopf algebras of van der Laan. We study some of its properties and review some examples of application. We get in particular Hopf algebras on heaps of pieces and retrieve some deformed versions of the noncommutative Faà di Bruno algebra introduced by Foissy.

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Acknowledgments

The authors would like to thank Jean-Christophe Novelli for his suggestions during the preparation of this paper. Moreover, the authors would like to thank the anonymous referee for their useful suggestions, which helped to improve the paper. The computations of this work have been done with the open-source mathematical software Sage [26] and one of its extensions, Sage-Combinat [27].

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  1. Laboratoire d’Informatique, du Traitement de l’Information et des Systèmes, Université de Rouen, Avenue de l’Université, 76801, Saint-Étienne-du-Rouvray Cedex, France

    Jean-Paul Bultel

  2. Laboratoire d’Informatique Gaspard-Monge, Université Paris-Est Marne-la-Vallée, 5 boulevard Descartes, Champs-sur-Marne, 77454, Marne-la-Vallée Cedex 2, France

    Samuele Giraudo

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  1. Jean-Paul Bultel
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Correspondence to Samuele Giraudo.

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Bultel, JP., Giraudo, S. Combinatorial Hopf algebras from PROs. J Algebr Comb 44, 455–493 (2016). https://doi.org/10.1007/s10801-016-0677-7

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  • DOI: https://doi.org/10.1007/s10801-016-0677-7

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