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Lie Theory for Quasi-Shuffle Bialgebras

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Periods in Quantum Field Theory and Arithmetic (ICMAT-MZV 2014)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 314))

Abstract

Many features of classical Lie theory generalize to the broader context of algebras over Hopf operads. However, this idea remains largely to be developed systematically. Quasi-shuffle algebras provide for example an interesting illustration of these phenomena, but have not been investigated from this point of view. The notion of quasi-shuffle algebras can be traced back to the beginnings of the theory of Rota-Baxter algebras, but was developed systematically only recently, starting essentially with Hoffman’s work, that was motivated by multizeta values (MZVs) and featured their bialgebra structure. Many partial results on the fine structure of quasi-shuffle bialgebras have been obtained since then but, besides the fact that each of these articles features a particular point of view, they fail to develop systematically a complete theory. This article builds on these various results and develops the analog theory, for quasi-shuffle algebras, of the theory of descent algebras and their relations to free Lie algebras for classical enveloping algebras.

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Acknowledgements

The authors were supported by the grant CARMA ANR-12-BS01-0017. We thank its participants and especially Jean-Christophe Novelli and Jean-Yves Thibon, for stimulating discussions on noncommutative symmetric functions and related structures. This article is, among others, a follow up of our joint works [17, 31]. We also thank the ICMAT Madrid for its hospitality.

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Authors and Affiliations

  1. LMPA Joseph Liouville–Université du Littoral Côte d’opale, Centre Universitaire de la Mi-Voix, 50, rue Ferdinand Buisson, CS 80699, 62228, Calais Cedex, France

    Loïc Foissy

  2. UMR 7351 CNRS–Université de Nice Parc Valrose, 06108, Nice Cedex 02, France

    Frédéric Patras

Authors
  1. Loïc Foissy
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  2. Frédéric Patras
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Correspondence to Loïc Foissy .

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Editors and Affiliations

  1. Institute of Mathematical Sciences (ICMAT), Spanish National Research Council (CSIC), Madrid, Spain

    José Ignacio Burgos Gil

  2. Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway

    Kurusch Ebrahimi-Fard

  3. Department of Mathematical Sciences, Durham University, Durham, UK

    Herbert Gangl

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Foissy, L., Patras, F. (2020). Lie Theory for Quasi-Shuffle Bialgebras. In: Burgos Gil, J., Ebrahimi-Fard, K., Gangl, H. (eds) Periods in Quantum Field Theory and Arithmetic. ICMAT-MZV 2014. Springer Proceedings in Mathematics & Statistics, vol 314. Springer, Cham. https://doi.org/10.1007/978-3-030-37031-2_19

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