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A Review on Comodule-Bialgebras

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Computation and Combinatorics in Dynamics, Stochastics and Control (Abelsymposium 2016)

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Abstract

We review some recent applications of the notion of comodule-bialgebra in several domains such as Combinatorics, Analysis and Quantum Field Theory.

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Notes

  1. 1.

    By an unfortunate conflict of terminology, composition of B-series corresponds to product of arborified moulds, whereas composition of the latter corresponds to substitution of B-series.

  2. 2.

    Only the vertices are decorated here: to be precise, an Ω-decorated forest is a pair (F, d) with F being a forest and \(d:\mathcal V(F)\to \varOmega \), where \(\mathcal V(F)\) stands for the set of vertices of F.

  3. 3.

    To be pronounced french-like, with stress on the last syllable. The last vowel (here, a or e) designates the type of symmetry considered.

  4. 4.

    Some confusion can arise in the literature around the words “label” and “decoration”. Here the decorations in Ω should not be confused with the labels, which are elements of the finite set X. Two distinct elements can bear the same decoration, but never the same label.

  5. 5.

    These are not Hopf algebras anymore strictly speaking because, due to the extra \(\mathbb Z^{d'}\)-decoration of the vertices, the coproducts are given by infinite linear combinations. This issue can be handled by working in the symmetric monoidal category of bi-graded vector spaces [4, Paragraph 2.3].

  6. 6.

    Private communication.

  7. 7.

    There is a subtle point here: strictly speaking, a covering subforest of height j with j > i − 1, which is a partition of a subset of \(\mathcal V_{\ge j}\), cannot be considered as a covering subforest of height ≥ i − 1, which is a partition of \(\mathcal V_{\ge i-1}\). Informally speaking, a covering subforest of height ≥ i − 1 covers \(\mathcal V_{\ge i-1}\) whereas a covering subforest of height j only covers a set \(\mathcal E\) such that \(\mathcal V_j\subset \mathcal E\subset \mathcal V_{\ge j}\), where \(\mathcal V_{j}\) stands for the set of height j vertices of t.

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Acknowledgements

I thank Kurusch Ebrahimi–Fard for his encouragements, as well as Yvain Bruned, Martin Hairer and Lorenzo Zambotti for introducing me to regularity structures. Special thanks to Yvain for illuminating discussions and for providing me the example in Sect. 9.

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Authors and Affiliations

  1. C.N.R.S.-UMR 6620, Université Clermont-Auvergne, Aubière, France

    Dominique Manchon

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  1. Dominique Manchon
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Correspondence to Dominique Manchon .

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Editors and Affiliations

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

    Elena Celledoni

  2. Department of Mathematics, University of Oslo, Oslo, Norway

    Giulia Di Nunno

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

    Kurusch Ebrahimi-Fard

  4. Department of Mathematics, University of Bergen, Bergen, Norway

    Hans Zanna Munthe-Kaas

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Manchon, D. (2018). A Review on Comodule-Bialgebras. In: Celledoni, E., Di Nunno, G., Ebrahimi-Fard, K., Munthe-Kaas, H. (eds) Computation and Combinatorics in Dynamics, Stochastics and Control. Abelsymposium 2016. Abel Symposia, vol 13. Springer, Cham. https://doi.org/10.1007/978-3-030-01593-0_20

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