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Hazewinkel wrote this article in 2005. Perhaps it's time for an update.

For example, updating item

34: Ordinary differential equations much work has been done on the underlying Hopf algebra (HA) of Lie-Butcher numerical methods for solving autonomous and non-autonomous ODE's (evolution equations) by , e.g., Ebrahimi-Fard, Hans Munthe-Kaas and their colleagues. See, e.g.,

i) "Butcher series: A story of rooted trees and numerical methods for evolution equations" by McLachlan, Modin, Munthe-Kaas, and Verdier

ii) "Lie-Butcher series, Geometry, Algebra and Computation" by Munthe-Kaas and K. Føllesdal

This is closely allied to item

81: Quantum theory

through the HA (combinatorial Faa di Bruno bialgebra, trees, Feynman diagrams) of renormalization to which Alain Connes, Christian Brouder, David Broadhurst, Dirk Kreimer, Kurush Ebrahimi-Fard, Hector Figueroa, Jose Gracia-Bondia, Hans Munthe-Kass, Loic Foissy, Karen Yeats, Paul-Hermann Balduf, et al. have contributed.

52: Convex polytopes (relabeled)

Hopf monoids have been introduced to explain the association of permutohedra and associahedra with compositional and multiplicative inversion and optimization:

"Hopf monoids and generalized permutahedra" by Marcelo Aguiar and Federico Ardila.

It would be motivational and useful in pursuing research in these topics if others would list some of their favorite interests under appropriate items and give associated authors and/or papers, particularly of an introductory nature.

Please feel free to note your own work (with no false modesty--if you are taking the time and effort to publish, you must feel it could be of interest to others, otherwise ...).

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60: Probability theory and stochastic processes

Hopf algebras feature in the work of Hairer and collaborators on the applications of the theory of regular structures to the solution of general singular SPDEs. See in particular:

  1. "Algebraic renormalisation of regularity structures", by Bruned, Hairer and Zambotti in Inventiones 2019.
  2. "An analytic BPHZ theorem for regularity structures", by Chandra and Hairer.
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  • $\begingroup$ ah beat me to it, but good references $\endgroup$ – Alexander Schmeding Jan 27 at 16:00
  • $\begingroup$ Related "Operads of (noncrossing) partitions, interacting bialgebras, and moment-cumulant relations" by Ebrahimi-Fard, Foissy, Joachim Kock, and Frédéric Patras. arxiv.org/abs/1907.01190 $\endgroup$ – Tom Copeland Jan 27 at 16:04
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    $\begingroup$ I think Ebrahimi-Fard, Patras and Speicher are now also investigating Hopf algebraic relations in free probability: arXiv:2001.03808: Wick polynomials in non-commutative probability $\endgroup$ – Alexander Schmeding Jan 27 at 16:29
  • $\begingroup$ @AlexanderSchmeding: +1 Yes. Hazewinkel in his 2004 article already mentioned this type of applications to free probability, combinatorics of cumulants etc. The applications to regularity structures are new, of course, because they were introduced by Hairer in 2013. $\endgroup$ – Abdelmalek Abdesselam Jan 27 at 18:06
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60: Probability theory,... There is even more movement in the stochastic PDE community with Hairer,s regularity structures. The groups appearing there are character groups of certain Hopf algebras and there is the massive work by Bruned, Hairer and Zambotti to uncover the algebraic framework, leading them to certain Hopf algebras: Algebraic renormalisatiom of regularity structures arxiv:1610.08468.

Also the whole topic of rough paths (Now Msc2020 60Lxx) is quite connected, as rough paths can be viewed as paths again in character groups of certain Hopf algebras. See e.g.

What does the group action of a rough path in a Lie group look like?

This can be found In most modern treatments in the guise of dealing with the tensor algebra and shuffle products. Some modern more algebraic developments are also included in the works of Ebrahimi-Fard, Manchon et al.

22Exx Infinite-dimensional Lie theory, turns out that character groups of Hopf algebras are often infinite-dimensional Lie groups (this provides Lie group structures for many well-known examples, such as the Butcher group from numerical analysis):

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  • $\begingroup$ Just came across this association in the last few days since it's related to noncrossing partitions and thence to compositional and multiplicative inversions and free probability. $\endgroup$ – Tom Copeland Jan 27 at 16:29
  • $\begingroup$ Since compositional inversion is related to vector fields and, therefore, to Lie algebras and groups, solns. to diff eqs, and trees, I always suspect there is a HA with its antipode and dual lurking around somewhere. $\endgroup$ – Tom Copeland Jan 27 at 16:48
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5. Combinatorics

Tree hook length formulas can be specified via a Hopf algebra.

Though I admit hook length formulas are not that interesting and may already fit under Hazewinkel's description of 5. Also the first paper doesn't actually use Hopf algebras and the second alludes to them.

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