arXiv subjects

Dominique Manchon

Publications and source records attributed to Dominique Manchon.

At least 19 recordsLinked to original sources

arXiv2026

Extended generalized permutahedra, and cointeracting bialgebras

A Hopf monoid structure on extended generalized permutahedra (EGP) was recently introduced by M.Aguiar and F.Ardila. We investigate the existence of a cointeracting bialgebra structure on EGP's. We show that a suitable notion of cointeraction exists, not in the classical comodule sense, but via the framework of measuring algebras. The comodule-type map assigns to each polyhedron the sum of pairs of face and tangent cone at the face. EGP's and affine cone EGP's form the cointeracting bimonoids in species with EGP as a third measuring structure. EGP's are in bijection to extended submodular functions. For an EGP, we also describe explicitly the submodular functions of its faces and tangent cones. The braid fan and its relation to preorders play a key role in this description.

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arXiv2026

The planar Hopf algebra of noncommutative multi-indices

We construct the planar Linares--Otto--Tempelmayr Hopf algebra, thereby filling the missing planar noncommutative multi-index corner in the square relating the LOT, Butcher--Connes--Kreimer, and Munthe-Kaas--Wright Hopf algebras. Starting from the free associative algebra on a weighted alphabet $\mathbb Z_{\ge -1}\times A$, we define an insertion-type product yielding a post-Lie structure on the Lie algebra generated by the linear span $V(A)$ of weight $-1$ monomials whose proper left prefixes all have nonnegative weight, and the Guin--Oudom construction then produces the planar LOT Hopf algebra. We introduce a planar tree fertility map from decorated planar rooted trees to monomials in $V(A)$, prove that it is a linear isomorphism, and obtain a natural Hopf algebra isomorphism with the Munthe-Kaas--Wright Hopf algebra. We further derive an explicit coproduct formula in terms of left-admissible cuts, establish the extraction-contraction coproduct, and construct a word symmetrization operator compatible with the classical tree symmetrization operator.

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arXiv2026

Une remarque sur l'arborification de Matula

Nous esquissons une application de l'arborification de Matula \`a l'\'etude de la fonction sommatoire des fonctions de M\" obius et de Liouville sur les entiers naturels - We sketch an application of Matula's arborification to the study of the partial sums of both M\" obius and Liouville function.

math.NT↗

arXiv2025

Sewing lemma and knitting lemma for metric spaces

We state and prove a sewing lemma in the general context of families of complete metric spaces indexed by an interval of the real line, encompassing the flow sewing lemma proved by I. Bailleul in 2015. A further generalisation to other metric parameter spaces P than intervals is moreover proposed, leading to a representation of the groupoid of thin-equivalent Lipschitz paths on P . Under a stronger hypothesis, we finally prove a two-dimensional version, the knitting lemma, which gives rise to a representation of the Lipschitz homotopy groupoid of the parameter space, without thinness condition.

math.CA↗

arXiv2025

Controlled rough paths: a general Hopf-algebraic setting

We set up controlled rough paths for a class of combinatorial Hopf algebras, encompassing shuffle, Butcher-Connes-Kreimer and Munthe-Kaas--Wright Hopf algebras. The class of controls we consider encompasses both H\"older continuous paths and (not necessarily continuous) paths with bounded $p$-variation. We prove existence and uniqueness of the solution of a lifted initial value problem in this general setting by applying the fixed point method in a suitable Banach space of controlled rough paths, and we prove a universal limit theorem addressing the robustness of the solution with respect to the parameters and the initial condition.

math.PR↗

arXiv2024

Rough differential equations and planarly branched universal limit theorem

The universal limit theorem is a central result in rough path theory, which has been proved for: (i) rough paths with roughness $\frac{1}{3}< \alpha \leq \frac{1}{2}$; (ii) geometric rough paths with roughness $0< \alpha \leq 1$; (iii) branched rough paths with roughness $0< \alpha \leq 1$. Planarly branched rough paths are natural generalizations of both rough paths and branched rough paths, in the sense that post-Lie algebras are generalizations of both Lie algebras and pre-Lie algebras. Here the primitive elements of the graded dual Hopf algebra of the Hopf algebra corresponding to the planarly branched rough paths (resp. rough paths, resp. branched rough paths) form a post-Lie (resp. Lie, resp. pre-Lie algebra). In this paper, we prove the universal limit theorem for planarly branched rough paths with roughness $\frac{1}{4}< \alpha \leq \frac{1}{3}$, via the method of Banach fixed point theorem.

math.PR↗

arXiv2024

Submodular functions, generalized permutahedra, conforming preorders, and cointeracting bialgebras

Submodular functions $z$ defined on the power set of a finite set are in bijection with generalized permutahedra $\egp(z)$. To any such $z$ we define a class of preorders, {\it conforming} preorders. We show the faces of $\egp(z)$ and the conforming preorders are in bijection. We investigate in detail this interplay between submodular functions and generalized permutahedra on one side, and conforming preorders on the other side, with many examples. In particular, the face poset structure of $\egp(z)$ correspond to two order relations $\lhd$ and $\btl$ on preorders, and we investigate their properties. Ardila and Aguiar \cite{AA2017} introduced a Hopf monoid of submodular functions/generalized permutahedra. We show there is a bimonoid of modular functions cointeracting in a non-standard way. By recent theory of L.Foissy \cite{Fo2022}, on double bialgebras we get a canonical polynomial associated to any submodular function.

math.CO↗

arXiv2024

Free Novikov algebras and the Hopf algebra of decorated multi-indices

We propose a combinatorial formula for the coproduct in a Hopf algebra of decorated multi-indices that recently appeared in the literature, which can be briefly described as the graded dual of the enveloping algebra of the free Novikov algebra generated by the set of decorations. Similarly to what happens for the Hopf algebra of rooted forests, the formula can be written in terms of admissible cuts. We also prove a combinatorial formula for the extraction-contraction coproduct for undecorated multi-indices, in terms of a suitable notion of covering subforest.

math.CO↗

arXiv2024

Free post-groups, post-groups from group actions, and post-Lie algebras

After providing a short review on the recently introduced notion of post-group by Bai, Guo, Sheng and Tang, we exhibit post-group counterparts of important post-Lie algebras in the literature, including the infinite-dimensional post-Lie algebra of Lie group integrators. The notion of free post-group is examined, and a group isomorphism between the two group structures associated to a free post-group is explicitly constructed.

math.QA↗

arXiv2023

On the free commutative monoid over a positive operad

We study algebraic structures on the free commutative twisted algebra generated by a positive operad $\mathbf q$, in the framework of vector species. Given a nonunital commutative twisted algebra structure $\mu$ on $\mathbf q$, we introduce the notion of $\mu$-compatible operad structure, leading to a nonunital operad structure on $\mathbf E \circ \mathbf q$, where $\mathbf E$ stands for the exponential species. Next, we define nested pre-Lie operads (NPL-operads), a weak form of the notion of operad, in which the nested associativity axiom is weakened down to a nested pre-Lie condition. This structure is new up to our knowledge. Several constructions of NPL-operads are presented. Finally, we define algebras over a NPL-operad, based on the notion of polynomial functions.

math.CO↗

arXiv2023

A twisted Hopf algebra of finite topological quandles

This paper describes some algebraic properties of the species of finite topological quandles. We construct two twisted bialgebra structures on this species, one of the first kind and one of the second kind. The obstruction for the structure to match the double twisted bialgebra axioms is explicitly described.

math.AT↗

arXiv2022

Algebraic deformation for (S)PDEs

We introduce a new algebraic framework based on the deformation of pre-Lie products. This allows us to provide a new construction of the algebraic objects at play in Regularity Structures in the work arXiv:1610.08468 and in arXiv:2005.01649 for deriving a general scheme for dispersive PDEs at low regularity. This construction also explains how the algebraic structure in arXiv:1610.08468 can be viewed as a deformation of the Butcher-Connes-Kreimer and the extraction-contraction Hopf algebras. We start by deforming various pre-Lie products via a Taylor deformation and then we apply the Guin-Oudom procedure which gives us an associative product whose adjoint can be compared with known coproducts. This work reveals that pre-Lie products and their deformation can be a central object in the study of (S)PDEs.

math.PR↗

arXiv2022

Post-Lie-Magnus expansion and BCH-recursion

We identify the Baker-Campbell-Hausdorff recursion driven by a weight$λ=1$ Rota-Baxter operator with the Magnus expansion relativeto the post-Lie structure naturally associated to the correspondingRota-Baxter algebra. Post-Lie Magnus expansion and BCH-recursionare reviewed before the proof of the main result.

math.RA↗

arXiv2020

The universal pre-Lie-Rinehart algebras of aromatic trees

We organize colored aromatic trees into a pre-Lie-Rinehart algebra (i.e. a flat torsion-free Lie-Rinehart algebra) endowed with a natural trace map, and show the freeness of this object among pre-Lie-Rinehart algebras with trace. This yields the algebraic foundations of aromatic B-series.

math.RA↗

arXiv2020

Doubling bialgebras of finite topologies

The species of finite topological spaces admits two graded bimonoid structures, recently defined by F. Fauvet, L. Foissy, and the second author. In this article, we define a doubling of this species in two different ways. We build a bimonoid structure on each of these species and describe a cointeraction between them. We also investigate two related associative products obtained by dualisation.

math.RA↗

arXiv2020

Planarly branched rough paths and rough differential equations on homogeneous spaces

The central aim of this work is to understand rough differential equations on homogeneous spaces. We focus on the formal approach, by giving an explicit expansion of the solution at each point of the real line in terms of decorated planar forests. For this we develop the notion of planarly branched rough paths, following M. Gubinelli's branched rough paths. The definition is similar to the one in the flat case, the main difference being the replacement of the Butcher--Connes--Kreimer Hopf algebra of non-planar rooted forests by the Munthe-Kaas--Wright Hopf algebra of planar rooted forests. We show how the latter permits to handle rough differential equations on homogeneous spaces using planarly branched rough paths, the same way branched rough paths are used in the context of rough differential equations on finite-dimensional vector spaces. An analogue of T. Lyons' extension theorem is proven. Finally, under analyticity assumptions on the coefficients and when the Hölder index of the driving path is equal to one, we show convergence of the planar forest expansion in a small time interval.

math.CA↗

arXiv2020

Families of algebraic structures

We give a general account of family algebras over a finitely presented linear operad, this operad together with its presentation naturally defining an algebraic structure on the set of parameters.

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arXiv2020

Free Rota-Baxter family algebras and Free (tri)dendriform family algebras

In this paper, we first construct the free Rota-Baxter family algebra generated by some set $X$ in terms of typed angularly $X$-decorated planar rooted trees. As an application, we obtain a new construction of the free Rota-Baxter algebra only in terms of angularly decorated planar rooted trees (not forests), which is quite different from the known construction via angularly decorated planar rooted forests by K. Ebrahimi-Fard and L. Guo. We then embed the free dendriform (resp. tridendriform) family algebra into the free Rota-Baxter family algebra of weight zero (resp. one). Finally, we prove that the free Rota-Baxter family algebra is the universal enveloping algebra of the free (tri)dendriform family algebra.

math.RA↗