arXiv subjects

Loïc Foissy

Publications and source records attributed to Loïc Foissy.

At least 19 recordsLinked to original sources

arXiv2026

Bialgebraic structures on boolean functions

We study several bialgebraic structures on boolean functions, that is to say maps defined on the set of subsets of a finite set $X$, taking the value $0$ on $\emptyset$. Examples of boolean functions are given by the indicator function of the hyperedges of a given hypergraph, or the rank function of a matroid. We give the species of boolean functions a two-parameters family of products and a coproduct, and this defines a two-parameters family of twisted bialgebras. We then try to define a second coproduct on boolean functions, based on contractions, in order to obtain a double bialgebra. We show that this is not possible on the whole species of boolean functions, but that there exists a maximal subspecies where this is possible. This subspecies being rather mysterious, we introduce rigid boolean functions and show that this subspecies has indeed a second coproduct, as wished, and that it contains rank functions of matroids and indicator functions associated to hypergraphs. As a consequence, we obtain a unique polynomial invariant on rigid boolean functions, which is a generalization of the chromatic polynomial of graphs.

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arXiv2025

Operads and bialgebras of multi-indices, and Novikov algebras

Noncommutative multi-indices are noncommutative monomials in a $\mathbb{N}$-indexed family of indeterminates. We define on them a $\mathbb{Z}$-graded operadic structure, with the help of a shifting derivation. Multi-indices of degree 0 are called populated: they form a suboperad, isomorphic to the operad of Novikov algebras. This operadic structure, and the relation between pre-Lie and Novikov algebras, induces two bialgebraic structure in cointeraction on commutative multi-indices. We show how to combinatorially embed this double bialgebra into the Connes-Kreimer Hopf algebra of rooted trees, with its two coproducts based, firstly on cuts, secondly, on contraction of edges, and how this embedding can be characterized by a Dyson-Schwinger equation. We also study the unique polynomial invariant compatible with the two bialgebraic structures on multi-indices and use to describe the antipode for the first coproduct.

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arXiv2025

Cocycle weighted infinitesimal bialgebras and pre-Lie algebras on rooted trees

The concept of weighted infinitesimal bialgebras provides an algebraic framework for understanding the non-homogeneous associative Yang-Baxter equation. In this paper, we endow the space of decorated planar rooted forests with a two-parameters family of coproducts, making it into a weighted infinitesimal bialgebra. A combinatorial characterization of the coproducts is given via the notion of forest biideals. Furthermore, by constructing a bilinear symmetric form and introducing a new grafting operation on rooted forests, we describe the associated dual products. We also introduce the notion of the pair-weight 1-cocycle condition and investigate the universal properties of decorated planar rooted forests satisfying this condition. This leads to the definition of a weighted $\Omega$-cocycle infinitesimal unitary bialgebra. As applications, we identify the initial object in the category of free cocycle infinitesimal unitary bialgebras on undecorated planar rooted forests, corresponding to the well-known noncommutative Connes-Kreimer Hopf algebra. In addition, we establish isomorphisms between different coproduct structures and construct a pre-Lie algebra structure on decorated planar rooted forests.

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arXiv2025

Noncrossing arithmetic

Higher-order notions of Kreweras complementation have appeared in the literature in the works of Krawczyk, Speicher, Mastnak, Nica, Arizmendi, Vargas, and others. While the theory has been developed primarily for specific applications in free probability, it also possesses an elegant, purely combinatorial core that is of independent interest. The present article aims at offering a simple account of various aspects of higher-order Kreweras complementation on the basis of elementary arithmetic, (co)algebraic, categorical and simplicial properties of noncrossing partitions. The main idea is to consider noncrossing partitions as providing an interesting noncommutative analogue of the interplay between the divisibility poset and the multiplicative monoid of positive integers. Just as the divisibility poset can be regarded as the decalage of the multiplicative monoid, we exhibit the lattice of noncrossing partitions as the decalage of a partial monoid structure on noncrossing partitions encoding higher-order Kreweras complements. While our results may be considered familiar, several of the viewpoints can be regarded as novel, offering an efficient approach both conceptually and computationally.

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arXiv2025

Construction of pre-and post-Lie algebras for stochastic PDEs

We give and study a construction of pre-Lie algebra structures on rooted trees whose edges and vertices are decorated, with a grafting product acting, through a map $\phi$, both on the decoration of the created edge and on the vertex that holds the grafting. We show that this construction gives a pre-Lie algebra if, and only if, the map $\phi$ satisfies a commutation relation, called tree-compatibility. We show how to extend this pre-Lie algebra structure to a post-Lie one by a semi-direct extension with another post-Lie algebra. We also define several constructions to obtain tree-compatible maps, and give examples, including a description of all tree-admissible maps when the space of decorations of the vertices is $2$-dimensional and the space of decorations of the edges is finite-dimensional. A particular example of such a construction is used by Bruned, Hairer and Zambotti for the study of stochastic partial differential equations: when no noise is involved, we show that the underlying tree-compatible map is the exponential of a simpler one and deduce an explicit isomorphism with a classical pre-Lie algebra of rooted trees; when a noise is involved, we obtain the underlying tree-compatible map as a direct sum. We also obtain with our formalism the post-Lie algebras described by Bruned and Katsetsiadis.

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arXiv2025

Cocommutative Com-PreLie bialgebras

A Com-PreLie bialgebra is a commutative bialgebra with an extra preLie product satisfying some compatibilities with the product and coproduct. We here give a classification of connected, cocommutative Com-PreLie bialgebras over a field of characteristic zero: we obtain a main family of symmetric algebras on a space V of any dimension, and another family available only if V is one-dimensional. We also explore the case of Com-PreLie bialgebras over a group algebra and over a tensor product of a group algebra and of a symmetric algebra.

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arXiv2025

Cointeraction on noncrossing partitions and related polynomial invariants

We study the structure of two cointeracting bialgebras on noncrossing partitions appearing in the theory of free probability. The first coproduct is given by separation of the blocks of the partitions into two parts, with respect to the nestings, while the second one is given by fusion of blocks. This structure implies the existence of a unique polynomial invariant respecting the product and both coproducts. We give a combinatorial interpretation of this invariant, study its values at -1 and use it for the computation of the antipode. We also give several results on its coefficients when applied to noncrossing partitions with no nesting. This leads to unexpected links with harmonic nested sums, Riordan arrays, composition of formal series and generalized Stirling numbers. This polynomial invariant is shown to be related to other ones, counting increasing or strictly increasing maps for the nesting order on noncrossing partitions, through the action of several characters.

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arXiv2025

Matrix symmetric and quasi-symmetric functions and noncommutative representation theory

A fundamental result by L. Solomon in algebraic combinatorics and representation theory states that Mackey formulas for products of characters of a symmetric group, or equivalently the computation of tensor products of representations thereof, can be lifted to the corresponding Solomon's descent algebra, a subalgebra of the group algebra with a very rich structure. Motivated by the structure of the product formula in these algebras and by other results and ideas in the field, we introduce and investigate in the present article a two dimensional analogue of descent algebras based on packed integer matrices that inherits most of their fundamental properties. One of the various bialgebra structures we introduce on packed integer matrices identifies with a bialgebra recently introduced by J. Diehl and L. Schmitz to define a two dimensional generalisation of Chen's iterated integrals signatures.

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arXiv2024

Cofree Com-PreLie algebras

A Com-PreLie bialgebra is a commutative bialgebra with an extra preLie product satisfying some compatibilities with the product and the coproduct. We here give examples of cofree Com-PreLie bialgebras, including all the ones such that the preLie product is homogeneous of degree $\ge$ --1. We also give a graphical description of free unitary Com-PreLie algebras, explicit their canonical bialgebra structure and exhibit with the help of a rigidity theorem certain cofree quotients, including the Connes-Kreimer Hopf algebra of rooted trees. We finally prove that the dual of these bialgebras are also enveloping algebras of preLie algebras, combinatorially described.

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arXiv2024

Hopf-algebraic structures on mixed graphs

We introduce two coproducts on mixed graphs (that is to say graphs with both oriented and unoriented edges), the first one by separation of the vertices into two parts, and the second one given by contraction and extractions of subgraphs. We show that, with the disjoint union product, this gives a double bialgebra, that is to say that the first coproduct makes it a Hopf algebra in the category of right comodules over the second coproduct. This structure implies the existence of a unique polynomial invariant on mixed graphs compatible with the product and both coproducts: we prove that it is the (strong) chromatic polynomial of Beck, Bogart and Pham.Using the action of the monoid of characters, we relate it to the weak chromatic polynomial, as well to Ehrhart polynomials and to a polynomial invariants related to linear extensions. As applications, we give an algebraic proof of the link between the values of the strong chromatic polynomial at negative values and acyclic orientations (a result due to Beck, Blado, Crawford, Jean-Louis and Young) and obtain a combinatorial description of the antipode of the Hopf algebra of mixed graphs.

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arXiv2024

The antipode of of a Com-PreLie Hopf algebra

We study the compatibility between the antipode and the preLie product of a Com-PreLie Hopf algebra, that is to say a commutative Hopf algebra with a complementary preLie product, compatible with the product and the coproduct in a certain sense. An example of such a Hopf algebra is the Connes-Kreimer Hopf algebra, with the preLie product given by graftingof forests, extending the free preLie product of grafting of rooted trees. This compatibility is then used to study the antipode of the Connes-Moscovici subalgebra, whichcan be defined with the help of this preLie product. The antipode of the generators of this subalgebra gives a family of combinatorial coefficients indexed by partitions,which can be computed with the help of iterated harmonic sums.

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arXiv2024

The Fortuin-Kasteleyn polynomial as a bialgebra morphism and applications to the Tutte polynomial

We compute an explicit formula for the antipode of the double bialgebra of graphs in terms of totally acyclic partial orientations, using some general results on double bialgebras. In analogy to what was already proven in Hopf-algebraic terms for the chromatic polynomial of a graph, we show that the Fortuin-Kasteleyn polynomial (a variant of the Tutte polynomial) is a morphism of the double algebra of graphs into that of polynomials, which generalizes the chromatic polynomial. When specialized at particular values, we give combinatorial interpretations of the Tutte polynomial of a graph, via covering graphs and covering forests, and of the Fortuin-Kasteleyn polynomial, via pairs of vertex--edge colorings. Finally we show that the map associating to a graph all its orientations is a Hopf morphism from the double bialgebra of graphs into that one of oriented graphs, allowing to give interpretations of the Fortuin-Kasteleyn polynomial when computed at negative values.

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arXiv2024

Commutative B\_infty -algebras are shuffle algebras

We here construct an explicit isomorphism between any commutative Hopf algebra which underlying coalgebra is the tensor coalgebra of a space $V$ and the shuffle algebra based on the same space. This isomorphism uses the commutative $B_\infty$ structure that governs the product and the eulerian idempotent, as well as the canonical projection on the space $V$. This generalizes Homan's isomorphism between commutative quasi-shuffle and shuffle algebras, which correspond to the case when the $B_\infty$ structure is given by an associative and commutative product. We develop several examples in details, including the Hopf algebra of finite topologies.

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arXiv2023

Primitive elements of a connected free bialgebra

We prove that the Lie algebra of primitive elements of a graded and connected bialgebra, free as an associative algebra, over a eld of characteristic zero, is a free Lie algebra. The main tool is a ltration, which allows to embed the associated graded Lie algebra into the Lie algebra of a free and cocommutative bialgebra. The result is then a consequence of Cartier-Quillen-Milnor-Moore's Shirshov-Witt's theorems.

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arXiv2023

A co-preLie structure from chronological loop erasure in graph walks

We show that the chronological removal of cycles from a walk on a graph, known as Lawler's loop-erasing procedure, generates a preLie co-algebra on the vector space spanned by the walks. In addition, we prove that the tensor and symmetric algebras of graph walks are graded Hopf algebras, provide their antipodes explicitly and recover the preLie co-algebra from a brace coalgebra on the tensor algebra of graph walks. Finally we exhibit sub-Hopf algebras associated to particular types of walks.

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arXiv2023

Bialgebras in cointeraction, the antipode and the eulerian idempotent

We give here a review of results about double bialgebras, that is to say bialgebras with two coproducts, the first one being a comodule morphism for the coaction induced by the second one. An accent is put on the case of connected bialgebras. The subjects of these results are the monoid of characters and their actions, polynomial invariants, the antipode and the eulerian idempotent. As examples, they are applied on a double bialgebra of graphs and on quasishuffle bialgebras. This includes a new proof of a combinatorial interpretation of the coefficients of the chromatic polynomial due to Greene and Zaslavsky.

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arXiv2023

Sequences of Trees and Higher-Order Renormalization Group Equations

We define a notion of higher order renormalization group equation and investigate when a sequence of trees satisfies such an equation. In the strongest sense, the sequence of trees satisfies a $k$th order renormalization group equation when applying any choice of Feynman rules results in a Green function satisfying a $k$th order renormalization group equation, and we characterize all such sequences of trees. We also make some comments on sequences of trees which require special choices of Feynman rules in order to satisfy a higher order renormalization group equation.

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arXiv2023

Hopf algebraic structures on hypergraphs and multi-complexes

Using the formalism of species and twisted objects, we introduce two structures of cointeracting bialgebras on hypergraphs, induced by two notions of induced sub-hypergraphs. We study the associated unique morphisms of cointeracting bialgebras from hypergraphs to the polynomial algebra in one indeterminate: in the first case, this gives the chromatic polynomial of a graph attached to the considered hypergraph. In the second case, we obtained Helgason's notion of chromatic polynomial of a hypergraph. We obtain Hopf-algebraic proves of results about the values of this chromatic polynomial in -1 or about its coefficients, with the help of the action of a monoid of characters. This allows to give multiplicity-free formulas for the antipodes of these objects, using various notions of acyclic orientations of hypergraphs. Mixing the two notions of induced sub-hypergraphs, we obtain a third Hopf algebra, firstly described by Aguiar and Ardila. We obtain negative results on the existence of a second coproduct making it a cointeracting bialgebra. Anyway, it is still possible to obtain a polynomial invariant from this structure, which is the chomatic polynomial described by Aval, Kharagbossian and Tanasa. We nally study Iovanov and Jaiung's Hopf algebra of multi-complexes, making it a cointeracting bialgebra which has for quotient one of the preceding cointeracting bialgebras of hypergraphs.

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