arXiv subjects

Loïc Foissy

Publications and source records attributed to Loïc Foissy.

At least 37 records · Page 2Linked to original sources

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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arXiv2023

Twisted bialgebras, cofreeness and cointeraction

We study twisted bialgebras and double twisted bialgebras, that is to say bialgebras in the category of linear species, or in the category of species in the category of coalgebras. We define the notion of cofree twisted coalgebra and generalize Hoffman's quasi-shuffle product, obtaining in particular a twisted bialgebra of set compositions Comp. Given a special character , this twisted bialgebra satisfies a terminal property, generalizing the one of the Hopf algebra of quasisymmetric functions proved by Aguiar, Bergeron and Sottile. We give Comp a second coproduct, making it a double twisted bialgebra, and prove that it is a terminal object in the category of double twisted bialgebras. Actions of characters on morphisms allow to obtain every twisted bialgebra morphisms from a connected double twisted bialgebra to Comp. These results are applied to examples based on graphs and on finite topologies, obtaining species versions of the chromatic symmetric series and chromatic polynomials, or of the Ehrhart polynomials. Moreover, through actions of monoids of characters, we obtain a Twisted bialgebraic interpretation of the duality principle.

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arXiv2023

Bialgebras overs another bialgebras and quasishuffle double bialgebras

Quasishuffle Hopf algebras, usually defined on a commutative monoid, can be more generally defined on any associative algebra V. If V is a commutative and cocommutative bialgebra, the associated quasishuffle bialgebra QSh(V) inherits a second coproduct $δ$ of contraction and extraction of words, cointeracting with the deconcatenation coproduct $Δ$, making QSh(V) a double bialgebra. In order to generalize the universal property of the Hopf algebra of quasisymmetric functions QSym (a particular case of quasishuffle Hopf algebra) as exposed by Aguiar, Bergeron and Sottile, we introduce the notion of double bialgebra over V. A bialgebra over V is a bialgebra in the category of right V-comodules and an extra condition is required on the second coproduct for double bialgebras over V. We prove that the quasishuffle bialgebra QSh(V) is a double bialgebra over V , and that it satisfies a universal property: for any bialgebra B over V and for any character $λ$ of B, under a connectedness condition, there exists a unique morphism $ϕ$ of bialgebras over V from B to QSh(V) such that $ε_δ\circ ϕ=λ$. When V is a double bialgebra over V , we obtain a unique morphism of double bialgebras over V from B to QSh(V), and show that this morphism $ϕ_1$ allows to obtain any morphism of bialgebra over V from B to QSh(V) thanks to an action of a monoid of characters. This formalism is applied to a double bialgebra of V-decorated graphs.

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arXiv2023

Contractions and extractions on twisted bialgebras and coloured Fock functors

We introduce a notion of extraction-contraction coproduct on twisted bialgebras, that is to say bialgebras in the category of linear species. If $P$ is a twisted bialgebra, a contraction-extraction coproduct sends $P[X]$ to $P[X/\sim]\otimes P[X]$ for any finite set $X$ and any equivalence relation $\sim$ on $X$, with a coassociativity constraint and compatibilities with the product and coproduct of $P$. We prove that if $P$ is a twisted bialgebra with an extraction-contraction coproduct, then $P\circ Com$ is a bialgebra in the category of coalgebraic species, that is to say species in the category of coalgebras.We then introduce a coloured version of the bosonic Fock functor. This induces a bifunctor which associates to any bialgebra $(V,\cdot,δ_V)$ and to any twisted bialgebra $P$ with an extraction-contraction coproduct a comodule-bialgebra $F_V[P]$: this object inherits a product $m$ and two coproducts $Δ$ and $δ$, such that $(F_V[P],m,Δ)$ is a bialgebra in the category of right $(F_V[P],m,δ)$-comodules.As an example, this is applied to the twisted bialgebra of graphs. The coloured Fock functors then allow to extend the construction of the double bialgebra of graphs to double bialgebras of graphs which vertices are decorated by elements of any bialgebra $V$. Other examples (on mixed graphs, hypergraphs, noncrossing partitions...) will be given in a series of forthcoming papers.

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arXiv2022

Typed angularly decorated planar rooted trees and generalized Rota-Baxter algebras

We introduce a generalization of parametrized Rota-Baxter algebras, which includes family and matching Rota-Baxter algebras. We study the structure needed on the set $Ω$ of parameters in order to obtain that free Rota-Baxter algebras are described in terms of typed and angularly decorated planar rooted trees: we obtain the notion of $λ$-extended diassociative semigroup, which includes sets (for matching Rota-Baxter algebras) and semigroups (for family Rota-Baxter algebras), and many other examples. We also describe free commutative $Ω$-Rota-Baxter algebras generated by a commutative algebra A in terms of typed words.

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arXiv2021

Reduced typed angularly decorated planar rooted trees and generalized tridendriform algebras

We introduce a generalization of tridendriform algebras, where each of the three products are replaced by a family of products indexed by a set $Ω$. We study the needed structure on $Ω$ for free $Ω$-tridendriform algebras to be built on Schr{ö}der trees (as it is the case in the classical case), with convenient decorations on their leaves. We obtain in this way extended triassociative semigroups. We describe commutative $Ω$-tridendriform algebras in terms of typed words. We also study links with generalizations of Rota-Baxter algebras and describe the Koszul duals of the corresponding operads.

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arXiv2021

Generalized associative algebras

We study diverse parametrized versions of the operad of associative algebra, where the parameter are taken in an associative semigroup $Ω$ (generalization of matching or family associative algebras) or in its cartesian square (two-parameters associative algebras). We give a description of the free algebras on these operads, study their formal series and prove that they are Koszul when the set of parameters is finite. We also study operadic morphisms between the operad of classical associative algebras and these objects, and links with other types of algebras (diassociative, dendriform, post-Lie...).

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arXiv2021

Enriched pre-Lie operads and freeness theorems

In this paper, we study the C-enriched pre-Lie operad defined by Calaque and Willwacher for any Hopf cooperad C to produce conceptual constructions of the operads acting on various deformation complexes. Maps between Hopf cooperads lead to maps between the corresponding enriched pre-Lie operads; we prove criteria for the module action of the domain on the codomain to be free, on the left and on the right. In particular, this implies a new functorial Poincaré--Birkhoff--Witt type theorem for universal enveloping brace algebras of pre-Lie algebras.

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arXiv2021

Chromatic polynomials and bialgebras of graphs

The chromatic polynomial is characterized as the unique polynomial invariant of graphs, compatible with two interacting bialgebras structures: the first coproduct is given by partitions of vertices into two parts, the second one by a contraction-extraction process. This gives Hopf-algebraic proofs of Rota's result on the signs of coefficients of chromatic polynomials and of Stanley's interpretation of the values at negative integers of chromatic polynomi-als. We also give non-commutative version of this construction, replacing graphs by indexed graphs and Q[X] by the Hopf algebra WSym of set partitions.

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arXiv2021

Algebraic structures on typed decorated rooted trees

Typed decorated trees are used by Bruned, Hairer and Zambotti to give a description of a renormalisation processon stochastic PDEs. We here study the algebraic structures on these objects: multiple prelie algebrasand related operads (generalizing a result by Chapoton and Livernet), noncommutative and cocommutative Hopf algebras (generalizing Grossman and Larson's construction),commutative and noncocommutative Hopf algebras (generalizing Connes and Kreimer's construction),bialgebras in cointeraction (generalizing Calaque, Ebrahimi-Fard and Manchon's result). We also define families of morphisms and in particular we prove that any Connes-Kreimer Hopf algebraof typed and decorated trees is isomorphic to a Connes-Kreimer Hopf algebra of non--typed and decoratedtrees (the set of decorations of vertices being bigger), through a contraction process,and finally obtain the Bruned-Hairer-Zambotti construction as a subquotient.

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arXiv2021

From non-unitary wheeled PROPs to smooth amplitudes and generalised convolutions

We introduce the concept of TRAP (Traces and Permutations), which can roughly be viewed as a wheeled PROP (Products and Permutations) without unit. TRAPs are equipped with a horizontal concatenation and partial trace maps. Continuous morphisms on an infinite dimensional topological space and smooth kernels (resp. smoothing operators) on a closed manifold form a TRAP but not a wheeled PROP. We build the free objects in the category of TRAPs as TRAPs of graphs and show that a TRAP can be completed to a unitary TRAP (or wheeled PROP). We further show that it can be equipped with a vertical concatenation, which on the TRAP of linear homomorphisms of a vector space, amounts to the usual composition. The vertical concatenation in the TRAP of smooth kernels gives rise to generalised convolutions. Graphs whose vertices are decorated by smooth kernels (resp. smoothing operators) on a closed manifold form a TRAP. From their universal properties we build smooth amplitudes associated with the graph.

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arXiv2020

Plane posets, special posets, and permutations

We study the self-dual Hopf algebra $\h\_{\SP}$ of special posets introduced by Malvenuto and Reutenauer and the Hopf algebra morphism from $\h\_{\SP}$ to to the Hopf algebra of free quasi-symmetric functions $\FQSym$ given by linear extensions. In particular, we construct two Hopf subalgebras both isomorphic to $\FQSym$; the first one is based on plane posets, the second one on heap-ordered forests. An explicit isomorphism between these two Hopf subalgebras is also defined with the help of two transformations on special posets. The restriction of the Hopf pairing of $\h\_{\SP}$ to these Hopf subalgebras and others is also studied, as well as certain isometries between them. These problems are solved using duplicial and dendriform structures.An error in Section 7 has been noticed by Darij Grinberg, and the text has been modified accordingly.

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arXiv2020

Quadri-algebras, preLie algebras, and the Catalan family of Lie idempotents

We compute the expansion of the Catalan family of Lie idempotents introduced in [Menous et al., Adv. Applied Math. 51 (2013), 177-22] on the PBW basis of the Lie module. It is found that the coefficient of a tree depends only on its number of left and right internal edges. In particular, the Catalan idempotents belong to a preLie algebra based on naked binary trees, of which we identify several Lie and preLie subalgebras.

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arXiv2020

Operads of (noncrossing) partitions, interacting bialgebras, and moment-cumulant relations

We establish and explore a relationship between two approaches to moment-cumulant relations in free probability theory: on one side the main approach, due to Speicher, given in terms of Möbius inversion on the lattice of noncrossing partitions, and on the other side the more recent non-commutative shuffle-algebra approach, where the moment-cumulant relations take the form of certain exponential-logarithm relations. We achieve this by exhibiting two operad structures on (noncrossing) partitions, different in nature: one is an ordinary, non-symmetric operad whose composition law is given by insertion into gaps between elements, the other is a coloured, symmetric operad with composition law expressing refinement of blocks. We show that these operad structures interact so as to make the corresponding incidence bialgebra of the former a comodule bialgebra for the latter. Furthermore, this interaction is compatible with the shuffle structure and thus unveils how the two approaches are intertwined. Moreover, the constructions and results are general enough to extend to ordinary set partitions.

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arXiv2019

Realizations of Hopf algebras of graphs by alphabets

We here give polynomial realizations of various Hopf algebras or bialgebras on Feynman graphs, graphs, posets or quasi-posets, that it to say injections of these objects into polynomial algebras generated by an alphabet. The alphabet here considered are totally quasi-ordered. The coproducts are given by doubling the alphabets; a second coproduct is defined by squaring the alphabets, and we obtain cointeracting bialgebras in the commutative case.

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arXiv2018

Surjections and double posets

The theory and structure of the Hopf algebra of surjections (known as WQSym, the Hopf algebra of word quasi-symmetric functions) parallels largely the one of bijections (known as FQSym or MR, the Hopf algebra of word quasi-symmetric functions or Malvenuto-Reutenauer Hopf algebra). The study of surjections from a picture and double poset theoretic point of view, which is the subject of the present article, seems instead new. The article is organized as follows. We introduce first a family of double posets, weak planar posets, that generalize the planar posets and are in bijection with surjections or, equivalently, packed words. The following sections investigate their Hopf algebraic properties, which are inherited from the Hopf algebra structure of double posets and their relations with WQSym

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arXiv2017

Extension of the product of a post-Lie algebra and application to the SISO feedback transformation group

We describe the both post-and pre-Lie algebra g SISO associated to the affine SISO feedback transformation group. We show that it is a member of a family of post-Lie algebras associated to representations of a particular solvable Lie algebra. We first construct the extension of the magmatic product of a post-Lie algebra to its enveloping algebra, which allows to describe free post-Lie algebras and is widely used to obtain the enveloping of g SISO and its dual.

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