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Loïc Foissy

Publications and source records attributed to Loïc Foissy.

69 records · Page 4Linked to original sources
arXiv2012

Bruhat order on plane posets and applications

A plane poset is a finite set with two partial orders, satisfying a certain incompatibility condition. The set PP of isoclasses of plane posets owns two products, and an infinitesimal Hopf algebra structure is defined on the vector space H_PP generated by PP, using the notion of biideals of plane posets. We here define a partial order on PP, making it isomorphic to the set of partitions with the weak Bruhat order. We prove that this order is compatible with both products of PP; moreover, it encodes a non degenerate Hopf pairing on the infinitesimal Hopf algebra H_PP.

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arXiv2012

Natural endomorphisms of shuffle algebras

We focus in this text on the adaptation to the study of shuffles of the main combinatorial tool in the theory of free Lie algebras, namely the existence of a universal algebra of endomorphisms for tensor and other cocommutative Hopf algebras: the family of Solomon's descent algebras of type A. We show that there exists similarly a natural endomorphism algebra for commutative shuffle algebras, which is a natural extension of the Malvenuto-Reutenauer Hopf algebra of permutations, or algebra of free quasi-symmetric functions. We study this new algebra for its own, establish freeness properties, study its generators, bases, and also feature its relations to the internal structure of shuffle algebras.

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arXiv2011

General Dyson-Schwinger equations and systems

We classify combinatorial Dyson-Schwinger equations giving a Hopf subalgebra of the Hopf algebra of Feynman graphs of the considered Quantum Field Theory. We first treat single equations with an arbitrary number (eventually infinite) of insertion operators. we distinguish two cases; in the first one, the Hopf subalgebra generated by the solution is isomorphic to the Faà di Bruno Hopf algebra or to the Hopf algebra of symmetric functions; in the second case, we obtain the dual of the enveloping algebra of a particular associative algebra (seen as a Lie algebra). We also treat systems with an arbitrary finite number of equations, with an arbitrary number of insertion operators, with at least one of degree 1 in each equation.

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arXiv2011

Free and cofree Hopf algebras

We first prove that a graded, connected, free and cofree Hopf algebra is always self-dual; then that two graded, connected, free and cofree Hopf algebras are isomorphic if, and only if, they have the same Poincaré-Hilbert formal series. If the characteristic of the base field is zero, we prove that the Lie algebra of the primitive elements of such an object is free, and we deduce a characterization of the formal series of free and cofree Hopf algebras by a condition of growth of the coefficients. We finally show that two graded, connected, free and cofree Hopf algebras are isomorphic as (non graded) Hopf algebras if, and only if, the Lie algebra of their primitive elements have the same number of generators.

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arXiv2011

Ordered forests and parking functions

We prove that the Hopf algebra of parking functions and the Hopf algebra of ordered forests are isomorphic, using a rigidity theorem for a particular type of bialgebras.

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arXiv2010

Systems of Dyson-Schwinger equations

We consider systems of combinatorial Dyson-Schwinger equations (briefly, SDSE) X_1=B^+_1(F_1(X_1,...,X_N))...X_N=B^+_N(F_N(X_1,...,X_N)) in the Connes-Kreimer Hopf algebra H_I of rooted trees decorated by I={1,...,N},where B^+_i is the operator of grafting on a root decorated by i, and F_1...,F_N are non-constant formal series.The unique solution X=(X_1,...,X_N) of this equation generates a graded subalgebra H_S of H_I. We characterize here all the families of formal series (F_1,...,F_N) such that H_S is a Hopf subalgebra. More precisely, we define three operations on SDSE (change of variables, dilatation and extension) and give two families of SDSE (cyclic and fundamental systems), and prove that any SDSE (S) such that H_S is Hopf is the concatenation of several fundamental or cyclic systems after the application of a change of variables, a dilatation and iterated extensions. We also describe the Hopf algebra H_S as the dual of the enveloping algebra of a Lie algebra g_S of one of the following types: 1. g_S is a Lie algebra of paths associated to a certain oriented graph. 2. g_S is an iterated extension of the Faà di Bruno Lie algebra. 3. g_S is an iterated extension of an abelian Lie algebra.

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arXiv2009

Free brace algebras are free prelie algebras

Let g be a free brace algebra. This structure implies that g is also a prelie algebra and a Lie algebra. It is already known that g is a free Lie algebra. We prove here that g is also a free prelie algebra, using a description of g with the help of planar rooted trees, a permutative product, and anipulations on the Poincaré-Hilbert series of g.

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arXiv2009

The operads of planar forests are Koszul

We describe the Koszul dual of two quadratic operads on planar forests introduced to study the infinitesimal Hopf algebra of planar rooted trees and prove that these operads are Koszul.

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arXiv2009

The infinitesimal Hopf algebra and the operads of planar forests

We introduce two operads which own the set of planar forests as a basis. With its usual product and two other products defined by different types of graftings, the algebra of planar rooted trees H becomes an algebra over these operads. The compatibility with the infinitesimal coproduct of H and these structures is studied. As an application, an inductive way of computing the dual basis of H for its infinitesimal pairing is given. Moreover, three Cartier-Quillen-Milnor-Moore theorems are given for the operads of planar forests and a rigidity theorem for one of them.

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arXiv2008

The infinitesimal Hopf algebra and the poset of planar forests

We introduce an infinitesimal Hopf algebra of planar trees, generalising the construction of the non-commutative Connes-Kreimer Hopf algebra. A non-degenerate pairing and a dual basis are defined, and a combinatorial interpretation of the pairing in terms of orders on the vertices of planar forests is given. Moreover, the coproduct and the pairing can also be described with the help of a partial order on the set of planar forests, making it isomorphic to the Tamari poset. As a corollary, the dual basis can be computed with a Möbius inversion.

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arXiv2007

Faà di Bruno subalgebras of the Hopf algebra of planar trees from combinatorial Dyson-Schwinger equations

We consider the combinatorial Dyson-Schwinger equation X=B^+(P(X)) in the non-commutative Connes-KreimerHopf algebra of planar rooted trees H, where B^+ is the operator of grafting on a root, and P a formal series. The unique solution X of this equation generates a graded subalgebra A_P of\H. We describe all the formal series P such that A_P is a Hopf subalgebra. We obtain in this way a 2-parameters family of Hopf subalgebras of H, organized into three isomorphism classes: a first one, restricted to a olynomial ring in one variable; a second one, restricted to the Hopf subalgebra of ladders, isomorphic to the Hopf algebra of quasi-symmetric functions; a last (infinite) one, which gives a non-commutative version of the Faà di Bruno Hopf algebra. By taking the quotient, the last classe gives an infinite set of embeddings of the Faà di Bruno algebra into the Connes-Kreimer Hopf algebra of rooted trees. Moreover, we give an embedding of the free Faà di Bruno Hopf algebra on D variables into a Hopf algebra of decorated rooted trees, togetherwith a non commutative version of this embedding.

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arXiv2007

L'algèbre des invariants d'un groupe de Coxeter agissant sur un mutiple de sa représentation standard

Let G be a Coxeter group of type A_n, B_n, D_n or I_2(N), or a complex reflection group of type G(de,e,n). Let V be its standard representation and let k be an integer greater than 2. Then G acts on S(V)^{\otimes k}. We show that the algebra of invariants (S(V)^{\otimes k})^G is a free (S(V)^G)^{\otimes k}-module of rank |G|^{k-1}, and that S(V)^{\otimes k} is not a free (S(V)^{\otimes k})^G-module.

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arXiv2007

Le groupe des traces de Poisson de la variete quotient h+h*/W en rang 2

Let $V$ be a symplectic space over $\mathbb{C}$, $dim_\mathbb{C} V=2l$, and let $G$ be a finite subgroup of $Sp(V)$. The invariant regular functions $\mathbb{C}[V]^G$ inherit a Poisson algebra structure and so the quotient variety ${\cal X}=V/G$ becomes then an affine algebraic Poisson variety. One can now consider the non commutative deformation of $\cal X$ given by the invariant algebra $A_l(\mathbb{C})^G$, where $A_l(\mathbb{C})$ stands for the Weyl algebra of rank $l$. There exist two families of natural examples of this situation. The first concerns wreath products of a finite subgroup of $SL(2,\mathbb{C})$ with an appropriate symmetric group acting on $(\mathbb{C}^2)^n$; the second family is constructed with a Weyl group $W$ acting on the double of the reflexion representation ${\mathfrak{h}}\oplus {\mathfrak{h}}^*$. A nice result of Berest, Etingof and Ginzburg establishes the finiteness of the dimension of $HP_0({\cal X})= \mathbb{C}[\cal X]/\{\mathbb{C}[\cal X], \mathbb{C}[\cal X]\}$, the Poisson trace group of $\cal X$. The purpose of this work is to compute this dimension in certain cases and in particular to compare it to the dimension of the usual trace group of the above mentioned non commutative deformation. The principal theorem establihed here is : {\bf Theorem.} With the above notations, we have the following equality: $$dim_\mathbb{C} HP_0({\mathfrak{h}\oplus {\mathfrak{h}}^*}/W)=dim_\mathbb{C} HH_0(A_l(\mathbb{C})^W).$$ Moreover, this common dimension is 1 in type $ A_2$, 2 in type $B_2$ and 3 in type $G_2$. We also give examples where the difference of these two dimensions is unbounded.

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arXiv2005

Bidendriform bialgebras, trees, and free quasi-symmetric functions

We introduce bidendriform bialgebras, which are bialgebras such that both product and coproduct can be split into two parts satisfying good compatibilities. For example, the Malvenuto-Reutenauer Hopf algebra and the non-commutative Connes-Kreimer Hopf algebras of planar decorated rooted trees are bidendriform bialgebras. We prove that all connected bidendriform bialgebras are generated by their primitive elements as a dendriform algebra bidendriform Milnor-Moore theorem) and then is isomorphic to a Connes-Kreimer Hopf algebra. As a corollary, the Hopf algebra of Malvenuto-Reutenauer is isomorphic to the Connes-kreimer Hopf algebra of planar rooted trees decorated by a certain set. We deduce that the Lie algebra of its primitive elements is free in characteristic zero (G. Duchamp, F. Hivert and J.-Y. Thibon conjecture).

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