Combinatorial structures of three vertices and Lie algebras

In this paper, we characterize digraphs of 3 vertices associated with Lie algebras according to isomorphism classes of these associated Lie algebras. At this respect, we introduce and implement two algorithmic methods: the first is devoted to draw the digraph associated with a given Lie algebra and the second allows us to determine if a given digraph is associated or not with a Lie algebra.Ministerio de Ciencia e InnovaciónFondo Europeo de Desarrollo Regiona

Combinatorial Hopf algebras

We define a "combinatorial Hopf algebra" as a Hopf algebra which is free (or cofree) and equipped with a given isomorphism to the free algebra over the indecomposables (resp. the cofree coalgebra over the primitives). The choice of such an isomorphism implies the existence a finer algebraic structure on the Hopf algebra and on the indecomposables (resp. the primitives). For instance a cofree-cocommutative right-sided combinatorial Hopf algebra is completely determined by its primitive part which is a pre-Lie algebra. The key example is the Connes-Kreimer Hopf algebra. The study of all these combinatorial Hopf algebra types gives rise to several good triples of operads. It involves the operads: dendriform, pre-Lie, brace, and variations of them.Comment: The second part, dealing with right-sided combinatorial Hopf algebras, has been completely modified in this new versio

Weighted infinitesimal bialgebras

As a uniform of two versions of infinitesimal bialgebras introduced respectively by Joni-Rota and Loday-Ronco, weighted infinitesimal bialgebras play an important role in mathematics and mathematical physics. In this paper, we introduce the concept of weighted infinitesimal Hopf modules and show that any module carries a natural structure of weighted infinitesimal unitary Hopf module over a weighted quasitriangular infinitesimal unitary bialgebra. We decorate planar rooted forests $H_{\mathrm{RT}}(X, \Omega)$ in a new way, and prove that the $H_{\mathrm{RT}}(X, \Omega)$, together with a coproduct $\Delta_{\epsilon}$ and grafting operations $\{ B^+_\omega \mid \omega\in \Omega\}$, is the free $\Omega$-cocycle infinitesimal unitary bialgebra (resp. Hopf algebra) of weight zero on a set $X$. A combinatorial description of $\Delta_{\epsilon}$ is given. As applications, we obtain the initial object in the category of cocycle infinitesimal unitary bialgebras (resp. Hopf algebras) on undecorated planar rooted forests, which is the object studied in the (noncommutative) Connes-Kreimer Hopf algebra. Finally, we derive two pre-Lie algebras from an arbitrary weighted infinitesimal bialgebra and weighted commutative infinitesimal bialgebra, respectively. The second construction generalizes the Gelfand-Dorfman Theorem on Novikov algebras.Comment: 44 pages; give the right reference

Trees, set compositions and the twisted descent algebra

We first show that increasing trees are in bijection with set compositions, extending simultaneously a recent result on trees due to Tonks and a classical result on increasing binary trees. We then consider algebraic structures on the linear span of set compositions (the twisted descent algebra). Among others, a number of enveloping algebra structures are introduced and studied in detail. For example, it is shown that the linear span of trees carries an enveloping algebra structure and embeds as such in an enveloping algebra of increasing trees. All our constructions arise naturally from the general theory of twisted Hopf algebras.Comment: 32 pages, 14 figures, references adde

Invariant tensors and the cyclic sieving phenomenon

We construct a large class of examples of the cyclic sieving phenomenon by expoiting the representation theory of semi-simple Lie algebras. Let $M$ be a finite dimensional representation of a semi-simple Lie algebra and let $B$ be the associated Kashiwara crystal. For $r\ge 0$, the triple $(X,c,P)$ which exhibits the cyclic sieving phenomenon is constructed as follows: the set $X$ is the set of isolated vertices in the crystal $\otimes^rB$; the map $c\colon X\rightarrow X$ is a generalisation of promotion acting on standard tableaux of rectangular shape and the polynomial $P$ is the fake degree of the Frobenius character of a representation of $\mathfrak{S}_r$ related to the natural action of $\mathfrak{S}_r$ on the subspace of invariant tensors in $\otimes^rM$. Taking $M$ to be the defining representation of $\mathrm{SL}(n)$ gives the cyclic sieving phenomenon for rectangular tableaux

Structures in Feynman Graphs -Hopf Algebras and Symmetries

We review the combinatorial structure of perturbative quantum field theory with emphasis given to the decomposition of graphs into primitive ones. The consequences in terms of unique factorization of Dyson--Schwinger equations into Euler products are discussed.Comment: 41 pages, epsf figures, talk given at the "Dennisfest", "Graphs and Patterns in Mathematics and Theoretical Physics, Stony Brook, June 2001, final version, to appear in the Proceeding

On the combinatorics of the Hopf algebra of dissection diagrams

In this article, we are interested in the Hopf algebra $\mathcal{H}_{D}$ of dissection diagrams introduced by Dupont in his thesis. We use the version with a parameter $x\in\mathbb{K}$. We want to study its underlying coalgebra. We conjecture it is cofree, except for a countable subset of $\mathbb{K}$. If $x=-1$ then we know there is no cofreedom. We easily see that $\mathcal{H}\_{D}$ is a free commutative right-sided combinatorial Hopf algebra according to Loday and Ronco. So, there exists a pre-Lie structure on its graded dual. Furthermore ${\mathcal{H}_{D}}^{\circledast}$ and the enveloping algebra of its primitive elements are isomorphic. Thus, we can equip ${\mathcal{H}\_{D}}^{\circledast}$ with a structure of Oudom and Guin. We focus on the pre-Lie structure on dissection diagrams and in particular on the pre-Lie algebra generated by the dissection diagram of degree $1$. We prove that it is not free. We express a Hopf algebra morphism between the Grossman and Larson Hopf algebra and ${\mathcal{H}_{D}}^{\circledast}$ by using pre-Lie and Oudom and Guin structures

Quadratic algebras related to the bihamiltonian operad

We prove the conjectures on dimensions and characters of some quadratic algebras stated by B.L.Feigin. It turns out that these algebras are naturally isomorphic to the duals of the components of the bihamiltonian operad.Comment: 23 pages, 5 figures; grammar and typesetting correcte

Tensor diagrams and cluster algebras

The rings of SL(V) invariants of configurations of vectors and linear forms in a finite-dimensional complex vector space V were explicitly described by Hermann Weyl in the 1930s. We show that when V is 3-dimensional, each of these rings carries a natural cluster algebra structure (typically, many of them) whose cluster variables include Weyl's generators. We describe and explore these cluster structures using the combinatorial machinery of tensor diagrams. A key role is played by the web bases introduced by G.Kuperberg.Comment: 73 pages, 66 figures; same results as in the earlier versions, changes in the expositio

Flows on rooted trees and the Narayana idempotents

Several generating series for flows on rooted trees are introduced, as elements in the group of series associated with the Pre-Lie operad. By combinatorial arguments, one proves identities that characterise these series. One then gives a complete description of the image of these series in the group of series associated with the Dendriform operad. This allows to recover the Lie idempotents in the descent algebras recently introduced by Menous, Novelli and Thibon. Moreover, one defines new Lie idempotents and conjecture the existence of some others.Comment: 37 pages ; 6 figure