Related papers: Rooted Tree Maps
It is known that the vector space spanned by labeled rooted trees forms a Hopf algebra. Let k be a field and let R be a commutative k-algebra. Let H denote the Hopf algebra of rooted trees labeled using derivations D in Der(R). In this…
We work with non-planar rooted trees which have a label set given by an arbitrary vector space $V$. By equipping $V$ with a complete locally convex topology, we show how a natural topology is induced on the tree algebra over $V$. In this…
We propose several constructions of commutative or cocommutative Hopf algebras based on various combinatorial structures, and investigate the relations between them. A commutative Hopf algebra of permutations is obtained by a general…
Arborified multiple zeta values are a generalization of multiple zeta values associated with rooted trees. There are two types of decorated rooted trees, corresponding respectively to the series and the integral expressions. Manchon…
We construct natural representations of the Connes-Kreimer Lie algebras on rooted trees/Feynman graphs arising from Hecke correspondences in the categories $\LRF, \LFG$ constructed by K. Kremnizer and the author. We thus obtain the…
The aim of this paper is an algebraic study of the Hopf algebra H_R of rooted trees, which was introduced in \cite{Kreimer1,Connes,Broadhurst,Kreimer2}. We first construct comodules over H_R from finite families of primitive elements.…
We construct an explicit Hopf algebra isomorphism from the algebra of heap-ordered trees to that of quasi-symmetric functions, generated by formal permutations, which is a lift of the natural projection of the Connes-Kreimer algebra of…
This paper generalizes the operadic construction of the Connes-Kreimer Hopf algebra of rooted trees by Moerdijk. Examples of Hopf algebras obtained in this way include the Loday-Ronco Hopf algebra of planar binary trees and the…
We review the relation between the Hopf algebra QSym of quasi-symmetric functions and the multiple zeta values, and then discuss a commutative diagram involving the Hopf algebra Sym of symmetric functions, the Hopf algebra dual NSym of…
We equip the graded polynomial algebra generated by nonplanar rooted binary trees with a Hopf algebra structure by defining a coproduct which disallows cutting both children of any given vertex, refining Connes-Kreimer's notion of…
Description of cocommutative Hopf algebras associated with families of trees. Applications include Cayley's theorem on the number of rooted trees with n nodes, and Catalan's theorem on the number of rooted ordered trees with n nodes.
Loday and Ronco defined an interesting Hopf algebra structure on the linear span of the set of planar binary trees. They showed that the inclusion of the Hopf algebra of non-commutative symmetric functions in the Malvenuto-Reutenauer Hopf…
Recent advances in stochastic PDEs, Hopf algebras of typed trees and integral equations have inspired the study of algebraic structures with replicating operations. To understand their algebraic and combinatorial nature, we first use rooted…
We construct symmetric monoidal categories $\LRF, \FD$ of rooted forests and Feynman graphs. These categories closely resemble finitary abelian categories, and in particular, the notion of Ringel-Hall algebra applies. The Ringel-Hall Hopf…
We exhibit a monoidal structure on the category of finite sets indexed by P-trees for a finitary polynomial endofunctor P. This structure categorifies the monoid scheme (over Spec N) whose semiring of functions is (a P-version of) the…
A commutative but not cocommutative graded Hopf algebra $\Hn$, based on ordered rooted trees, is studied. This Hopf algebra generalizes the Hopf algebraic structure of unordered rooted trees $\Hc$, developed by Butcher in his study of…
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…
We prove that there is an isomorphism between the Hopf Algebra of Feynman diagrams and the Hopf algebra corresponding to the Homogenous Multiple Zeta Value ring H in C<<X,Y>> . In other words, Feynman diagrams evaluate to Multiple Zeta…
A non-commutative, planar, Hopf algebra of rooted trees was proposed in L. Foissy, Bull. Sci. Math. 126 (2002) 193-239. In this paper we propose such a non-commutative Hopf algebra for graphs. In order to define a non-commutative product we…
Consider the coradical filtrations of the Hopf algebras of planar binary trees of Loday and Ronco and of permutations of Malvenuto and Reutenauer. We give explicit isomorphisms showing that the associated graded Hopf algebras are dual to…