Les algèbres de Hopf des arbres enracinés décorés, I

Foissy, Loïc

doi:10.1016/s0007-4497(02)01108-9

Cited by 135 publications

(193 citation statements)

References 15 publications

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“…The Connes-Kreimer Hopf algebra of rooted trees is commutative while the Hopf algebra of rooted trees considered here are noncommutative, as in the case of Foissy and Holtkamp [14,23]. The approaches and results in the commutative and noncommutative cases are similar.…”

mentioning

confidence: 85%

“…This construction has various generalizations, including the noncommutative and decorated cases [14,24,28,29].…”

Section: Operated Hopf Algebras Of Decorated Forestsmentioning

confidence: 99%

“…The intrinsic connection of Hopf algebras with combinatorics was first revealed in the pioneering work of Joni and Rota [25]. Since then, many Hopf algebras has been built on various combinatorial objects, especially trees and rooted trees, such as those of Connes-Kreimer [8,27], Foissy and Holtkamp [14,24], Loday-Ranco [32] and Grossman-Larson [17]. Hopf algebras were also built from free objects in various contexts, such as free associative algebras and the enveloping algebras of Lie algebras.…”

Section: Introductionmentioning

confidence: 99%

“…We recall the construction [14,24] of the noncommutative Connes-Kreimer Hopf algebra H RT := H RT (X). A subforest of a planar rooted tree T ∈ T(X) is the forest consisting of a set of vertices of T together with their descents and edges connecting all these vertices.…”

mentioning

confidence: 99%

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Hopf algebras of rooted forests, cocyles, and free Rota-Baxter algebras

Zhang

Gao

Guo

2016

Journal of Mathematical Physics

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Abstract. The Hopf algebra and the Rota-Baxter algebra are the two algebraic structures underlying the algebraic approach of Connes and Kreimer to renormalization of perturbative quantum field theory. In particular the Hopf algebra of rooted trees serves as the "baby model" of Feynman graphs in their approach and can be characterized by certain universal properties involving a Hochschild 1-cocycle. Decorated rooted trees have also been applied to study Feynman graphs. We will continue the study of universal properties of various spaces of decorated rooted trees with such a 1-cocycle, leading to the concept of a cocycle Hopf algebra. We further apply the universal properties to equip a free Rota-Baxter algebra with the structure of a cocycle Hopf algebra or a cocycle bialgebra.

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mentioning

confidence: 85%

“…This construction has various generalizations, including the noncommutative and decorated cases [14,24,28,29].…”

Section: Operated Hopf Algebras Of Decorated Forestsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Hopf algebras of rooted forests, cocyles, and free Rota-Baxter algebras

Zhang

Gao

Guo

2016

Journal of Mathematical Physics

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“…This is exactly what arborification/coarborification does. Once the original definitions of Ecalle are translated into a Hopf-algebraic setting, with the use of Connes-Kreimer Hopf algebra CK and its graded dual, it is possible to recognize that the arborification transform is nothing else that a property of factorisation of characters between Hopf algebras (we perform this at the same time for the shuffle and quasishuffle cases), using the fact that CK is an initial object for Hochschild cohomology for a particular category of cogebras ( [7], [14], [15]). …”

Section: Introductionmentioning

confidence: 99%

Ecalle's arborification-coarborification transforms and Connes-Kreimer Hopf algebra

Fauvet¹,

Menous²

2017

Ann. Sci. École Norm. Sup.

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We give a natural and complete description of Ecalle's mould-comould formalism within a Hopf-algebraic framework. The arborification transform thus appears as a factorization of characters, involving the shuffle or quasishuffle Hopf algebras, thanks to a universal property satisfied by Connes-Kreimer Hopf algebra. We give a straightforward characterization of the fundamental process of homogeneous coarborification, using the explicit duality between decorated Connes-Kreimer and GrossmanLarson algebras. Finally, we introduce a new Hopf algebra that systematically underlies the calculations for the normalization of local dynamical systems.

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Nonsymmetric Operads

Giraudo¹

2018

Nonsymmetric Operads in Combinatorics

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The natural Hopf algebra N • O of an operad O is a Hopf algebra whose bases are indexed by some words on O. We introduce new bases of these Hopf algebras deriving from free operads via new lattice structures on their basis elements. We construct polynomial realizations of N • O by using alphabets of variables endowed with unary and binary relations. By specializing our polynomial realizations, we discover links between N • O and the Hopf algebra of word quasi-symmetric functions of Hivert, the decorated versions of the Connes-Kreimer Hopf algebra of Foissy, the Faà di Bruno Hopf algebra and its deformations, and the noncommutative multi-symmetric functions Hopf algebras of Novelli and Thibon.

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Les algèbres de Hopf des arbres enracinés décorés, I

Cited by 135 publications

References 15 publications

Hopf algebras of rooted forests, cocyles, and free Rota-Baxter algebras

Hopf algebras of rooted forests, cocyles, and free Rota-Baxter algebras

Ecalle's arborification-coarborification transforms and Connes-Kreimer Hopf algebra

Nonsymmetric Operads

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