Comparison of Hopf algebras on trees

Holtkamp, Ralf

doi:10.1007/s00013-003-0796-y

Cited by 76 publications

(96 citation statements)

References 11 publications

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“…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.…”

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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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“…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%

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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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“…For other instances of strong relationship between bialgebras and trees (generators of the free magmatic algebra), the reader can look at (nonexhaustive list) [1,3,8,9,[11][12][13]19]. At the suggestion of one of the referees, we also give the following references on tree enumeration and combinatorial approaches to the Cartier-MilnorMoore and Poincaré-Birkhoff-Witt theorems: [2,[16][17][18]21].…”

Section: )mentioning

confidence: 99%

Coassociative magmatic bialgebras and the Fine numbers

2007

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We prove a structure theorem for the connected coassociative magmatic bialgebras. The space of primitive elements is an algebra over an operad called the primitive operad. We prove that the primitive operad is magmatic generated by n − 2 operations of arity n. The dimension of the space of all the n-ary operations of this primitive operad turns out to be the Fine number F n−1 . In short, the triple of operads (As, Mag, MagFine) is good.

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“…A non commutative version, the Hopf algebra of planar rooted trees, is introduced in [3,5]. We furthemore introduce in [4] an infinitesimal version of this object, replacing admissible cuts by left admissible cuts: this last object is here denoted by H. Similarly with the Hopf case, H is a self-dual object and it owns a non-degenerate, symmetric Hopf pairing, denoted by −, − .…”

Section: Introductionmentioning

confidence: 99%

The Infinitesimal Hopf Algebra and the Operads of Planar Forests

Foissy¹

2009

International Mathematics Research Notices

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HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. The infinitesimal Hopf algebra and the operads of planar forestsLoïc Foissy To cite this version:Loïc Foissy. The infinitesimal Hopf algebra and the operads of planar forests. 2009. hal-00353236The infinitesimal Hopf algebra and the operads of planar forests ABSTRACT. 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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Comparison of Hopf algebras on trees

Cited by 76 publications

References 11 publications

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

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

Coassociative magmatic bialgebras and the Fine numbers

The Infinitesimal Hopf Algebra and the Operads of Planar Forests

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