Relating the Connes–Kreimer and Grossman–Larson Hopf Algebras Built on Rooted Trees

By Florin Panaite. 2000

66 citations recorded by the source corpus.

We prove that there is a Hopf duality between two Hopf algebras built on rooted trees: the Connes–Kreimer Hopf algebra HR which controls the renormalization in quantum field theory, and the Grossman–Larson Hopf algebra A introduced ten years ago through some 'differential' and combinatorial reason. We then study two natural operators on A, inspired by similar ones introduced by Connes and Kreimer for HR.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/4e4153381c6e0f06f029aa464d7c042654cc7741

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Citations in the corpus

Combinatorics of rooted trees and Hopf algebras (2002, methodology): “Kreimer’s Hopf algebra HK admits a derivation called the growth operator, which is important in describing the relation of this algebra to another Hopf algebra studied earlier by A. Connes and H. Moscovici [4].”
Finite dimensional comodules over the Hopf algebra of rooted trees (2001, background): “It is shown in [8, 12] that U(L) and the Grossman-Larson Hopf algebra on rooted trees are isomorphic (see [6, 7]).”
Combinatorial Hopf algebras (2008, background): “les in the literature appear as free-commutative (rather than cofree-cocommutative). For instance the dual of SGV (K), that is the GrossmanLarson Hopf algebra, is the Connes-Kreimer Hopf algebra (cf. [39, 20]). It can be constructed either directly as in [11], or by means of the cofree pre-Lie coalg”