scholarly journals Generalizing the Connes Moscovici Hopf algebra to contain all rooted trees

2015 ◽  
Vol 56 (4) ◽  
pp. 042302
Author(s):  
Susama Agarwala ◽  
Colleen Delaney
Keyword(s):  
Hopf Algebra ◽  
Rooted Trees

scholarly journals A combinatorial non-commutative Hopf algebra of graphs

2014 ◽  
Vol Vol. 16 no. 1 (Combinatorics) ◽  
Author(s):  
Adrian Tanasa ◽  
Gerard Duchamp ◽  
Loïc Foissy ◽  
Nguyen Hoang-Nghia ◽  
Dominique Manchon
Keyword(s):  
Hopf Algebra ◽  
Rooted Trees ◽  
Quantum Field ◽  
International Audience ◽  
The One ◽  
Planar Rooted Trees

Combinatorics International audience A non-commutative, planar, Hopf algebra of planar rooted trees was defined independently by one of the authors in Foissy (2002) and by R. Holtkamp in Holtkamp (2003). In this paper we propose such a non-commutative Hopf algebra for graphs. In order to define a non-commutative product we use a quantum field theoretical (QFT) idea, namely the one of introducing discrete scales on each edge of the graph (which, within the QFT framework, corresponds to energy scales of the associated propagators). Finally, we analyze the associated quadri-coalgebra and codendrifrom structures.

scholarly journals Hopf algebras on planar trees and permutations

2021 ◽  
Author(s):  
Diego Arcis ◽  
Sebastián Márquez
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Rooted Trees ◽  
Labeled Trees ◽  
Different Types ◽  
Structure Lead ◽  
Planar Rooted Trees

We endow the space of rooted planar trees with the structure of a Hopf algebra. We prove that variations of such a structure lead to Hopf algebras on the spaces of labeled trees, [Formula: see text]-trees, increasing planar trees and sorted trees. These structures are used to construct Hopf algebras on different types of permutations. In particular, we obtain new characterizations of the Hopf algebras of Malvenuto–Reutenauer and Loday–Ronco via planar rooted trees.

scholarly journals On the Lie enveloping algebra of a pre-Lie algebra

2008 ◽  
Vol 2 (1) ◽  
pp. 147-167 ◽  
Author(s):  
J.-M. Oudom ◽  
D. Guin
Keyword(s):  
Hopf Algebra ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Rooted Trees ◽  
Enveloping Algebra ◽  
Associative Product ◽  
Planar Rooted Trees ◽  
Symmetric Brace Algebras

AbstractWe construct an associative product on the symmetric module S(L) of any pre-Lie algebra L. It turns S(L) into a Hopf algebra which is isomorphic to the enveloping algebra of LLie. Then we prove that in the case of rooted trees our construction gives the Grossman-Larson Hopf algebra, which is known to be the dual of the Connes-Kreimer Hopf algebra. We also show that symmetric brace algebras and pre-Lie algebras are the same. Finally, we give a similar interpretation of the Hopf algebra of planar rooted trees.

Counterterms in the context of the universal Hopf algebra of renormalization

2014 ◽  
Vol 29 (08) ◽  
pp. 1450045 ◽  
Author(s):  
Ali Shojaei-Fard
Keyword(s):  
Hopf Algebra ◽  
Rooted Trees ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
New Interpretation

The manuscript discovers a new interpretation of counterterms of renormalizable Quantum Field Theories in terms of formal expansions of decorated rooted trees.

scholarly journals ON THE ANTIPODE OF KREIMER'S HOPF ALGEBRA

2001 ◽  
Vol 16 (22) ◽  
pp. 1427-1434 ◽  
Author(s):  
HÉCTOR FIGUEROA ◽  
JOSÉ M. GRACIA-BONDÍA
Keyword(s):  
Hopf Algebra ◽  
Rooted Trees ◽  
New Formula

We give a new formula for the antipode of the algebra of rooted trees, directly in terms of the bialgebra structure. The equivalence, proved in this paper, among the three available formulas for the antipode, reflects the equivalence among the Bogoliubov–Parasiuk–Hepp, Zimmermann, and Dyson–Salam renormalization schemes.

scholarly journals Algebraic Structures on Typed Decorated Rooted Trees

2021 ◽  
Author(s):  
Loïc Foissy ◽  
◽  
Keyword(s):  
Hopf Algebra ◽  
Lie Algebras ◽  
Hopf Algebras ◽  
Rooted Trees ◽  
Stochastic Pdes ◽  
Algebraic Structures ◽  
Contraction Process

Typed decorated trees are used by Bruned, Hairer and Zambotti to give a description of a renormalisation process on stochastic PDEs. We here study the algebraic structures on these objects: multiple pre-Lie algebras and related operads (generalizing a result by Chapoton and Livernet), noncommutative and cocommutative Hopf algebras (generalizing Grossman and Larson's construction), commutative and noncocommutative Hopf algebras (generalizing Connes and Kreimer's construction), bialgebras in cointeraction (generalizing Calaque, Ebrahimi-Fard and Manchon's result). We also define families of morphisms and in particular we prove that any Connes-Kreimer Hopf algebra of typed and decorated trees is isomorphic to a Connes-Kreimer Hopf algebra of non-typed and decorated trees (the set of decorations of vertices being bigger), through a contraction process, and finally obtain the Bruned-Hairer-Zambotti construction as a subquotient.

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