A noncommutative symmetric system over the Grossman-Larson Hopf algebra of labeled rooted trees

Zhao, Wenhua

doi:10.1007/s10801-007-0100-5

Cited by 10 publications

(23 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We have the following result, which is by now a classical one, and for which various proofs are available ( [15], [22], [20], [33] …”

Section: Hopf Algebras Of Treesmentioning

confidence: 96%

“…The graded dual of CK H will play a crucial role in the sequel and is strongly related to the Grossman-Larson Hopf algebra GL H (see [18], [19], [20] and [33]). The algebra GL H is the linear span of rooted trees whose vertices (except the root) are decorated by H (see [15]) : using 0 to note the absence of decoration, any such tree can be written B Where I is any subset of {1, .…”

Section: Hopf Algebras Of Treesmentioning

confidence: 99%

“…Proof The proof is based on the following result of Grossman and Larson (see [33], [34]): Let τ the map from GL H to [x, ∂ x ] defined by…”

Section: Homogeneous Coarborificationmentioning

confidence: 99%

“…In the reverse direction, Wenhua Zhao (see [33], [34], [35]) has for his part rediscovered the constructions of coarborification and then obtained in effect the mechanisms of arborification by dualizing and going to CK. Notably, Zhao's results concern in fact both plain and contracting arborification.…”

Section: Remarkmentioning

confidence: 99%

See 3 more Smart Citations

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

Fauvet¹,

Menous²

2017

Ann. Sci. École Norm. Sup.

View full text Add to dashboard Cite

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.

show abstract

“…We have the following result, which is by now a classical one, and for which various proofs are available ( [15], [22], [20], [33] …”

Section: Hopf Algebras Of Treesmentioning

confidence: 96%

Section: Hopf Algebras Of Treesmentioning

confidence: 99%

“…Proof The proof is based on the following result of Grossman and Larson (see [33], [34]): Let τ the map from GL H to [x, ∂ x ] defined by…”

Section: Homogeneous Coarborificationmentioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

See 2 more Smart Citations

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

Fauvet¹,

Menous²

2017

Ann. Sci. École Norm. Sup.

View full text Add to dashboard Cite

show abstract

“…In the second part, Sections 3 and 4, we review some of the main results that will appear in the sequels [34,36,37]. The main reasons for our including Sections 3 and 4 in this paper are as follows.…”

mentioning

confidence: 99%

Noncommutative symmetric systems over associative algebras

Zhao

2007

Journal of Pure and Applied Algebra

Self Cite

View full text Add to dashboard Cite

This paper is the first of a sequence of papers [W. Zhao, Differential operator specializations of noncommutative symmetric functions (submitted for publication). math.CO/0509134; W. Zhao, Noncommutative symmetric functions and the inversion problem (submitted for publication). math.CV/0509135; W. Zhao, A NCS system over the Grossman-Larson Hopf algebra of labeled rooted trees (submitted for publication). math.CO/0509136; W. Zhao, NCS systems over differential operator algebras and the Grossman-Larson Hopf algebra of labeled rooted trees (submitted for publication). math.CO/0509138. preprint] on the NCS (noncommutative symmetric) systems over differential operator algebras in commutative or noncommutative variables [W. Zhao, Differential operator specializations of noncommutative symmetric functions (submitted for publication). math.CO/0509134]; the NCS systems over the Grossman-Larson

show abstract

Two interacting Hopf algebras of trees: A Hopf-algebraic approach to composition and substitution of B-series

Calaque

Ebrahimi‐Fard

Manchon

2011

Advances in Applied Mathematics

109

View full text Add to dashboard Cite

Hopf algebra structures on rooted trees are by now a well-studied object, especially in the context of combinatorics. In this work we consider a Hopf algebra H by introducing a coproduct on a (commutative) algebra of rooted forests, considering each tree of the forest (which must contain at least one edge) as a Feynman-like graph without loops. The primitive part of the graded dual is endowed with a pre-Lie product defined in terms of insertion of a tree inside another. We establish a surprising link between the Hopf algebra H obtained this way and the well-known Connes-Kreimer Hopf algebra of rooted trees H CK by means of a natural H-bicomodule structure on H CK . This enables us to recover recent results in the field of numerical methods for differential equations due to Chartier, Hairer and Vilmart as well as Murua.

show abstract

A noncommutative symmetric system over the Grossman-Larson Hopf algebra of labeled rooted trees

Cited by 10 publications

References 25 publications

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

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

Noncommutative symmetric systems over associative algebras

Two interacting Hopf algebras of trees: A Hopf-algebraic approach to composition and substitution of B-series

Contact Info

Product

Resources

About