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Article Dans Une Revue Advances in Applied Mathematics Année : 2011

Two Hopf algebras of trees interacting

Résumé

Hopf algebra structures on rooted trees are by now a well-studied object, especially in the context of combinatorics. They are essentially characterized by the coproduct map. In this work we define yet another Hopf algebra H by introducing a new coproduct on a (commutative) algebra of rooted forests, considering each tree of the forest (which must contain at least one edge) as a Feyman-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 by means of a natural H-bicomodule structure on the latter. This enables us to recover results in the field of numerical methods for differential equations due to Chartier, Hairer and Vilmart as well as Murua.

Dates et versions

hal-00288313 , version 1 (16-06-2008)

Identifiants

Citer

Damien Calaque, Kurusch Ebrahimi-Fard, Dominique Manchon. Two Hopf algebras of trees interacting: A Hopf-algebraic approach to composition and substitution of B-series. Advances in Applied Mathematics, 2011, 47 (2), pp.282-308. ⟨10.1016/j.aam.2009.08.003⟩. ⟨hal-00288313⟩
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