Two interacting Hopf algebras of trees

Details Two interacting Hopf algebras of trees Two interacting Hopf algebras of trees

2008-06-13

arxiv.org/abs/0806.2238v4

Advances in Applied Mathematics 47 (2011) 282-308

Mathematics

Combinatorics

Error in antipode formula (section 7) corrected. Erratum submitted

Scientific paper

10.1016/j.aam.2009.08.003

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 by means of a natural H-bicomodule structure on the latter. 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.

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