Relating two Hopf algebras built from an operad

Frédéric Chapoton (1) , Muriel Livernet (2)

1 ICJ - Institut Camille Jordan
2 LAGA - Laboratoire Analyse, Géométrie et Applications

Starting from an operad, one can build a family of posets. From this family of posets, one can define an incidence Hopf algebra. By another construction, one can also build a group directly from the operad. We then consider its Hopf algebra of functions. We prove that there exists a surjective morphism from the latter Hopf algebra to the former one. This is illustrated by the case of an operad built on rooted trees, the $\NAP$ operad, where the incidence Hopf algebra is identified with the Connes-Kreimer Hopf algebra of rooted trees.

Mots clés: Posets; Hopf algebras; Operads; Rooted trees

Domaines: Anneaux et algèbres [math.RA]; Combinatoire [math.CO]

Frédéric Chapoton, Muriel Livernet. Relating two Hopf algebras built from an operad. International Mathematics Research Notices, 2007, 2007, rnm131, 27 p. ⟨10.1093/imrn/rnm131⟩. ⟨hal-00165245v2⟩