Relating Two Hopf Algebras Built from An Operad

Abstract

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.where I is a finite set and ≃ runs over the set of equivalence relations on I. Note that this monoidal functor is not symmetric.The data of a species P is equivalent to the data of a collection of sets P(n) with actions of the symmetric groups. The set P(n) can be defined as P ({1, . . . , n}), with the obvious action of the symmetric group S n . The other way round, one can recover the set P(I) as a colimit. Set-operadsA set-operad P is a monoid with unit in the monoidal category of species for the tensor product •. This means the data of a morphism of species γ : P • P → P,

A rooted-treesq-series lifting a one-parameter family of Lie idempotents

“…There is a group associated to each operad; see Appendix A and [Chapoton 2002a;2007/08;van der Laan 2003;Chapoton and Livernet 2007]. We will need the group G PL associated to the pre-Lie operad.…”

“…This can also be found in [Chapoton 2002a;2007/08;van der Laan 2003;Chapoton and Livernet 2007]. We use in this section the definition of the notion operad by a multiple composition map γ , which is equivalent (using the unit) to the definition using the single compositions • i that we have used elsewhere in the article.…”

“…One starting point of this article was the existence of a one-parameter family of Lie idempotents belonging to the descent algebras of the symmetric groups [Duchamp et al 1994;. As Lie idempotents, these are in particular Lie elements.…”

“…On this subject, the reader may also consult [Chapoton 2002a;2007/08;van der Laan 2003;Chapoton and Livernet 2007].…”

Combinatorial Hopf algebras from PROs

“…One of these is a construction [vdL04] taking an operad O as input and producing a bialgebra H (O) as output, which is called the natural bialgebra of O. This construction has been studied in some recent works: in [CL07], it is shown that H can be rephrased in terms of an incidence Hopf algebra of a certain family of posets, and in [ML14], a general formula for its antipode is established. Let us also cite [Fra08] in which this construction is considered to study series of trees.…”