On the combinatorics of the Hopf algebra of dissection diagrams

Abstract

In this article, we study the Hopf algebra $$\mathcal {H}_{\tiny \textsc {D}}$$ of dissection diagrams introduced by Dupont in his thesis, more precisely we focus on its underlying coalgebra. We use the version with a parameter x in the base field. We conjecture it is cofree if $$x=1$$ or x is not a root of unity. If $$x=-1$$, then we know there is no cofreeness. Since $$\mathcal {H}_{\tiny \textsc {D}}$$ is a free-commutative right-sided combinatorial Hopf algebra as defined by Loday and Ronco, then there exists a pre-Lie structure on the primitives of its graded dual. Furthermore $$\mathcal {H}_{\tiny \textsc {D}}^{\circledast }$$ and the enveloping algebra of its primitive elements are isomorphic. Thus, we can equip $$\mathcal {H}_{\tiny \textsc {D}}^{\circledast }$$ with a structure of Oudom–Guin. We focus on the pre-Lie structure on dissection diagrams and in particular on the pre-Lie algebra generated by the dissection diagram of degree 1. We prove that it is not free. We express a Hopf algebra morphism between the Grossman–Larson Hopf algebra and $$\mathcal {H}_{\tiny \textsc {D}}^{\circledast }$$ by using pre-Lie and Oudom–Guin structures.