On the combinatorics of the Hopf algebra of dissection diagrams

Abstract

In this article, we are interested in the Hopf algebra ${\mathcal{H}}_D$ of dissection diagrams introduced by Dupont in his thesis. We use the version with a parameter $x\in \mathbb{K}$. We want to study its underlying coalgebra. We conjecture it is cofree, except for a countable subset of $\mathbb{K}$. If $x=-1$ then we know there is no cofreedom. We easily see that $\mathcal{H}\_D$ is a free commutative right-sided combinatorial Hopf algebra according to Loday and Ronco. So, there exists a pre-Lie structure on its graded dual. Furthermore ${{\mathcal{H}}_D}^{⊛}$ and the enveloping algebra of its primitive elements are isomorphic. Thus, we can equip ${\mathcal{H}\_D}^{⊛}$ with a structure of Oudom and 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 and Larson Hopf algebra and ${{\mathcal{H}}_D}^{⊛}$ by using pre-Lie and Oudom and Guin structures.