A survey on the Hopf algebra of dissection diagrams
Résumé
In this article, we are interested in the Hopf algebra H D of dissection diagrams introduced by Dupont in his thesis. We use the version with a parameter x ∈ K. We want to study its underlying coalgebra. We conjecture it is cofree, except for a countable subset of K. If x = −1 then we know there is no cofreedom. We easily see that 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 H D ⊛ and the enveloping algebra of its primitive elements are isomorphic. Thus, we can equip 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 H D ⊛ by using pre-Lie and Oudom and Guin structures.
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