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.

中文翻译：

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剖析图的Hopf代数的组合

在本文中，我们研究了Dupont在其论文中引入的解剖图的Hopf代数\（\ mathcal {H} _ {\ tiny \ textsc {D}} \），更准确地说，我们专注于其底层的代数。我们在基本字段中使用带有参数x的版本。我们猜想，如果\（x = 1 \）或x不是统一根，则它是cofree的。如果\（x = -1 \），那么我们知道没有共自由度。由于\（\ mathcal {H} _ {\ tiny \ textsc {D}} \）是Loday和Ronco定义的自由可交换的右侧组合Hopf代数，因此在（）的原语上存在pre-Lie结构其分级为双重。此外\（\ mathcal {H} _ {\ tiny \ textsc {D}} ^ {\ circledast} \）其原始元素的包络代数是同构的。因此，我们可以为\（\ mathcal {H} _ {\ tiny \ textsc {D}} ^ {\ circledast} \装备Oudom–Guin结构。我们将重点放在解剖图上的pre-Lie结构上，尤其是在1度解剖图上生成的pre-Lie代数上。我们证明它不是自由的。我们使用前李和Oudom–Guin结构表示Grossman–Larson Hopf代数和\（\ mathcal {H} _ {\ tiny \ textsc {D}} ^ {\ circledast} \）之间的Hopf代数同态。