From left modules to algebras over an operad: application to combinatorial Hopf algebras

Livernet, Muriel

Annales mathématiques Blaise Pascal, Tome 17 (2010) no. 1, pp. 47-96.

Nous étudions en détail le foncteur oubli de la catégorie des $𝕊$-modules dans la catégorie des espaces vectoriels gradués. Cela nous permet de généraliser les résultats de Patras et Reutenauer obtenus dans le cadre associatif à toute opérade $𝒫$ : les $𝒫$-algèbres dans la catégorie des $𝕊$-modules deviennent des $𝒫$-algèbres dans la catégorie des espaces vectoriels gradués. Il en est de même pour les $𝒫$-algèbres de Hopf lorsque l’opérade $𝒫$ est une opérade de Hopf. De plus, si l’opérade est régulière, alors on obtient deux structures de $𝒫$-algèbres (de Hopf) dans la catégorie des espaces vectoriels gradués. Comme application, nous montrons qu’un certain nombre d’opérades de Hopf donne lieu à des algèbres de Hopf combinatoires connues. Le fait que ces algèbres de Hopf soient libres ou colibres est une conséquence directe de la théorie des opérades.

The purpose of this paper is two fold: we study the behaviour of the forgetful functor from $𝕊$-modules to graded vector spaces in the context of algebras over an operad and derive the construction of combinatorial Hopf algebras. As a byproduct we obtain freeness and cofreeness results for those Hopf algebras.

Let $𝒪$ denote the forgetful functor from $𝕊$-modules to graded vector spaces. Left modules over an operad $𝒫$ are treated as $𝒫$-algebras in the category of $𝕊$-modules. We generalize the results obtained by Patras and Reutenauer in the associative case to any operad $𝒫$: the functor $𝒪$ sends $𝒫$-algebras to $𝒫$-algebras. If $𝒫$ is a Hopf operad the functor $𝒪$ sends Hopf $𝒫$-algebras to Hopf $𝒫$-algebras. If the operad $𝒫$ is regular one gets two different structures of Hopf $𝒫$-algebras in the category of graded vector spaces. We develop the notion of unital infinitesimal $𝒫$-bialgebras and prove freeness and cofreeness results for Hopf algebras built from Hopf operads. Finally, we prove that many combinatorial Hopf algebras arise from our theory, as it is the case for various Hopf algebras defined on the faces of the permutohedra and associahedra.

Classification : 18D50, 16W30, 16A06
Mots clés : $\mathbb{S}$-module, operad, twisted bialgebra, free associative algebra, combinatorial Hopf algebra

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