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Muriel Livernet
From left modules to algebras over an operad: application to combinatorial Hopf algebras
Annales mathématiques Blaise Pascal, 17 no. 1 (2010), p. 47-96, doi: 10.5802/ambp.278
Article PDF | Reviews MR 2674654 | Zbl 1206.18010 | 1 citation in Cedram
Class. Math.: 18D50, 16W30, 16A06
Keywords: $\mathbb{S}$-module, operad, twisted bialgebra, free associative algebra, combinatorial Hopf algebra

Résumé - Abstract

The purpose of this paper is two fold: we study the behaviour of the forgetful functor from $\mathbb{S}$-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 $\mathcal{O}$ denote the forgetful functor from $\mathbb{S}$-modules to graded vector spaces. Left modules over an operad $\mathcal{P}$ are treated as $\mathcal{P}$-algebras in the category of $\mathbb{S}$-modules. We generalize the results obtained by Patras and Reutenauer in the associative case to any operad $\mathcal{P}$: the functor $\mathcal{O}$ sends $\mathcal{P}$-algebras to $\mathcal{P}$-algebras. If $\mathcal{P}$ is a Hopf operad the functor $\mathcal{O}$ sends Hopf $\mathcal{P}$-algebras to Hopf $\mathcal{P}$-algebras. If the operad $\mathcal{P}$ is regular one gets two different structures of Hopf $\mathcal{P}$-algebras in the category of graded vector spaces. We develop the notion of unital infinitesimal $\mathcal{P}$-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.

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