Shuffle bialgebras

Ronco, María

Annales de l'Institut Fourier, Tome 61 (2011) no. 3, pp. 799-850.

Le but de ce travail est l’étude des espaces d’éléments primitifs de certaines algèbres de Hopf combinatoires, dont les espaces vectoriels sous-jacents admettent des bases indexées par des sous ensembles de l’ensemble des applications entre ensembles finis. Pour donner une description précise de ces objets nous introduisons la notion d’algèbre shuffle, qui correspond à un type d’algèbre colorée pour laquelle les compositions ne sont pas toujours définies. Nous définissons des bigèbres dans ce contexte et nous calculons leurs sous espaces d’éléments primitifs. Ces espaces d’éléments primitifs peuvent être décrits en terme des générateurs et relations comme des exemples d’autres types d’algèbres colorées.

The goal of our work is to study the spaces of primitive elements of some combinatorial Hopf algebras, whose underlying vector spaces admit linear basis labelled by subsets of the set of maps between finite sets. In order to deal with these objects we introduce the notion of shuffle algebras, which are coloured algebras where composition is not always defined. We define bialgebras in this framework and compute the subpaces of primitive elements associated to them. These spaces of primitive elements have natural structure of some type of coloured algebras, which we describe in terms of generators and relations.

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