The dual of infinitesimal unitary Hopf algebras and planar rooted forests

Abstract

We study the infinitesimal (in the sense of Joni and Rota) bialgebra $H_{RT}$ of planar rooted trees introduced in a previous work of two of the authors, whose coproduct is given by deletion of a vertex. We prove that its dual $H_{RT}^*$ is isomorphic to a free non unitary algebra, and give two free generating sets. Giving $H_{RT}$ a second product, we make it an infinitesimal bialgebra in the sense of Loday and Ronco, which allows to explicitly construct a projector onto its space of primitive elements, which freely generates $H_{RT}$.

Keywords

• Infinitesimal bialgebra
• planar rooted trees
• graftings
• 1-cocycle

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