The dual of infinitesimal unitary Hopf algebras and planar rooted forests

Year 2023, Volume: 33 Issue: 33, 178 - 204, 09.01.2023

Xiaomeng Wang Loic Foıssy Xing Gao

https://doi.org/10.24330/ieja.1220707

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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