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International Mathematics Research Notices Advance Access published online on August 27, 2009
International Mathematics Research Notices, doi:10.1093/imrn/rnp132
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© The Author 2009. Published by Oxford University Press. All rights reserved. For Permissions, please e-mail: journals.permissions@oxfordjournals.org

The Infinitesimal Hopf Algebra and the Operads of Planar Forests

Loïc Foissy

Laboratoire de Mathématiques, FRE3111, Université de Reims Moulin de la Housse - BP 1039 - 51687 REIMS Cedex 2, France

Correspondence: Correspondence to be sent to: loic.foissy{at}univ-reims.fr

We introduce two operads based on the set of planar forests. With its usual product and two other products defined by different types of graftings, the algebra of planar rooted trees Formula becomes an algebra over these operads. The compatibility with the infinitesimal coproduct of Formula and these structures is studied. As an application, an inductive way of computing the dual basis of Formula for its infinitesimal pairing is given. Moreover, three Cartier–Quillen–Milnor–Moore theorems are given for the operads of planar forests and a rigidity theorem for one of them.


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