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Journal of Lie Theory 20 (2010), No. 1, 003--015
Copyright Heldermann Verlag 2010

A Combinatorial Basis for the Free Lie Algebra of the Labelled Rooted Trees

Nantel Bergeron
Dept. of Mathematics and Statistics, York University, Toronto, Ontario M3J 1P3, Canada

Muriel Livernet
LAGA et CNRS, Institut Galilée, Université Paris 13, 99 Av. J.-B. Clément, 93430 Villetaneuse, France


\def\calT{{\cal T}} \def\calF{{\cal F}} \def\Lie{{\cal {L}}{\it ie}} \def\N{{\Bbb N}} The pre-Lie operad is an operad structure on the species $\calT$ of labelled rooted trees. A result of F. Chapoton shows that the pre-Lie operad is a free twisted Lie algebra over a field of characteristic zero, that is $\calT = \Lie \circ \calF$ for some species $\calF$. Indeed Chapoton proves that any section of the indecomposables of the pre-Lie operad, viewed as a twisted Lie algebra, gives such a species $\calF$. \par In this paper, we first construct an explicit vector space basis of $\calF[S]$ when $S$ is a linearly ordered set. We deduce the associated explicit species $\calF$, solution to the equation $\calT = \Lie \circ \calF$. As a corollary the graded vector space $(\calF[\{1,\ldots,n\}])_{n\in\N}$ forms a sub non-symmetric operad of the pre-Lie operad $\calT$.

Keywords: Free Lie algebra, rooted tree, pre-Lie operad, Lyndon word.

MSC: 18D, 05E, 17B

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