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
bergeron@mathstat.yorku.ca
Muriel Livernet
LAGA et CNRS, Institut Galilée, Université Paris 13, 99 Av. J.-B. Clément, 93430 Villetaneuse, France
livernet@math.univ-paris13.fr
[Abstract-pdf]
\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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