A combinatorial basis for the free Lie algebra of the labelled rooted trees

Nantel Bergeron (1) , Muriel Livernet (2)

1 Department of Mathematics and Statistics [Toronto]
2 LAGA - Laboratoire Analyse, Géométrie et Applications

The pre-Lie operad can be realized as a space T of labelled rooted trees. A result of F. Chapoton shows that the pre-Lie operad is a free twisted Lie algebra. That is, the S-module T is obtained as the plethysm of the S-module Lie with an S-module F. In the context of species, we construct an explicit basis of F. This allows us to give a new proof of Chapoton's results. Moreover it permits us to show that F forms a sub nonsymmetric operad of the pre-Lie operad T.

Mots clés: Lyndon word; free Lie algebra; rooted tree; pre-Lie operad

Domaines: Anneaux et algèbres [math.RA]; Combinatoire [math.CO]

Nantel Bergeron, Muriel Livernet. A combinatorial basis for the free Lie algebra of the labelled rooted trees. 2007. ⟨hal-00165987⟩