Free pre-Lie algebras are free as Lie algebras

Frédéric Chapoton (1)

1 ICJ - Institut Camille Jordan

We prove that free pre-Lie algebras, when considered as Lie algebras, are free. Working in the category of S-modules, we define a natural filtration on the space of generators. We also relate the symmetric group action on generators with the structure of the anticyclic PreLie operad.

Mots clés: operad; pre-Lie algebra; Lie algebra; tree; spectral sequence

Domaines: Anneaux et algèbres [math.RA]; Algèbres quantiques [math.QA]

Frédéric Chapoton. Free pre-Lie algebras are free as Lie algebras. Canadian Mathematical Bulletin, 2010, 53 (3), pp.425-437. ⟨10.4153/CMB-2010-063-2⟩. ⟨hal-00142155⟩