Free Pre-Lie Algebras are Free as Lie Algebras

Source abstract

Abstract We prove that the -module PreLie is a free Lie algebra in the category of -modules and can therefore be written as the composition of the -module Lie with a new -module X . This implies that free pre-Lie algebras in the category of vector spaces, when considered as Lie algebras, are free on generators that can be described using X . Furthermore, we define a natural filtration on the -module X . We also obtain a relationship between X and the -module coming from the anticyclic structure of the PreLie operad.