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Free Pre-Lie Algebras are Free as Lie Algebras

Published online by Cambridge University Press:  20 November 2018

Frédéric Chapoton
Frédéric Chapoton*
Affiliation:
Université de Lyon, Université Lyon 1, Institut Camille Jordan, Villeurbanne, France e-mail: [email protected]
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Abstract

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We prove that the $\mathfrak{S}$-module PreLie is a free Lie algebra in the category of $\mathfrak{S}$-modules and can therefore be written as the composition of the $\mathfrak{S}$-module Lie with a new $\mathfrak{S}$-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 $\mathfrak{S}$-module $X$. We also obtain a relationship between $X$ and the $\mathfrak{S}$-module coming from the anticyclic structure of the PreLie operad.

Keywords

18D5017B0118G4005C05
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 53 , Issue 3 , 01 September 2010 , pp. 425 - 437
Copyright
Copyright © Canadian Mathematical Society 2010

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