Geodesic
Free Pre-Lie Algebras are Free as Lie Algebras
Canadian mathematical bulletin, Tome 53 (2010) no. 3, pp. 425-437

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DOI

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.
DOI : 10.4153/CMB-2010-063-2
Mots-clés : 18D50, 17B01, 18G40, 05C05 
Chapoton, Frédéric. Free Pre-Lie Algebras are Free as Lie Algebras. Canadian mathematical bulletin, Tome 53 (2010) no. 3, pp. 425-437. doi: 10.4153/CMB-2010-063-2
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     author = {Chapoton, Fr\'ed\'eric},
     title = {Free {Pre-Lie} {Algebras} are {Free} as {Lie} {Algebras}},
     journal = {Canadian mathematical bulletin},
     pages = {425--437},
     year = {2010},
     volume = {53},
     number = {3},
     doi = {10.4153/CMB-2010-063-2},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2010-063-2/}
}
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