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Search: MSC category 18D50 ( Operads [See also 55P48] )

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1. CMB 2010 (vol 53 pp. 425)

Chapoton, Frédéric
Free Pre-Lie Algebras are Free as Lie Algebras
We prove that the $\mathfrak{S}$-module $\operatorname{PreLie}$ 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 $\operatorname{Lie}$ 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 $\operatorname{PreLie}$ operad.

Categories:18D50, 17B01, 18G40, 05C05

2. CMB 2007 (vol 50 pp. 182)

Chapoton, Frédéric
On the Coxeter Transformations for Tamari Posets
A relation between the anticyclic structure of the dendriform operad and the Coxeter transformations in the Grothendieck groups of the derived categories of modules over the Tamari posets is obtained.

Categories:18D50, 18E30, 06A11

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