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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: chapoton@math.univ-lyon1.fr
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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

References

[1] Burde, D., Left-symmetric algebras, or pre-Lie algebras in geometry and physics. Cent. Eur. J. Math. 4(2006), no. 3, 323–357. doi:10.2478/s11533-006-0014-9Google Scholar
[2] Cayley, A., On the analytical forms called trees. Amer. J. Math. 4(1881), 266–268. doi:10.2307/2369158Google Scholar
[3] Chauve, C., Dulucq, S., and Guibert, O., Enumeration of some labelled trees. In: Formal power series and algebraic combinatorics (Moscow, 2000), Springer, Berlin, 2000, pp. 146–157.Google Scholar
[4] Corless, R. M., Gonnet, G. H., Hare, D. E. G., Jeffrey, D. J., and Knuth, D. E., On the LambertW function. Adv. Comput. Math. 5(1996), no. 4, 329–359. doi:10.1007/BF02124750Google Scholar
[5] Chapoton, F., On some anticyclic operads. Algebr. Geom. Topol. 5(2005), 53–69. doi:10.2140/agt.2005.5.53Google Scholar
[6] Chapoton, F., Hyperarbres, arbres enracinés et partitions pointées. Homology, Homotopy Appl. 9(2007), no. 1, 193–212.Google Scholar
[7] Chapoton, F. and Livernet, M., Pre-Lie algebras and the rooted trees operad. Internat. Math. Res. Notices 2001, no. 8, 395–408.Google Scholar
[8] Foissy, L., Finite-dimensional comodules over the Hopf algebra of rooted trees. J. Algebra 255, no. 1, 89–120. doi:10.1016/S0021-8693(02)00110-2Google Scholar
[9] Jantzen, J. C., Representations of algebraic groups. Second ed., Mathematical Surveys and Monographs, 107, American Mathematical Society, Providence, RI, 2003.Google Scholar
[10] Livernet, M., From left modules to algebras over an operad: application to combinatorial Hopf algebras. To appear, Ann. Math. Blaise Pascal.Google Scholar
[11] Loday, J.-L., Stasheff, J. D., and Voronov, A. A., eds., Operads: proceedings of renaissance conferences. In: Papers from the Special Session on Moduli Spaces, Operads and Representation Theory held at the AMS Meeting in Hartford, CT, March 4–5, 1995, and from the Conference on Operads and Homotopy Algebra held in Luminy, May 29–June 2, 1995. Contemporary Mathematics, 202, American Mathematical Society, Providence, RI, 1997.Google Scholar
[12] Priddy, S. B., Koszul resolutions. Trans. Amer. Math. Soc. 152(1970), 39–60. doi:10.2307/1995637Google Scholar
[13] Stover, C. R., The equivalence of certain categories of twisted Lie and Hopf algebras over a commutative ring. J. Pure Appl. Algebra 86(1993), no. 3, 289–326. doi:10.1016/0022-4049(93)90106-4Google Scholar