The symmetric operation in a free preLie algebra is magmatic
Authors:
Nantel Bergeron and JeanLouis Loday
Journal:
Proc. Amer. Math. Soc. 139 (2011), 15851597
MSC (2010):
Primary 16W30, 17A30, 18D50, 81R60
Published electronically:
December 15, 2010
MathSciNet review:
2763748
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Abstract: A preLie product is a binary operation whose associator is symmetric in the last two variables. As a consequence its antisymmetrization is a Lie bracket. In this paper we study the symmetrization of the preLie product. We show that it does not satisfy any other universal relation than commutativity. This means that the map from the free commutativemagmatic algebra to the free preLie algebra induced by the symmetrization of the preLie product is injective. This result is in contrast with the associative case, where the symmetrization gives rise to the notion of a Jordan algebra. We first give a selfcontained proof. Then we give a proof which uses the properties of dendriform and duplicial algebras.
 1.
Nantel
Bergeron and Muriel
Livernet, The nonsymmetric operad preLie is free, J. Pure
Appl. Algebra 214 (2010), no. 7, 1165–1172. MR 2586994
(2011h:18009), 10.1016/j.jpaa.2009.10.003
 2.
Frédéric
Chapoton and Muriel
Livernet, PreLie algebras and the rooted trees operad,
Internat. Math. Res. Notices 8 (2001), 395–408. MR 1827084
(2002e:17003), 10.1155/S1073792801000198
 3.
V. Dotsenko, Freeness theorems for operads via Groebner basis (2010), 15 pp., arXiv: 0907:4958
 4.
Muriel
Livernet, A rigidity theorem for preLie algebras, J. Pure
Appl. Algebra 207 (2006), no. 1, 1–18. MR 2244257
(2007g:17001), 10.1016/j.jpaa.2005.10.014
 5.
JeanLouis
Loday, Algèbres ayant deux opérations associatives
(digèbres), C. R. Acad. Sci. Paris Sér. I Math.
321 (1995), no. 2, 141–146 (French, with
English and French summaries). MR 1345436
(96f:16013)
 6.
JeanLouis
Loday, Dialgebras, Dialgebras and related operads, Lecture
Notes in Math., vol. 1763, Springer, Berlin, 2001,
pp. 7–66. MR 1860994
(2002i:17004), 10.1007/3540453288_2
 7.
JeanLouis
Loday, Generalized bialgebras and triples of operads,
Astérisque 320 (2008), x+116 (English, with English
and French summaries). MR 2504663
(2010f:18007)
 8.
J.L. Loday, B. Vallette, Algebraic operads, in preparation.
 9.
M.
Markl, Lie elements in preLie algebras, trees and cohomology
operations, J. Lie Theory 17 (2007), no. 2,
241–261. MR 2325698
(2008j:17009)
 10.
María
Ronco, Eulerian idempotents and MilnorMoore theorem for certain
noncocommutative Hopf algebras, J. Algebra 254
(2002), no. 1, 152–172. MR 1927436
(2003f:16064), 10.1016/S00218693(02)000972
 11.
Martin
Markl, Steve
Shnider, and Jim
Stasheff, Operads in algebra, topology and physics,
Mathematical Surveys and Monographs, vol. 96, American Mathematical
Society, Providence, RI, 2002. MR 1898414
(2003f:18011)
 12.
M. Ronco, Shuffle bialgebras, to appear in Annales Institut Fourier.
 13.
K.
A. Zhevlakov, A.
M. Slin′ko, I.
P. Shestakov, and A.
I. Shirshov, Rings that are nearly associative, Pure and
Applied Mathematics, vol. 104, Academic Press, Inc. [Harcourt Brace
Jovanovich, Publishers], New YorkLondon, 1982. Translated from the Russian
by Harry F. Smith. MR 668355
(83i:17001)
 1.
 N. Bergeron, M. Livernet, The nonsymmetric operad preLie is free, Journal of Pure and Applied Algebra 214 (2010), 11651172. MR 2586994
 2.
 F. Chapoton, M. Livernet, PreLie algebras and the rooted trees operad. Internat. Math. Res. Notices 2001, no. 8, 395408. MR 1827084 (2002e:17003)
 3.
 V. Dotsenko, Freeness theorems for operads via Groebner basis (2010), 15 pp., arXiv: 0907:4958
 4.
 M. Livernet, A rigidity theorem for preLie algebras, J. Pure Appl. Algebra, 207, 1 (2006), 118. MR 2244257 (2007g:17001)
 5.
 J.L. Loday, Algèbres ayant deux opérations associatives (digèbres). C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), no. 2, 141146. MR 1345436 (96f:16013)
 6.
 J.L. Loday, Dialgebras, in ``Dialgebras and related operads'', Springer Lecture Notes in Math. 1763 (2001), 766. MR 1860994 (2002i:17004)
 7.
 J.L. Loday, Generalized bialgebras and triples of operads, Astérisque 320 (2008), x+116 pp. MR 2504663 (2010f:18007)
 8.
 J.L. Loday, B. Vallette, Algebraic operads, in preparation.
 9.
 M. Markl, Lie elements in preLie algebras, trees and cohomology operations. J. Lie Theory 17 (2007), no. 2, 241261. MR 2325698 (2008j:17009)
 10.
 M. Ronco, Eulerian idempotents and MilnorMoore theorem for certain noncocommutative Hopf algebras. J. Algebra 254 (2002), no. 1, 152172. MR 1927436 (2003f:16064)
 11.
 M. Markl, S. Shnider, J. Stasheff, Operads in algebra, topology and physics. Mathematical Surveys and Monographs, 96. American Mathematical Society, Providence, RI, 2002. MR 1898414 (2003f:18011)
 12.
 M. Ronco, Shuffle bialgebras, to appear in Annales Institut Fourier.
 13.
 K.A. Zhevlakov, A.M. Slinko, I.P. Shestakov, A.I. Shirshov, Rings that are nearly associative. Pure and Applied Mathematics, 104. Academic Press, Inc., 1982. MR 668355 (83i:17001)
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Additional Information
Nantel Bergeron
Affiliation:
Department of Mathematics and Statistics, York University, Toronto, Ontario M3J 1P3, Canada
Email:
bergeron@yorku.ca
JeanLouis Loday
Affiliation:
Institut de Recherche Mathématique Avancée, CNRS et Université de Strasbourg, 7 rue R. Descartes, 67084 Strasbourg Cedex, France
Email:
loday@math.unistra.fr
DOI:
http://dx.doi.org/10.1090/S000299392010108134
Keywords:
PreLie algebra,
Jordan algebra,
magmatic algebra,
dendriform algebra,
duplicial algebra,
operad,
planar tree
Received by editor(s):
May 25, 2010
Published electronically:
December 15, 2010
Communicated by:
Jim Haglund
Article copyright:
© Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
