Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-24T18:36:36.315Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Lie enveloping algebra of a pre-Lie algebra

Published online by Cambridge University Press:  28 May 2008

J.-M. Oudom  and
D. Guin
J.-M. Oudom
Affiliation:
[email protected], Université Montpellier II, case 051, Place Eugène Bataillon, 34 095 Montpellier, France
D. Guin
Affiliation:
[email protected], Université Montpellier II, case 051, Place Eugène Bataillon, 34 095 Montpellier, France
Get access

Abstract

We construct an associative product on the symmetric module S(L) of any pre-Lie algebra L. It turns S(L) into a Hopf algebra which is isomorphic to the enveloping algebra of LLie. Then we prove that in the case of rooted trees our construction gives the Grossman-Larson Hopf algebra, which is known to be the dual of the Connes-Kreimer Hopf algebra. We also show that symmetric brace algebras and pre-Lie algebras are the same. Finally, we give a similar interpretation of the Hopf algebra of planar rooted trees.

Keywords

brace algebraHopf algebrapre-Lie algebrarooted tree
Type
Research Article
Information
Journal of K-Theory , Volume 2 , Issue 1 , August 2008 , pp. 147 - 167
Copyright
Copyright © ISOPP 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Balavoine, D., Deformations of algebras over a quadratic operad, Contemp. Math. 202, Amer. Math. Soc., Providence (1997), 207–234Google Scholar
2.Brouder, C. and Frabetti, A., QED Hopf algebras on planar binary trees, preprint, QA 0112043Google Scholar
3.Chapoton, F. and Livernet, M., Pre-Lie algebras and the rooted trees operad, IMRN 8 (2001), 395–408CrossRefGoogle Scholar
4.Chapoton, F., Algèbres pré-Lie et algèbres de Hopf liées à la renormalisation, CRAS 332, Série I (2001), 681–684Google Scholar
5.Chapoton, F., Un théorème de Cartier-Milnor-Moore-Quillen pour les digèbres dendriformes et les algèbres braces, JPAA 168 (2002), 1–18Google Scholar
6.Connes, A. and Kreimer, D., Hopf algebras, renormalization and noncommutative geometry, Comm. Math. Phys. 199, n° 1 (1998), 203–242CrossRefGoogle Scholar
7.Connes, A. and Kreimer, D., Insertion and Elimination: the doubly infinite Lie algebra of Feynman graphs, Ann. Henri Poincaré 3, no. 3 (2002), 411–433CrossRefGoogle Scholar
8.Foissy, L., Les algèbres de Hopf des arbres enracinés décorés, thèse de l'université de Reims, (2002)CrossRefGoogle Scholar
9.Foissy, L., Les algèbres de Hopf des arbres enracinés, II, Bull. Sci. Math. 126 (2002), 249–288CrossRefGoogle Scholar
10.Gerstenhaber, M., The cohomology structure of an associative ring, Ann. Math. 78 (1963), 267–288Google Scholar
11.Getzler, and Jones, , Operads, homotopy algebra and iterated integrals for double loop spaces, preprint, hep-th 9403055Google Scholar
12.Grossman, R. and Larson, R.-G., Hopf algebraic structure of families of trees, J. Alg. 126, n° 1 (1989), 184–210CrossRefGoogle Scholar
13.Hinich, V., Tamarkin's proof of Kontsevich formality theorem, preprint, QA 0003052Google Scholar
14.Hoffman, M., Combinatorics of rooted trees a Hopf algebras, Trans. Amer. Math. Soc. 355 , n° 9 (2003), 3795–3811CrossRefGoogle Scholar
15.Holtkamp, R., Comparaison of Hopf algebras on trees, Arch. Math. (Basel) 80, n°4 (2003), 368–383Google Scholar
16.Kontsevich, M., Deformation quantization of Poisson manifolds, I, preprint, q-alg 9709040Google Scholar
17.Lada, T. and Markl, M., Symmetric brace algebras, preprint, QA 0307054Google Scholar
18.Loday, J.-L., La renaissance des opérades, Séminaire Bourbaki, Vol. 1994/95. Astérisque 237, Exp. No. 792, 3 (1996), 47–74Google Scholar
19.Loday, J.-L. and Ronco, M., Hopf algebra of the planar binary trees, Adv. Math. 139, n° 2 (1998), 293–309Google Scholar
20.Matsushima, Y., Affine structures on complex manifolds, Osaka J. Math. 5 (1968), 215–222Google Scholar
21.Moerdijk, I., On the Connes-Kreimer construction of Hopf algebras, Contemp. Math. 271 (2001), 311–321CrossRefGoogle Scholar
22.Nijenhuis, A., Sur une classe de propriétés communes à quelques types différents d'algèbres, Enseignement Math. (2) 14 (1968), 225–277Google Scholar
23.Panaite, F., Relating the Connes-Kreimer and the Grossman-Larson Hopf algebras built on rooted trees, Let. Math. Phy. 51, n° 3 (2000), 211–219CrossRefGoogle Scholar
24.Ronco, M., Eulerian idempotents an Milnor-Moore theorem for certain non cocommutative Hopf algebras, J. Alg. 254 (2002), 152–172CrossRefGoogle Scholar
25.Tamarkin, D., Another proof of M. Kontsevich formality theorem dor , preprint, QA 9803025Google Scholar
26.van der Laan, P., Some Hopf algebras of trees, preprint, QA 0106244Google Scholar