[On the Lie enveloping algebra of a pre-Lie algebra]
We construct an associative product on the symmetric module of any pre-Lie algebra L. It turns into a Hopf algebra which is isomorphic to the envelopping algebra of . Then we prove that in the case of rooted trees our construction is dual to that of Connes and Kreimer. 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.
Nous construisons un produit associatif sur le module symétrique de toute algèbre pré-Lie L qui en en fait une algèbre de Hopf isomorphe à . Nous montrons ensuite, que dans le cas des arbres enracinés, notre construction est duale à celle de Connes et Kreimer. Nous montrons aussi que les structures d'algèbres braces symétriques et d'algèbres pré-Lie sont identiques. Enfin, nous donnons une interprétation analogue de l'algèbre de Hopf des arbres plans enracinés.
Accepted:
Published online:
Jean-Michel Oudom 1; Daniel Guin 1
@article{CRMATH_2005__340_5_331_0, author = {Jean-Michel Oudom and Daniel Guin}, title = {Sur l'alg\`ebre enveloppante d'une alg\`ebre {pr\'e-Lie}}, journal = {Comptes Rendus. Math\'ematique}, pages = {331--336}, publisher = {Elsevier}, volume = {340}, number = {5}, year = {2005}, doi = {10.1016/j.crma.2005.01.010}, language = {fr}, }
Jean-Michel Oudom; Daniel Guin. Sur l'algèbre enveloppante d'une algèbre pré-Lie. Comptes Rendus. Mathématique, Volume 340 (2005) no. 5, pp. 331-336. doi : 10.1016/j.crma.2005.01.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.01.010/
[1] QED Hopf algebras on planar binary trees (preprint) | arXiv
[2] Pre-Lie algebras and the rooted trees operad, Int. Math. Res. Notices, Volume 8 (2001), pp. 395-408
[3] Un théorème de Cartier–Milnor–Moore–Quillen pour les digèbres dendriformes et les algèbres braces, J. Pure Appl. Algebra, Volume 168 (2002), pp. 1-18
[4] Hopf algebras, renormalization and noncommutative geometry, Commun. Math. Phys., Volume 199 (1998) no. 1, pp. 203-242
[5] L. Foissy, Les algèbres de Hopf des arbres enracinés décorés, thèse de l'université de Reims, 2002
[6] Hopf algebraic structure of families of trees, J. Algebra, Volume 126 (1989) no. 1, pp. 184-210
[7] Tamarkin's proof of Kontsevich formality theorem (preprint) | arXiv
[8] Combinatorics of rooted trees and Hopf algebras, Trans. Amer. Math. Soc., Volume 355 (2003) no. 9, pp. 3795-3811
[9] Symmetric brace algebras with application to particles of high spin (preprint) | arXiv
[10] Hopf algebra of the planar binary trees, Adv. Math., Volume 139 (1998) no. 2, pp. 293-309
[11] On the Connes–Kreimer construction of Hopf algebras, Contemp. Math., Volume 271 (2001), pp. 311-321
[12] Relating the Connes–Kreimer and the Grossman–Larson Hopf algebras built on rooted trees, Lett. Math. Phys., Volume 51 (2000) no. 3, pp. 211-219
[13] Eulerian idempotents and Milnor–Moore theorem for certain non cocommutative Hopf algebras, J. Algebra, Volume 254 (2002), pp. 152-172
[14] An another proof of M. Kontsevich formality theorem for (preprint) | arXiv
[15] Some Hopf algebras of trees (preprint) | arXiv
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