Construction de certaines opérades et bigèbres associées aux polytopes de Stasheff et hypercubes
HTML articles powered by AMS MathViewer
- by Frédéric Chapoton PDF
- Trans. Amer. Math. Soc. 354 (2002), 63-74 Request permission
Abstract:
Stasheff polytopes, introduced by Stasheff in his study of $H$-spaces, are linked to associativity. The direct sum of their cellular complexes is the underlying complex of the $A_{\infty }$ operad which describes homotopy associative algebras. In particular, there exists a quasi-isomorphism $A_{\infty }\rightarrow \operatorname {As}$.
Here, we define on the direct sum of their dual cellular complexes the structure of a differential graded operad. This construction extends the dendriform operad of Loday, which corresponds to the vertices of the polytopes. We also define the structure of a differential graded operad on the direct sum of the dual cellular complexes of the hypercubes. We define a quasi-isomorphism from $\operatorname {As}$ to each of these operads.
We also define non-differential variants of the two preceding operads and a morphism from $\operatorname {As}$ to each of these operads. We show that the free algebras have a coproduct which turns them into bialgebras.
Résumé. Les polytopes de Stasheff, introduits pour l’étude des $H$-espaces, sont liés à l’associativité. La somme directe de leurs complexes cellulaires forme le complexe sous-jacent à l’opérade $A_\infty$ qui décrit les algèbres associatives à homotopie près. En particulier, il existe un quasi-isomorphisme $A_\infty \to \operatorname {As}$.
Ici, on munit la somme directe des duaux de leurs complexes cellulaires d’une structure d’opérade différentielle graduée. Cette construction généralise l’opérade des algèbres dendriformes de Loday, qui correspond aux sommets des polytopes. On munit aussi la somme directe des duaux des complexes cellulaires des hypercubes d’une structure d’opérade différentielle graduée. On définit un quasi-isomorphisme de $\operatorname {As}$ dans chacune de ces deux opérades.
On construit également des variantes non différentielles des deux opérades précédentes. On définit un morphisme de $\operatorname {As}$ dans chacune de ces opérades et on montre que les algèbres libres sont munies d’un coproduit coassociatif qui en fait des bigèbres.
References
- Frédéric Chapoton, Algèbres de Hopf des permutahèdres, associahèdres et hypercubes, Adv. Math. 150 (2000), no. 2, 264–275 (French, with English summary). MR 1749253, DOI 10.1006/aima.1999.1868
- Frédéric Chapoton, Bigèbres différentielles graduées associées aux permutoèdres, associaèdres et hypercubes, Annales de l’Institut Fourier 50 (2000), 1127–1153.
- Alain Connes and Dirk Kreimer, Hopf algebras, renormalization and noncommutative geometry, Comm. Math. Phys. 199 (1998), no. 1, 203–242. MR 1660199, DOI 10.1007/s002200050499
- Maxim Kontsevich and Yan Soibelman, Deformations of algebras over operads and Deligne’s conjecture, preprint QA/0001151, january 2000.
- Jean-Louis 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
- Jean-Louis Loday, Overview on Leibniz algebras, dialgebras and their homology, Cyclic cohomology and noncommutative geometry (Waterloo, ON, 1995) Fields Inst. Commun., vol. 17, Amer. Math. Soc., Providence, RI, 1997, pp. 91–102. MR 1478704, DOI 10.5802/aif.1604
- —, Dialgebras, Prépublication de l’IRMA (Strasbourg) Vol 14, 1999.
- Jean-Louis Loday and María O. Ronco, Hopf algebra of the planar binary trees, Adv. Math. 139 (1998), no. 2, 293–309. MR 1654173, DOI 10.1006/aima.1998.1759
- Clauda Malvenuto and Christophe Reutenauer, Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra 177 (1995), no. 3, 967–982. MR 1358493, DOI 10.1006/jabr.1995.1336
- María Ronco, Primitive elements in a free dendriform algebra. New trends in Hopf algebra theory (La Falda, 1999), Contemporary Math. 267, A.M.S., Providence, RI, 2000, pp. 245–253.
- James Dillon Stasheff, Homotopy associativity of $H$-spaces. I, II, Trans. Amer. Math. Soc. 108 (1963), 275-292; ibid. 108 (1963), 293–312. MR 0158400, DOI 10.1090/S0002-9947-1963-0158400-5
Additional Information
- Frédéric Chapoton
- Affiliation: Equipe Analyse Algébrique, Case 82, Institut de Mathématiques, 175 Rue du Chevaleret 75013 Paris, France
- Email: chapoton@math.jussieu.fr
- Received by editor(s): April 21, 2000
- Published electronically: April 24, 2001
- © Copyright 2001 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 354 (2002), 63-74
- MSC (2000): Primary 18D50, 52B11; Secondary 16W30
- DOI: https://doi.org/10.1090/S0002-9947-01-02809-4
- MathSciNet review: 1859025