The Diagonal of the Stasheff Polytope

  • Jean-Louis LodayEmail author
Part of the Progress in Mathematics book series (PM, volume 287)


We construct an A-infinity structure on the tensor product of two A-infinity algebras by using the simplicial decomposition of the Stasheff polytope. The key point is the construction of an operad AA-infinity based on the simplicial Stasheff polytope. The operad AA-infinity admits a coassociative diagonal and the operad A-infinity is a retract by deformation of it. We compare these constructions with analogous constructions due to Saneblidze–Umble and Markl–Shnider based on the Boardman–Vogt cubical decomposition of the Stasheff polytope.

Key words

Stasheff polytope Associahedron Operad Bar–cobar construction Cobarconstruction A-infinity algebra AA-infinity algebra Diagonal 



I thank Bruno Vallette for illuminating discussions on the algebras up to homotopy and Samson Saneblidze for sharing his drawings with me some years ago. Thanks to Emily Burgunder, Martin Markl, Samson Saneblidze, Jim Stasheff and Ron Umble for their comments on the previous versions of this paper. I warmly thank the referee for his careful reading and his precious comments which helped me to improve this text.This work is partially supported by the French agency ANR.


  1. 1.
    Boardman, J.M., Vogt, R.M.: Homotopy invariant algebraic structures on topological spaces. Lecture Notes in Mathematics, vol. 347, p. 257. Springer, Berlin (1973)Google Scholar
  2. 2.
    Gaberdiel, M., Zwiebach, B.: Tensor constructions of open string theories. I. Foundations. Nucl. Phys. B 505(3), 569–624 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Geyer, W.: On Tamari lattices. Discrete Math. 133(1–3), 99–122 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Kadeishvili, T., Saneblidze, S.: The twisted Cartesian model for the double path fibration. ArXiv math.AT/0210224Google Scholar
  5. 5.
    Keller, B.: Introduction to A-infinity algebras and modules. Homology Homotopy Appl. 3(1), 1–35 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Loday, J.-L.: Arithmetree. J. Algebra 258(1), 275–309 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Loday, J.-L.: Realization of the Stasheff polytope. Arch. Math. (Basel) 83(3), 267–278 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Loday, J.-L.: Parking functions and triangulation of the associahedron. Proceedings of the Street’s fest. Contemp. Math. AMS 431, 327–340 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    MacLane, S.: Homology. Die Grundlehren der mathematischen Wissenschaften, Bd, vol. 114, p. 422. Academic, New York (1963)Google Scholar
  10. 10.
    Markl, M., Shnider, S., Stasheff, J.: Operads in algebra, topology and physics. Mathematical Surveys and Monographs vol. 96, p. 349. American Mathematical Society, Providence, RI (2002)Google Scholar
  11. 11.
    Markl, M., Shnider, S.: Associahedra, cellular W-construction and products of A -algebras. Trans. Am. Math. Soc. 358(6), 2353–2372 (2006) (electronic)Google Scholar
  12. 12.
    Prouté, A.: A -structures, modèle minimal de Baues-Lemaire et homologie des fibrations. Thèse d’Etat. Université Paris VII (1984)Google Scholar
  13. 13.
    Saneblidze, S., Umble, R.: A Diagonal on the Associahedra, preprint. ArXiv math.AT/0011065Google Scholar
  14. 14.
    Saneblidze, S., Umble, R.: Diagonals on the permutahedra, multiplihedra and associahedra. Homology Homotopy Appl. 6(1), 363–411 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Stasheff, J.D.: Homotopy associativity of H-spaces. I, II. Trans. Am. Math. Soc. 108, 275–292; 293–312 (1963)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Institut de Recherche Mathématique AvancéeCNRS et Université de StrasbourgStrasbourg CedexFrance

Personalised recommendations