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Higher Structures in Geometry and Physics pp 269–292Cite as

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The Diagonal of the Stasheff Polytope

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Part of the book series: Progress in Mathematics ((PM,volume 287))

Abstract

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.

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References

  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. Gaberdiel, M., Zwiebach, B.: Tensor constructions of open string theories. I. Foundations. Nucl. Phys. B 505(3), 569–624 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  3. Geyer, W.: On Tamari lattices. Discrete Math. 133(1–3), 99–122 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  4. Kadeishvili, T., Saneblidze, S.: The twisted Cartesian model for the double path fibration. ArXiv math.AT/0210224

    Google Scholar 

  5. Keller, B.: Introduction to A-infinity algebras and modules. Homology Homotopy Appl. 3(1), 1–35 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Loday, J.-L.: Arithmetree. J. Algebra 258(1), 275–309 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Loday, J.-L.: Realization of the Stasheff polytope. Arch. Math. (Basel) 83(3), 267–278 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Loday, J.-L.: Parking functions and triangulation of the associahedron. Proceedings of the Street’s fest. Contemp. Math. AMS 431, 327–340 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. MacLane, S.: Homology. Die Grundlehren der mathematischen Wissenschaften, Bd, vol. 114, p. 422. Academic, New York (1963)

    Google Scholar 

  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. 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. Prouté, A.: A ∞ -structures, modèle minimal de Baues-Lemaire et homologie des fibrations. Thèse d’Etat. Université Paris VII (1984)

    Google Scholar 

  13. Saneblidze, S., Umble, R.: A Diagonal on the Associahedra, preprint. ArXiv math.AT/0011065

    Google Scholar 

  14. Saneblidze, S., Umble, R.: Diagonals on the permutahedra, multiplihedra and associahedra. Homology Homotopy Appl. 6(1), 363–411 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  15. Stasheff, J.D.: Homotopy associativity of H-spaces. I, II. Trans. Am. Math. Soc. 108, 275–292; 293–312 (1963)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

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.

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Authors and Affiliations

  1. Institut de Recherche Mathématique Avancée, CNRS et Université de Strasbourg, 7 rue R. Descartes, 67084, Strasbourg Cedex, France

    Jean-Louis Loday

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  1. Jean-Louis Loday
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Correspondence to Jean-Louis Loday .

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Editors and Affiliations

  1. Universität Zürich, Institut für Mathematik, Winterthurerstr. 190, Zürich, CH-8057, Switzerland

    Alberto S. Cattaneo

  2. Loyola University, Chicago, Department of Mathematics, Chicago, 60626, Illinois, USA

    Anthony Giaquinto

  3. Pennsylvania State University, Department of Mathematics, Eberly College of Science, University Park, 16802, Pennsylvania, USA

    Ping Xu

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Loday, JL. (2011). The Diagonal of the Stasheff Polytope. In: Cattaneo, A., Giaquinto, A., Xu, P. (eds) Higher Structures in Geometry and Physics. Progress in Mathematics, vol 287. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-4735-3_13

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