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Nonsymmetric Operads

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Nonsymmetric Operads in Combinatorics

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Abstract

This chapter introduces nonsymmetric operads. Our presentation relies on the framework of graded collections and graded spaces introduced in the previous chapters. We consider here also set-operads, algebras over operads, free operads, presentations by generators and relations, Koszul duality and Koszulity of operads. At the end of the chapter, several examples of operads on a large family of combinatorial collections are provided.

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References

  1. M. Aguiar, M. Livernet, The associative operad and the weak order on the symmetric groups. J. Homotopy Relat. Str. 2(1), 57–84 (2007)

    MathSciNet  MATH  Google Scholar 

  2. J. Alm, D. Petersen, Brown’s dihedral moduli space and freedom of the gravity operad. Ann. Sci. Éc. Norm. Supér. 50(5), 1081–1122 (2017)

    Article  MathSciNet  Google Scholar 

  3. A. Björner, A.M. Garsia, R.P. Stanley, An Introduction to Cohen-Macaulay Partially Ordered Sets. Nato Science Series C: - Mathematical and Physical Sciences, vol. 83 (Reidel, Dordrecht 1982)

    Google Scholar 

  4. N. Bergeron, M. Livernet, The non-symmetric operad pre-Lie is free. J. Pure Appl. Algebra 214(7), 1165–1172 (2010)

    Article  MathSciNet  Google Scholar 

  5. J.M. Boardman, R.M. Vogt, Homotopy invariant algebraic structures on topological spaces. Lecture Notes in Mathematics, vol. 347 (Springer, Berlin, 1973)

    Book  Google Scholar 

  6. C. Chenavier, C. Cordero, S. Giraudo, Generalizations of the associative operad and convergent rewrite systems. Higher Dimens. Rewriting Algebra (2018). https://doi.org/10.29007/mfnh

  7. P. Caron, J.-C. Champarnaud, L. Mignot, Multi-bar and multi-tilde regular operators. J. Autom. Lang. Comb. 16(1), 11–26 (2011)

    MathSciNet  MATH  Google Scholar 

  8. F. Chapoton, S. Giraudo, Enveloping operads and bicoloured noncrossing configurations. Exp. Math. 23(3), 332–349 (2014)

    Article  MathSciNet  Google Scholar 

  9. F. Chapoton, On some anticyclic operads. Algebr. Geom. Topol. 5, 53–69 (2005)

    Article  MathSciNet  Google Scholar 

  10. F. Chapoton, Sur le nombre d’intervalles dans les treillis de Tamari. Sém. Lothar. Combin. B55f, 18 (2006)

    Google Scholar 

  11. F. Chapoton, The anticyclic operad of moulds. Int. Math. Res. Not. 20, 1–36 (2007); Art. ID rnm078

    Google Scholar 

  12. F. Chapoton, Operads and algebraic combinatorics of trees. Sém. Lothar. Combin. B58c, 27 (2008)

    Google Scholar 

  13. F. Chapoton, Flows on rooted trees and the Menous-Novelli-Thibon idempotents. Math. Scand. 115(1), 20–61 (2014)

    Article  MathSciNet  Google Scholar 

  14. F. Chapoton, M. Livernet, Pre-Lie algebras and the rooted trees operad. Int. Math. Res. Not. 8, 395–408 (2001)

    Article  MathSciNet  Google Scholar 

  15. F. Chapoton, M. Livernet, Relating two Hopf algebras built from an operad. Int. Math. Res. Not. 2007(24), 1–27 (2007); Art. ID rnm131

    Google Scholar 

  16. V. Dotsenko, A. Khoroshkin, Gröbner bases for operads. Duke Math. J. 153(2), 363–396 (2010)

    Article  MathSciNet  Google Scholar 

  17. M. Gerstenhaber, The cohomology structure of an associative ring. Ann. Math. 78, 267–288 (1963)

    Article  MathSciNet  Google Scholar 

  18. E. Getzler, Two-dimensional topological gravity and equivariant cohomology. Commun. Math. Phys. 163(3), 473–489 (1994)

    Article  MathSciNet  Google Scholar 

  19. S. Giraudo, Combinatorial operads from monoids. J. Algebra Comb. 41(2), 493–538 (2015)

    Article  MathSciNet  Google Scholar 

  20. S. Giraudo, Operads from posets and Koszul duality. Eur. J. Comb. 56C, 1–32 (2016)

    Article  MathSciNet  Google Scholar 

  21. S. Giraudo, Pluriassociative algebras I: the pluriassociative operad. Adv. Appl. Math. 77, 1–42 (2016)

    Article  MathSciNet  Google Scholar 

  22. S. Giraudo, Combalgebraic structures on decorated cliques, in Formal Power Series and Algebraic Combinatorics (2017)

    Google Scholar 

  23. V. Ginzburg, M.M. Kapranov, Koszul duality for operads. Duke Math. J. 76(1), 203–272 (1994)

    Article  MathSciNet  Google Scholar 

  24. S. Giraudo, J.-G. Luque, L. Mignot, F. Nicart, Operads, quasiorders, and regular languages. Adv. Appl. Math. 75, 56–93 (2016)

    Article  MathSciNet  Google Scholar 

  25. E. Hoffbeck, A Poincaré-Birkhoff-Witt criterion for Koszul operads. Manuscripta Math. 131(1-2), 87–110 (2010)

    Article  MathSciNet  Google Scholar 

  26. P. Leroux, L-algebras, triplicial-algebras, within an equivalence of categories motivated by graphs. Comm. Algebra 39(8), 2661–2689 (2011)

    Article  MathSciNet  Google Scholar 

  27. M. Livernet, A rigidity theorem for pre-Lie algebras. J. Pure Appl. Algebra 207(1), 1–18 (2006)

    Article  MathSciNet  Google Scholar 

  28. J.-G. Luque, L. Mignot, F. Nicart, Some combinatorial operators in language theory. J. Autom. Lang. Comb. 18(1), 27–52 (2013)

    MathSciNet  MATH  Google Scholar 

  29. J.-L. Loday, La renaissance des opérades. Séminaire Bourbaki 37(792), 47–74 (1996)

    MATH  Google Scholar 

  30. J.-L. Loday, Dialgebras, in Dialgebras and Related Operads. Lecture Notes in Mathematics, vol. 1763 (Springer, Berlin, 2001), pp. 7–66

    Google Scholar 

  31. J.-L. Loday, Inversion of integral series enumerating planar trees. Sém. Lothar. Combin. 53, 16 (2005); Art. B53d

    Google Scholar 

  32. J.-L. Loday, Generalized bialgebras and triples of operads. Astérisque 320, 1–114 (2008)

    MathSciNet  MATH  Google Scholar 

  33. J.-L. Loday, On the operad of associative algebras with derivation. Georgian Math. J. 17(2), 347–372 (2010)

    MathSciNet  MATH  Google Scholar 

  34. J.-L. Loday, M. Ronco, Algèbres de Hopf colibres. C. R. Math. 337(3), 153–158 (2003)

    Article  MathSciNet  Google Scholar 

  35. J.-L. Loday, M. Ronco, Trialgebras and families of polytopes, in Homotopy Theory: Relations with Algebraic Geometry, Group Cohomology, and Algebraic K-Theory. Contemporary Mathematics, vol. 346 (American Mathematical Society, Providence, 2004), pp. 369–398

    Google Scholar 

  36. J.-L. Loday, M. Ronco, On the structure of cofree Hopf algebras. J. Reine Angew. Math. 592, 123–155 (2006)

    MathSciNet  MATH  Google Scholar 

  37. J.-L. Loday, B. Vallette, Algebraic Operads. Grundlehren der mathematischen Wissenschaften, vol. 346 (Springer, Heidelberg, 2012), pp. xxiv+634

    Book  Google Scholar 

  38. M. Markl, Operads and PROPs, in Handbook of Algebra, vol. 5 (Elsevier/North-Holland, Amsterdam, 2008), pp. 87–140

    MATH  Google Scholar 

  39. J.P. May, The Geometry of Iterated Loop Spaces. Lectures Notes in Mathematics, vol. 271 (Springer, Berlin, 1972), pp. viii+175

    Google Scholar 

  40. M.A. Méndez, Set Operads in Combinatorics and Computer Science. SpringerBriefs in Mathematics (Springer, Cham, 2015), pp. xvi+129

    Google Scholar 

  41. M. Méndez, J. Yang, Möbius species. Adv. Math. 85(1), 83–128 (1991)

    Article  MathSciNet  Google Scholar 

  42. S.B. Priddy, Koszul resolutions. Trans. Am. Math. Soc. 152, 39–60 (1970)

    Article  MathSciNet  Google Scholar 

  43. A. Saïdi, The pre-Lie operad as a deformation of NAP. J. Algebra Appl. 13(1), 23 (2014)

    Article  MathSciNet  Google Scholar 

  44. N.J.A. Sloane, The on-line encyclopedia of integer sequences (1996). https://oeis.org/

  45. B. Vallette, Homology of generalized partition posets. J. Pure Appl. Algebra 208(2), 699–725 (2007)

    Article  MathSciNet  Google Scholar 

  46. D. Yau, Colored Operads. Graduate Studies in Mathematics (American Mathematical Society, Providence, 2016)

    Google Scholar 

  47. G.W. Zinbiel, Encyclopedia of types of algebras 2010, in Operads and Universal Algebra. Nankai Series in Pure, Applied Mathematics and Theoretical Physics, vol. 9 (World Scientific Publishing, Hackensack, 2012), pp. 217–297

    Google Scholar 

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Authors and Affiliations

  1. University of Paris-Est, Marne-la-Vallee, France

    Samuele Giraudo

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  1. Samuele Giraudo
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Giraudo, S. (2018). Nonsymmetric Operads. In: Nonsymmetric Operads in Combinatorics. Springer, Cham. https://doi.org/10.1007/978-3-030-02074-3_5

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  • DOI: https://doi.org/10.1007/978-3-030-02074-3_5

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  • Online ISBN: 978-3-030-02074-3

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