Advertisement

Methods to Prove Koszulity of an Operad

  • Jean-Louis Loday
  • Bruno Vallette
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 346)

Abstract

This chapter extends to the operadic level the various methods, obtained in Chap.  4, to prove that algebras are Koszul. They rely either on rewriting systems, PBW and Gröbner bases, distributive laws (Diamond Lemma), or combinatorics (partition poset method). The notion of shuffle operad plays a key role in this respect. We also introduce the Manin products constructions for operads.

Keywords

Maximal Chain White Product Black Product Symmetric Monoidal Category Koszul Duality 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. AM99.
    G. Arone and M. Mahowald, The Goodwillie tower of the identity functor and the unstable periodic homotopy of spheres, Invent. Math. 135 (1999), no. 3, 743–788. MathSciNetzbMATHCrossRefGoogle Scholar
  2. Bac76.
    K. Baclawski, Homology and combinatorics of ordered sets, Ph.D. thesis, Harvard Univ., Cambridge, Mass., 1976. Google Scholar
  3. BC04.
    J. C. Baez and A. S. Crans, Higher-dimensional algebra. VI. Lie 2-algebras, Theory Appl. Categ. 12 (2004), 492–538 (electronic). MathSciNetzbMATHGoogle Scholar
  4. Bec69.
    Jon Beck, Distributive laws, Sem. on Triples and Categorical Homology Theory (ETH, Zürich, 1966/67), Springer, Berlin, 1969, pp. 119–140. CrossRefGoogle Scholar
  5. BGS82.
    A. Björner, A. M. Garsia, and R. P. Stanley, An introduction to Cohen-Macaulay partially ordered sets, Ordered sets (Banff, Alta., 1981), NATO Adv. Study Inst. Ser. C: Math. Phys. Sci., vol. 83, Reidel, Dordrecht, 1982, pp. 583–615. CrossRefGoogle Scholar
  6. BM08.
    D. V. Borisov and Y. I. Manin, Generalized operads and their inner cohomomorphisms, Geometry and dynamics of groups and spaces, Progr. Math., vol. 265, Birkhäuser, Basel, 2008, pp. 247–308. CrossRefGoogle Scholar
  7. Chi05.
    Michael Ching, Bar constructions for topological operads and the Goodwillie derivatives of the identity, Geom. Topol. 9 (2005), 833–933. MathSciNetzbMATHCrossRefGoogle Scholar
  8. CK98.
    A. Connes and D. Kreimer, Hopf algebras, renormalization and noncommutative geometry, Comm. Math. Phys. 199 (1998), no. 1, 203–242. MathSciNetzbMATHCrossRefGoogle Scholar
  9. CL07.
    —, Relating two Hopf algebras built from an operad, Int. Math. Res. Not. IMRN (2007), no. 24. Google Scholar
  10. Cur12.
    P.-L. Curien, Operads, clones and distributive laws, Proc. Int. Conf., in Nankai Series in Pure, Applied Mathematics and Theoretical Physics 9 (2012), 25–49. Google Scholar
  11. DK07.
    V. Dotsenko and A. Khoroshkin, Character formulas for the operad of a pair of compatible brackets and for the bi-Hamiltonian operad, Funktsional. Anal. i Prilozhen. 41 (2007), no. 1, 1–22, 96. MathSciNetzbMATHCrossRefGoogle Scholar
  12. DK10.
    —, Gröbner bases for operads, Duke Math. J. 153 (2010), no. 2, 363–396. MathSciNetzbMATHCrossRefGoogle Scholar
  13. Dot07.
    Vladimir Dotsenko, An operadic approach to deformation quantization of compatible Poisson brackets. I, J. Gen. Lie Theory Appl. 1 (2007), no. 2, 107–115 (electronic). MathSciNetzbMATHCrossRefGoogle Scholar
  14. EFG05.
    K. Ebrahimi-Fard and L. Guo, On products and duality of binary, quadratic, regular operads, J. Pure Appl. Algebra 200 (2005), no. 3, 293–317. MathSciNetzbMATHCrossRefGoogle Scholar
  15. Far79.
    F. D. Farmer, Cellular homology for posets, Math. Japon. 23 (1978/79), no. 6, 607–613. MathSciNetGoogle Scholar
  16. FM97.
    T. F. Fox and M. Markl, Distributive laws, bialgebras, and cohomology, Operads: Proceedings of Renaissance Conferences (Hartford, CT/Luminy, 1995), Contemp. Math., vol. 202, Amer. Math. Soc., Providence, RI, 1997, pp. 167–205. CrossRefGoogle Scholar
  17. Fre04.
    —, Koszul duality of operads and homology of partition posets, in “Homotopy theory and its applications (Evanston, 2002)”, Contemp. Math. 346 (2004), 115–215. MathSciNetCrossRefGoogle Scholar
  18. GK94.
    V. Ginzburg and M. Kapranov, Koszul duality for operads, Duke Math. J. 76 (1994), no. 1, 203–272. MathSciNetzbMATHCrossRefGoogle Scholar
  19. GK95b.
    V. Ginzburg and M. Kapranov, Erratum to: “Koszul duality for operads”, Duke Math. J. 80 (1995), no. 1, 293. MathSciNetzbMATHCrossRefGoogle Scholar
  20. Hof10c.
    —, A Poincaré-Birkhoff-Witt criterion for Koszul operads, Manuscripta Math. 131 (2010), no. 1-2, 87–110. MathSciNetzbMATHCrossRefGoogle Scholar
  21. Liv06.
    —, A rigidity theorem for pre-Lie algebras, J. Pure Appl. Algebra 207 (2006), no. 1, 1–18. MathSciNetCrossRefGoogle Scholar
  22. LR04.
    J.-L. Loday and M. O. Ronco, Trialgebras and families of polytopes, Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic K-theory, Contemp. Math., vol. 346, Amer. Math. Soc., Providence, RI, 2004, pp. 369–398. MR 2066507 (2006e:18016) CrossRefGoogle Scholar
  23. Mar96a.
    —, Distributive laws and Koszulness, Ann. Inst. Fourier (Grenoble) 46 (1996), no. 2, 307–323. MathSciNetzbMATHCrossRefGoogle Scholar
  24. Mén89.
    M. Méndez, Monoides, c-monoides, especies de Möbius y coàlgebras, Ph.D. thesis, Universidad Central de Venezuela, http://www.ivic.gob.ve/matematicas/?mod=mmendez.htm, 1989.
  25. Mer04.
    —, Operads, deformation theory and F-manifolds, Frobenius manifolds, Aspects Math., E36, Vieweg, Wiesbaden, 2004, pp. 213–251. Google Scholar
  26. Mer05.
    —, Nijenhuis infinity and contractible differential graded manifolds, Compos. Math. 141 (2005), no. 5, 1238–1254. MathSciNetCrossRefGoogle Scholar
  27. ML95.
    —, Homology, Classics in Mathematics, Springer-Verlag, Berlin, 1995, Reprint of the 1975 edition. zbMATHGoogle Scholar
  28. MY91.
    M. Méndez and J. Yang, Möbius species, Adv. Math. 85 (1991), no. 1, 83–128. MathSciNetzbMATHCrossRefGoogle Scholar
  29. Ron11.
    —, Shuffle algebras, Annales Instit. Fourier 61 (2011), no. 1, 799–850. MathSciNetzbMATHCrossRefGoogle Scholar
  30. Sta97a.
    R. P. Stanley, Enumerative combinatorics. Vol. 1, Cambridge Studies in Advanced Mathematics, vol. 49, Cambridge University Press, Cambridge, 1997, With a foreword by Gian-Carlo Rota, Corrected reprint of the 1986 original. zbMATHCrossRefGoogle Scholar
  31. Str08.
    Henrik Strohmayer, Operads of compatible structures and weighted partitions, J. Pure Appl. Algebra 212 (2008), no. 11, 2522–2534. MathSciNetzbMATHCrossRefGoogle Scholar
  32. Str09.
    —, Operad profiles of Nijenhuis structures, Differential Geom. Appl. 27 (2009), no. 6, 780–792. MathSciNetzbMATHCrossRefGoogle Scholar
  33. Str10.
    —, Prop profile of bi-Hamiltonian structures, J. Noncommut. Geom. 4 (2010), no. 2, 189–235. MathSciNetzbMATHCrossRefGoogle Scholar
  34. Val07a.
    Bruno Vallette, Homology of generalized partition posets, J. Pure Appl. Algebra 208 (2007), no. 2, 699–725. MathSciNetzbMATHCrossRefGoogle Scholar
  35. Val07b.
    —, A Koszul duality for props, Trans. of Amer. Math. Soc. 359 (2007), 4865–4993. MathSciNetzbMATHCrossRefGoogle Scholar
  36. Val08.
    —, Manin products, Koszul duality, Loday algebras and Deligne conjecture, J. Reine Angew. Math. 620 (2008), 105–164. MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Jean-Louis Loday
    • 1
  • Bruno Vallette
    • 2
  1. 1.IRMA, CNRS et Université de StrasbourgStrasbourgFrance
  2. 2.Lab. de Mathématiques J.A. DieudonnéUniversité de Nice-Sophia AntipolisNiceFrance

Personalised recommendations