(Co)Homology of Algebras over an Operad

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


In this chapter, we introduce the André–Quillen cohomology and homology for algebras over an operad, which provides us with homological invariants. It plays a role in many classification problems, like for instance deformation theory. We use the resolutions provided by the Koszul duality theory to make explicit small chain complexes which computes it.


Algebra Structure Vertex Operator Algebra Abelian Extension Cochain Complex Infinitesimal Deformation 
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.


  1. And74.
    M. André, Homologie des algèbres commutatives, Springer-Verlag, Berlin, 1974, Die Grundlehren der mathematischen Wissenschaften, Band 206. MATHCrossRefGoogle Scholar
  2. Bal97.
    David Balavoine, Deformations of algebras over a quadratic operad, Operads: Proceedings of Renaissance Conferences (Hartford, CT/Luminy, 1995), Contemp. Math., vol. 202, Amer. Math. Soc., Providence, RI, 1997, pp. 207–234. CrossRefGoogle Scholar
  3. Bal98.
    —, Homology and cohomology with coefficients, of an algebra over a quadratic operad, J. Pure Appl. Algebra 132 (1998), no. 3, 221–258. MathSciNetMATHCrossRefGoogle Scholar
  4. Bar68.
    M. Barr, Harrison homology, Hochschild homology and triples, J. Algebra 8 (1968), 314–323. MathSciNetMATHCrossRefGoogle Scholar
  5. BB69.
    M. Barr and J. Beck, Homology and standard constructions, Sem. on Triples and Categorical Homology Theory (ETH, Zürich, 1966/67), Springer, Berlin, 1969, pp. 245–335. CrossRefGoogle Scholar
  6. BFLS98.
    G. Barnich, R. Fulp, T. Lada, and J. Stasheff, The sh Lie structure of Poisson brackets in field theory, Comm. Math. Phys. 191 (1998), no. 3, 585–601. MathSciNetMATHCrossRefGoogle Scholar
  7. BJT97.
    H.-J. Baues, M. Jibladze, and A. Tonks, Cohomology of monoids in monoidal categories, Operads: Proceedings of Renaissance Conferences (Hartford, CT/Luminy, 1995) (Providence, RI), Contemp. Math., vol. 202, Amer. Math. Soc., 1997, pp. 137–165. CrossRefGoogle Scholar
  8. BMR04.
    H.-J. Baues, E. G. Minian, and B. Richter, Crossed modules over operads and operadic cohomology, K-Theory 31 (2004), no. 1, 39–69. MathSciNetMATHCrossRefGoogle Scholar
  9. CFK01.
    I. Ciocan-Fontanine and M. Kapranov, Derived Quot schemes, Ann. Sci. École Norm. Sup. (4) 34 (2001), no. 3, 403–440. MathSciNetMATHGoogle Scholar
  10. Con85.
    Alain Connes, Noncommutative differential geometry, Inst. Hautes Études Sci. Publ. Math. (1985), no. 62, 257–360. Google Scholar
  11. Dzh99.
    A. Dzhumadil’daev, Cohomologies and deformations of right-symmetric algebras, J. Math. Sci. (New York) 93 (1999), no. 6, 836–876, Algebra, 11. MathSciNetMATHCrossRefGoogle Scholar
  12. Fra01.
    Alessandra Frabetti, Dialgebra (co)homology with coefficients, Dialgebras and related operads, Lecture Notes in Math., vol. 1763, Springer, Berlin, 2001, pp. 67–103. CrossRefGoogle Scholar
  13. 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
  14. Fre06.
    —, Théorie des opérades de Koszul et homologie des algèbres de Poisson, Ann. Math. Blaise Pascal 13 (2006), no. 2, 237–312. MathSciNetMATHCrossRefGoogle Scholar
  15. Fre09a.
    —, Modules over operads and functors, Lecture Notes in Mathematics, vol. 1967, Springer-Verlag, Berlin, 2009. MATHCrossRefGoogle Scholar
  16. GCTV09.
    I. Galvez-Carrillo, A. Tonks, and B. Vallette, Homotopy Batalin-Vilkovisky algebras, Journal Noncommutative Geometry (2009), arXiv:0907.2246, 49 pp.
  17. Ger63.
    M. Gerstenhaber, The cohomology structure of an associative ring, Ann. of Math. (2) 78 (1963), 267–288. MathSciNetMATHCrossRefGoogle Scholar
  18. Ger64.
    —, On the deformation of rings and algebras, Ann. of Math. (2) 79 (1964), 59–103. MathSciNetMATHCrossRefGoogle Scholar
  19. GH00.
    P. G. Goerss and M. J. Hopkins, André-Quillen (co)-homology for simplicial algebras over simplicial operads, Une dégustation topologique [Topological morsels]: homotopy theory in the Swiss Alps (Arolla, 1999), Contemp. Math., vol. 265, Amer. Math. Soc., Providence, RI, 2000, pp. 41–85. CrossRefGoogle Scholar
  20. GJ94.
    E. Getzler and J. D. S. Jones, Operads, homotopy algebra and iterated integrals for double loop spaces, hep-th/9403055 (1994).
  21. GK94.
    V. Ginzburg and M. Kapranov, Koszul duality for operads, Duke Math. J. 76 (1994), no. 1, 203–272. MathSciNetMATHCrossRefGoogle Scholar
  22. GV95.
    M. Gerstenhaber and A. A. Voronov, Homotopy G-algebras and moduli space operad, Internat. Math. Res. Notices (1995), no. 3, 141–153 (electronic). Google Scholar
  23. Har62.
    D. K. Harrison, Commutative algebras and cohomology, Trans. Amer. Math. Soc. 104 (1962), 191–204. MathSciNetMATHCrossRefGoogle Scholar
  24. Hin97.
    Vladimir Hinich, Homological algebra of homotopy algebras, Comm. Algebra 25 (1997), no. 10, 3291–3323, see also Erratum arXiv:math/0309453. MathSciNetMATHCrossRefGoogle Scholar
  25. HM10.
    J. Hirsch and J. Millès, Curved Koszul duality theory, arXiv:1008.5368 (2010).
  26. Hof10a.
    Eric Hoffbeck, Γ-homology of algebras over an operad, Algebr. Geom. Topol. 10 (2010), no. 3, 1781–1806. MR 2683753 (2011f:16022) MathSciNetMATHCrossRefGoogle Scholar
  27. Hof10b.
    —, Obstruction theory for algebras over an operad, ArXiv e-prints (2010). Google Scholar
  28. Kar87.
    Max Karoubi, Homologie cyclique et K-théorie, Astérisque (1987), no. 149, 147. Google Scholar
  29. Kel05.
    —, Deformation quantization after Kontsevich and Tamarkin, Déformation, quantification, théorie de Lie, Panor. Synthèses, vol. 20, Soc. Math. France, Paris, 2005, pp. 19–62. Google Scholar
  30. KS10.
    —, Deformation theory. I [Draft], http://www.math.ksu.edu/~soibel/Book-vol1.ps, 2010.
  31. Liv98a.
    Muriel Livernet, Homotopie rationnelle des algèbres sur une opérade, Prépublication de l’Institut de Recherche Mathématique Avancée [Prepublication of the Institute of Advanced Mathematical Research], 1998/32, Université Louis Pasteur Département de Mathématique Institut de Recherche Mathématique Avancée, Strasbourg, 1998, Thèse, Université Louis Pasteur (Strasbourg I), Strasbourg, 1998. Google Scholar
  32. Liv99.
    —, On a plus-construction for algebras over an operad, K-Theory 18 (1999), no. 4, 317–337. MathSciNetCrossRefGoogle Scholar
  33. Lod98.
    —, Cyclic homology, second ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 301, Springer-Verlag, Berlin, 1998, Appendix E by María O. Ronco, Chap. 13 by the author in collaboration with Teimuraz Pirashvili. Google Scholar
  34. LP93.
    J.-L. Loday and T. Pirashvili, Universal enveloping algebras of Leibniz algebras and (co)homology, Math. Ann. 296 (1993), no. 1, 139–158. MathSciNetMATHCrossRefGoogle Scholar
  35. LTV10.
    P. Lambrechts, V. Turchin, and I. Volić, The rational homology of the space of long knots in codimension >2, Geom. Topol. 14 (2010), no. 4, 2151–2187. MathSciNetMATHCrossRefGoogle Scholar
  36. Mar92.
    Martin Markl, A cohomology theory for A(m)-algebras and applications, J. Pure Appl. Algebra 83 (1992), no. 2, 141–175. MathSciNetMATHCrossRefGoogle Scholar
  37. Mil10.
    Joan Millès, The Koszul complex is the cotangent complex, arXiv:1004.0096 (2010).
  38. Mil11.
    —, André-Quillen cohomology of algebras over an operad, Adv. Math. 226 (2011), no. 6, 5120–5164. MR 2775896 MathSciNetCrossRefGoogle Scholar
  39. ML98.
    —, Categories for the working mathematician, second ed., Graduate Texts in Mathematics, vol. 5, Springer-Verlag, New York, 1998. MATHGoogle Scholar
  40. MS02.
    J. E. McClure and J. H. Smith, A solution of Deligne’s Hochschild cohomology conjecture, Recent progress in homotopy theory (Baltimore, MD, 2000), Contemp. Math., vol. 293, Amer. Math. Soc., Providence, RI, 2002, pp. 153–193. CrossRefGoogle Scholar
  41. MV09a.
    S. A. Merkulov and B. Vallette, Deformation theory of representations of prop(erad)s. I, J. Reine Angew. Math. 634 (2009), 51–106. MathSciNetMATHGoogle Scholar
  42. MV09b.
    —, Deformation theory of representations of prop(erad)s. II, J. Reine Angew. Math. 636 (2009), 123–174. MathSciNetMATHGoogle Scholar
  43. NR66.
    A. Nijenhuis and R. W. Richardson, Jr., Cohomology and deformations in graded Lie algebras, Bull. Amer. Math. Soc. 72 (1966), 1–29. MathSciNetMATHCrossRefGoogle Scholar
  44. NR67.
    —, Deformations of Lie algebra structures, J. Math. Mech. 17 (1967), 89–105. MathSciNetMATHGoogle Scholar
  45. Qui70.
    —, On the (co-) homology of commutative rings, Applications of Categorical Algebra (Proc. Sympos. Pure Math., Vol. XVII, New York, 1968), Amer. Math. Soc., Providence, RI, 1970, pp. 65–87. Google Scholar
  46. Rez96.
    C. W. Rezk, Spaces of algebra structures and cohomology of operads, Ph.D. thesis, MIT, 1996. Google Scholar
  47. Rob03.
    Alan Robinson, Gamma homology, Lie representations and E multiplications, Invent. Math. 152 (2003), no. 2, 331–348. MathSciNetMATHCrossRefGoogle Scholar
  48. RW02.
    A. Robinson and S. Whitehouse, Operads and Γ-homology of commutative rings, Math. Proc. Cambridge Philos. Soc. 132 (2002), no. 2, 197–234. MathSciNetMATHCrossRefGoogle Scholar
  49. Sin06.
    D. P. Sinha, Operads and knot spaces, J. Amer. Math. Soc. 19 (2006), no. 2, 461–486 (electronic). MathSciNetMATHCrossRefGoogle Scholar
  50. Sin09.
    —, The topology of spaces of knots: cosimplicial models, Amer. J. Math. 131 (2009), no. 4, 945–980. MathSciNetMATHCrossRefGoogle Scholar
  51. SS85.
    M. Schlessinger and J. Stasheff, The Lie algebra structure of tangent cohomology and deformation theory, J. Pure Appl. Algebra 38 (1985), no. 2-3, 313–322. MathSciNetMATHCrossRefGoogle Scholar
  52. Sta93.
    —, The intrinsic bracket on the deformation complex of an associative algebra, J. Pure Appl. Algebra 89 (1993), no. 1-2, 231–235. MathSciNetMATHCrossRefGoogle Scholar
  53. Tou04.
    V. Tourtchine, On the homology of the spaces of long knots, Advances in topological quantum field theory, NATO Sci. Ser. II Math. Phys. Chem., vol. 179, Kluwer Acad. Publ., Dordrecht, 2004, pp. 23–52. CrossRefGoogle Scholar
  54. vdL02.
    P. van der Laan, Operads up to Homotopy and Deformations of Operad Maps, arXiv:math.QA/0208041 (2002).
  55. Wil07.
    S. O. Wilson, Free Frobenius algebra on the differential forms of a manifold, arXiv:0710.3550 (2007).

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