Examples of Algebraic Operads

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

Abstract

In Chap.  9, we studied in detail the operad Ass encoding the associative algebras. It is a paradigm for nonsymmetric operads, symmetric operads, cyclic operads. In this chapter we present several other examples of operads. First, we present the two other “graces”, the operads Com and Lie encoding respectively the commutative (meaning commutative and associative) algebras, and the Lie algebras. Second, we introduce more examples of binary quadratic operads: Poisson, Gerstenhaber, pre-Lie, Leibniz, Zinbiel, dendriform, magmatic, several variations like Jordan algebra, divided power algebra, Batalin–Vilkovisky algebra. Then we present various examples of operads involving higher ary-operations: homotopy algebras, infinite-magmatic, brace, multibrace, Jordan triples, Lie triples. The choice is dictated by their relevance in various parts of mathematics: differential geometry, noncommutative geometry, harmonic analysis, algebraic combinatorics, theoretical physics, computer science. Of course, this list does not exhaust the examples appearing in the existing literature. The reader may have a look at the cornucopia of types of algebras 2012 to find more examples.

Keywords

Hopf Algebra Associative Algebra Jordan Algebra Hochschild Cohomology Planar Algebra 
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.
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References

  1. AL04.
    M. Aguiar and J.-L. Loday, Quadri-algebras, J. Pure Appl. Algebra 191 (2004), no. 3, 205–221. MathSciNetMATHGoogle Scholar
  2. AP10.
    M. Ammar and N. Poncin, Coalgebraic approach to the Loday infinity category, stem differential for 2n-ary graded and homotopy algebras, Ann. Inst. Fourier (Grenoble) 60 (2010), no. 1, 355–387. MathSciNetMATHGoogle Scholar
  3. BB09.
    M. A. Batanin and C. Berger, The lattice path operad and Hochschild cochains, Alpine perspectives on algebraic topology, Contemp. Math., vol. 504, Amer. Math. Soc., Providence, RI, 2009, pp. 23–52. Google Scholar
  4. BD04.
    A. Beilinson and V. Drinfeld, Chiral algebras, American Mathematical Society Colloquium Publications, vol. 51, American Mathematical Society, Providence, RI, 2004. MATHGoogle Scholar
  5. BE74.
    M. G. Barratt and Peter J. Eccles, Γ+ -structures. I. A free group functor for stable homotopy theory, Topology 13 (1974), 25–45. MathSciNetMATHGoogle Scholar
  6. Ber96.
    Clemens Berger, Opérades cellulaires et espaces de lacets itérés, Ann. Inst. Fourier (Grenoble) 46 (1996), no. 4, 1125–1157. MathSciNetMATHGoogle Scholar
  7. BF04.
    C. Berger and B. Fresse, Combinatorial operad actions on cochains, Math. Proc. Cambridge Philos. Soc. 137 (2004), no. 1, 135–174. MathSciNetMATHGoogle Scholar
  8. BGG78.
    I. N. Bernšteĭn, I. M. Gel’fand, and S. I. Gel’fand, Algebraic vector bundles on P n and problems of linear algebra, Funktsional. Anal. i Prilozhen. 12 (1978), no. 3, 66–67. MathSciNetMATHGoogle Scholar
  9. BL11.
    N. Bergeron and J.-L. Loday, The symmetric operation in a free pre-Lie algebra is magmatic, Proc. Amer. Math. Soc. 139 (2011), no. 5, 1585–1597. MathSciNetMATHGoogle Scholar
  10. 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. Google Scholar
  11. BR10.
    E. Burgunder and M. Ronco, Tridendriform structure on combinatorial Hopf algebras, J. Algebra 324 (2010), no. 10, 2860–2883. MR 2725205 (2011m:16079) MathSciNetMATHGoogle Scholar
  12. BV81.
    I. A. Batalin and G. A. Vilkovisky, Gauge algebra and quantization, Phys. Lett. B 102 (1981), no. 1, 27–31. MathSciNetGoogle Scholar
  13. CE48.
    C. Chevalley and S. Eilenberg, Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc. 63 (1948), 85–124. MathSciNetMATHGoogle Scholar
  14. CG08.
    X. Z. Cheng and E. Getzler, Transferring homotopy commutative algebraic structures, J. Pure Appl. Algebra 212 (2008), no. 11, 2535–2542. MathSciNetMATHGoogle Scholar
  15. Cha01b.
    —, Un endofoncteur de la catégorie des opérades, Dialgebras and related operads, Lecture Notes in Math., vol. 1763, Springer, Berlin, 2001, pp. 105–110. Google Scholar
  16. Cha02.
    —, Opérades différentielles graduées sur les simplexes et les permutoèdres, Bull. Soc. Math. France 130 (2002), no. 2, 233–251. MathSciNetMATHGoogle Scholar
  17. CK98.
    A. Connes and D. Kreimer, Hopf algebras, renormalization and noncommutative geometry, Comm. Math. Phys. 199 (1998), no. 1, 203–242. MathSciNetMATHGoogle Scholar
  18. CL01.
    F. Chapoton and M. Livernet, Pre-Lie algebras and the rooted trees operad, Internat. Math. Res. Notices (2001), no. 8, 395–408. Google Scholar
  19. CLP02.
    J. M. Casas, J.-L. Loday, and T. Pirashvili, Leibniz n-algebras, Forum Math. 14 (2002), no. 2, 189–207. MathSciNetMATHGoogle Scholar
  20. Coh76.
    F. R. Cohen, The homology of Open image in new window -spaces, The homology of iterated loop spaces (Cohen, Frederick R., Lada, Thomas J. and May, J. Peter), Lecture Notes in Mathematics, Vol. 533, Springer-Verlag, Berlin, 1976, pp. vii+490. Google Scholar
  21. Cos07.
    Kevin Costello, Topological conformal field theories and Calabi–Yau categories, Adv. Math. 210 (2007), no. 1, 165–214. MathSciNetMATHGoogle Scholar
  22. DCV11.
    G. Drummond-Cole and B. Vallette, Minimal model for the Batalin–Vilkovisky operad, ArXiv:1105.2008 (2011).
  23. Del93.
    Pierre Deligne, Letter to Stasheff, Gerstenhaber, May, Schechtman, Drinfeld. Google Scholar
  24. DK09.
    —, Free resolutions via Gröbner bases, arXiv:0912.4895 (2009).
  25. DM69.
    P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. (1969), no. 36, 75–109. Google Scholar
  26. Dok09.
    Ioannis Dokas, Zinbiel algebras and commutative algebras with divided powers, Glasg. Math. J. 51 (2009), no. 10, 1–11. Google Scholar
  27. DTT07.
    V. Dolgushev, D. Tamarkin, and B. Tsygan, The homotopy Gerstenhaber algebra of Hochschild cochains of a regular algebra is formal, J. Noncommut. Geom. 1 (2007), no. 1, 1–25. MathSciNetGoogle Scholar
  28. Dup76.
    Johan L. Dupont, Simplicial de Rham cohomology and characteristic classes of flat bundles, Topology 15 (1976), no. 3, 233–245. MathSciNetMATHGoogle Scholar
  29. Dup78.
    —, Curvature and characteristic classes, Lecture Notes in Mathematics, Vol. 640, Springer-Verlag, Berlin, 1978. MATHGoogle Scholar
  30. 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. MathSciNetMATHGoogle Scholar
  31. EFG07.
    —, Coherent unit actions on regular operads and Hopf algebras, Theory Appl. Categ. 18 (2007), 348–371. MathSciNetMATHGoogle Scholar
  32. FBZ04.
    E. Frenkel and D. Ben-Zvi, Vertex algebras and algebraic curves, second ed., Mathematical Surveys and Monographs, vol. 88, American Mathematical Society, Providence, RI, 2004. MATHGoogle Scholar
  33. 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. Google Scholar
  34. FOOO09a.
    K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono, Lagrangian intersection Floer theory: anomaly and obstruction. Part I, AMS/IP Studies in Advanced Mathematics, vol. 46, American Mathematical Society, Providence, RI, 2009. Google Scholar
  35. FOOO09b.
    —, Lagrangian intersection Floer theory: anomaly and obstruction. Part II, AMS/IP Studies in Advanced Mathematics, vol. 46, American Mathematical Society, Providence, RI, 2009. Google Scholar
  36. Fre00.
    Benoit Fresse, On the homotopy of simplicial algebras over an operad, Trans. Amer. Math. Soc. 352 (2000), no. 9, 4113–4141. MathSciNetMATHGoogle Scholar
  37. Fre06.
    —, Théorie des opérades de Koszul et homologie des algèbres de Poisson, Ann. Math. Blaise Pascal 13 (2006), no. 2, 237–312. MathSciNetMATHGoogle Scholar
  38. Fre10.
    —, Koszul duality complexes for the cohomology of iterated loop spaces of spheres, ArXiv e-prints (2010). Google Scholar
  39. Fre11a.
    —, Iterated bar complexes of E-infinity algebras and homology theories, Algebr. Geom. Topol. 11 (2011), no. 2, 747–838. MathSciNetMATHGoogle Scholar
  40. Fre11b.
    —, Koszul duality of E n -operads, Selecta Math. (N.S.) 17 (2011), no. 2, 363–434. MR 2803847 MathSciNetMATHGoogle Scholar
  41. Gan03.
    Wee Liang Gan, Koszul duality for dioperads, Math. Res. Lett. 10 (2003), no. 1, 109–124. MathSciNetMATHGoogle Scholar
  42. GCTV09.
    I. Galvez-Carrillo, A. Tonks, and B. Vallette, Homotopy Batalin-Vilkovisky algebras, Journal Noncommutative Geometry (2009), arXiv:0907.2246, 49 pp.
  43. Ger63.
    M. Gerstenhaber, The cohomology structure of an associative ring, Ann. of Math. (2) 78 (1963), 267–288. MathSciNetMATHGoogle Scholar
  44. Get94.
    E. Getzler, Batalin-Vilkovisky algebras and two-dimensional topological field theories, Comm. Math. Phys. 159 (1994), no. 2, 265–285. MathSciNetMATHGoogle Scholar
  45. Get95.
    —, Operads and moduli spaces of genus 0 Riemann surfaces, The moduli space of curves (Texel Island, 1994). Progr. Math., vol. 129, Birkhäuser Boston, Boston, MA, 1995, pp. 199–230. Google Scholar
  46. Get09.
    —, Lie theory for nilpotent L ∞ -algebras, Ann. of Math. (2) 170 (2009), no. 1, 271–301. MathSciNetMATHGoogle Scholar
  47. Gin06.
    V. Ginzburg, Calabi-Yau algebras, arXiv:math/0612139 (2006).
  48. GJ94.
    E. Getzler and J. D. S. Jones, Operads, homotopy algebra and iterated integrals for double loop spaces, hep-th/9403055 (1994).
  49. GK94.
    V. Ginzburg and M. Kapranov, Koszul duality for operads, Duke Math. J. 76 (1994), no. 1, 203–272. MathSciNetMATHGoogle Scholar
  50. GK95a.
    E. Getzler and M. M. Kapranov, Cyclic operads and cyclic homology, Geometry, topology, & physics, Conf. Proc. Lecture Notes Geom. Topology, IV, Int. Press, Cambridge, MA, 1995, pp. 167–201. Google Scholar
  51. GK98.
    E. Getzler and M. M. Kapranov, Modular operads, Compositio Math. 110 (1998), no. 1, 65–126. MathSciNetMATHGoogle Scholar
  52. GL89.
    R. Grossman and R. G. Larson, Hopf-algebraic structure of families of trees, J. Algebra 126 (1989), no. 1, 184–210. MathSciNetMATHGoogle Scholar
  53. GM88.
    W. M. Goldman and J. J. Millson, The deformation theory of representations of fundamental groups of compact Kähler manifolds, Inst. Hautes Études Sci. Publ. Math. (1988), no. 67, 43–96. Google Scholar
  54. Gne97.
    A. Victor Gnedbaye, Opérades des algèbres (k+1)-aires, Operads: Proceedings of Renaissance Conferences (Hartford, CT/Luminy, 1995), Contemp. Math., vol. 202, 1997, pp. 83–113. Google Scholar
  55. GS10.
    J. Giansiracusa and P. Salvatore, Formality of the framed little 2-discs operad and semidirect products, Homotopy theory of function spaces and related topics, Contemp. Math., vol. 519, Amer. Math. Soc., Providence, RI, 2010, pp. 115–121. Google Scholar
  56. 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
  57. Har62.
    D. K. Harrison, Commutative algebras and cohomology, Trans. Amer. Math. Soc. 104 (1962), 191–204. MathSciNetMATHGoogle Scholar
  58. Hin03.
    —, Tamarkin’s proof of Kontsevich formality theorem, Forum Math. 15 (2003), no. 4, 591–614. MathSciNetMATHGoogle Scholar
  59. HKR62.
    G. Hochschild, B. Kostant, and A. Rosenberg, Differential forms on regular affine algebras, Trans. Amer. Math. Soc. 102 (1962), 383–408. MathSciNetMATHGoogle Scholar
  60. HM10.
    J. Hirsch and J. Millès, Curved Koszul duality theory, arXiv:1008.5368 (2010).
  61. Hol06.
    Ralf Holtkamp, On Hopf algebra structures over free operads, Adv. Math. 207 (2006), no. 2, 544–565. MathSciNetMATHGoogle Scholar
  62. HS93.
    V. Hinich and V. Schechtman, Homotopy Lie algebras, I. M. Gel’fand Seminar, Adv. Soviet Math., vol. 16, 93, pp. 1–28. Google Scholar
  63. Hua97.
    Yi-Zhi Huang, Two-dimensional conformal geometry and vertex operator algebras, Progress in Mathematics, vol. 148, Birkhäuser Boston Inc., Boston, MA, 1997. MATHGoogle Scholar
  64. HW95.
    P. Hanlon and M. Wachs, On Lie k-algebras, Adv. Math. 113 (1995), no. 2, 206–236. MathSciNetMATHGoogle Scholar
  65. Jac62.
    Nathan Jacobson, Lie algebras, Interscience Tracts in Pure and Applied Mathematics, No. 10, Interscience Publishers (a division of John Wiley & Sons), New York–London, 1962. MATHGoogle Scholar
  66. Jon99.
    Vaughan F. R. Jones, Planar algebras, i, arXiv:math/9909027 (1999).
  67. Jon10.
    —, Quadratic tangles in planar algebras, arXiv:1007.1158 (2010).
  68. Kac98.
    Victor Kac, Vertex algebras for beginners, second ed., University Lecture Series, vol. 10, American Mathematical Society, Providence, RI, 1998. MATHGoogle Scholar
  69. Kad82.
    —, The algebraic structure in the homology of an A(∞)-algebra, Soobshch. Akad. Nauk Gruzin. SSR 108 (1982), no. 2, 249–252 (1983). MathSciNetGoogle Scholar
  70. Kad88.
    —, The structure of the A(∞)-algebra, and the Hochschild and Harrison cohomologies, Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR 91 (1988), 19–27. MathSciNetGoogle Scholar
  71. Kas93.
    Takuji Kashiwabara, On the homotopy type of configuration complexes, Algebraic topology (Oaxtepec, 1991), Contemp. Math., vol. 146, 1993, pp. 159–170. Google Scholar
  72. Kau07.
    R. M. Kaufmann, On spineless cacti, Deligne’s conjecture and Connes-Kreimer’s Hopf algebra, Topology 46 (2007), no. 1, 39–88. MathSciNetMATHGoogle Scholar
  73. Kau08.
    —, A proof of a cyclic version of Deligne’s conjecture via Cacti, Mathematical Research Letters 15 (2008), no. 5, 901–921. MathSciNetMATHGoogle Scholar
  74. 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
  75. Klj74.
    A. A. Kljačko, Lie elements in a tensor algebra, Sibirsk. Mat. Ž. 15 (1974), 1296–1304, 1430. MR 0371961 (51 #8178) MathSciNetGoogle Scholar
  76. KM94.
    M. Kontsevich and Yu. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164 (1994), no. 3, 525–562. MathSciNetMATHGoogle Scholar
  77. Knu83.
    Finn F. Knudsen, The projectivity of the moduli space of stable curves. II. The stacks M g,n, Math. Scand. 52 (1983), no. 2, 161–199. MathSciNetMATHGoogle Scholar
  78. Kon03.
    —, Deformation quantization of Poisson manifolds, Lett. Math. Phys. 66 (2003), no. 3, 157–216. MathSciNetMATHGoogle Scholar
  79. Kos85.
    —, Crochet de Schouten-Nijenhuis et cohomologie, Astérisque (1985), no. Numero Hors Serie, 257–271, The mathematical heritage of Élie Cartan (Lyon, 1984). MR 837203 (88m:17013) Google Scholar
  80. KP11.
    O. Kravchenko and M. Polyak, Diassociative algebras and Milnor’s invariants for tangles, Lett. Math. Phys. 95 (2011), no. 3, 297–316. MR 2775128 (2012a:57018) MathSciNetMATHGoogle Scholar
  81. Kra00.
    Olga Kravchenko, Deformations of Batalin-Vilkovisky algebras, Poisson geometry (Warsaw, 1998), Banach Center Publ., vol. 51, Polish Acad. Sci., Warsaw, 2000, pp. 131–139. Google Scholar
  82. KS96.
    Yvette Kosmann-Schwarzbach, From Poisson algebras to Gerstenhaber algebras, Ann. Inst. Fourier (Grenoble) 46 (1996), no. 5, 1243–1274. MathSciNetMATHGoogle Scholar
  83. KS00.
    M. Kontsevich and Y. Soibelman, Deformations of algebras over operads and the Deligne conjecture, Conférence Moshé Flato 1999, Vol. I (Dijon), Math. Phys. Stud., vol. 21, Kluwer Acad. Publ., Dordrecht, 2000, pp. 255–307. Google Scholar
  84. KS04.
    Yvette Kosmann-Schwarzbach, Derived brackets, Lett. Math. Phys. 69 (2004), 61–87. MathSciNetMATHGoogle Scholar
  85. KS09.
    M. Kontsevich and Y. Soibelman, Notes on A ∞ -algebras, A ∞ -categories and non-commutative geometry, Homological mirror symmetry, Lecture Notes in Phys., vol. 757, Springer, Berlin, 2009, pp. 153–219. Google Scholar
  86. KS10.
    —, Deformation theory. I [Draft], http://www.math.ksu.edu/~soibel/Book-vol1.ps, 2010.
  87. Lis52.
    William G. Lister, A structure theory of Lie triple systems, Trans. Amer. Math. Soc. 72 (1952), 217–242. MathSciNetMATHGoogle Scholar
  88. LM05.
    —, Symmetric brace algebras, Appl. Categ. Structures 13 (2005), no. 4, 351–370. MathSciNetMATHGoogle Scholar
  89. Lod06.
    Jean-Louis Loday, Inversion of integral series enumerating planar trees, Sém. Lothar. Combin. 53 (2004/06), Art. B53d, 16 pp. (electronic). MR MR2180779 (2006k:05016) Google Scholar
  90. Lod89.
    —, Opérations sur l’homologie cyclique des algèbres commutatives, Invent. Math. 96 (1989), no. 1, 205–230. MathSciNetGoogle Scholar
  91. Lod93.
    —, Une version non commutative des algèbres de Lie: les algèbres de Leibniz, Enseign. Math. (2) 39 (1993), no. 3-4, 269–293. MathSciNetGoogle Scholar
  92. Lod95.
    —, Cup-product for Leibniz cohomology and dual Leibniz algebras, Math. Scand. 77 (1995), no. 2, 189–196. MathSciNetGoogle Scholar
  93. Lod97.
    —, Overview on Leibniz algebras, dialgebras and their homology, Cyclic cohomology and noncommutative geometry (Waterloo, ON, 1995), Fields Inst. Commun., vol. 17, Amer. Math. Soc., Providence, RI, 1997, pp. 91–102. Google Scholar
  94. 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
  95. Lod01.
    —, Dialgebras, Dialgebras and related operads, Lecture Notes in Math., vol. 1763, Springer, Berlin, 2001, pp. 7–66. Google Scholar
  96. Lod02.
    —, Arithmetree, J. Algebra 258 (2002), no. 1, 275–309, Special issue in celebration of Claudio Procesi’s 60th birthday. MathSciNetGoogle Scholar
  97. Lod04b.
    —, Scindement d’associativité et algèbres de Hopf, Actes des Journées Mathématiques à la Mémoire de Jean Leray, Sémin. Congr., vol. 9, Soc. Math. France, Paris, 2004, pp. 155–172. Google Scholar
  98. Lod06.
    —, Completing the operadic butterfly, Georgian Math. J. 13 (2006), no. 4, 741–749. MathSciNetGoogle Scholar
  99. Lod07.
    —, On the algebra of quasi-shuffles, Manuscripta Math. 123 (2007), no. 1, 79–93. MathSciNetGoogle Scholar
  100. Lod08.
    —, Generalized bialgebras and triples of operads, Astérisque (2008), no. 320, x+116. Google Scholar
  101. Lod10.
    —, On the operad of associative algebras with derivation, Georgian Math. J. 17 (2010), no. 2, 347–372. MathSciNetGoogle Scholar
  102. LP93.
    J.-L. Loday and T. Pirashvili, Universal enveloping algebras of Leibniz algebras and (co)homology, Math. Ann. 296 (1993), no. 1, 139–158. MathSciNetMATHGoogle Scholar
  103. 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) Google Scholar
  104. LR06.
    —, On the structure of cofree Hopf algebras, J. Reine Angew. Math. 592 (2006), 123–155. MathSciNetMATHGoogle Scholar
  105. LR10.
    —, Combinatorial Hopf algebras, Clay Mathematics Proceedings 12 (2010), 347–384. MathSciNetGoogle Scholar
  106. LR12.
    —, Permutads, Journal of Combinatorial Theory A (2012). Google Scholar
  107. LS93.
    T. Lada and J. Stasheff, Introduction to SH Lie algebras for physicists, Internat. J. Theoret. Phys. 32 (1993), no. 7, 1087–1103. MathSciNetMATHGoogle Scholar
  108. Mac95.
    I. G. Macdonald, Symmetric functions and Hall polynomials, second ed., Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1995, With contributions by A. Zelevinsky, Oxford Science Publications. MATHGoogle Scholar
  109. Man99a.
    Marco Manetti, Deformation theory via differential graded Lie algebras, Algebraic Geometry Seminars, 1998–1999 (Italian) (Pisa), Scuola Norm. Sup., Pisa, 1999, pp. 21–48. Google Scholar
  110. Man99b.
    Yu. I. Manin, Frobenius manifolds, quantum cohomology, and moduli spaces, American Mathematical Society Colloquium Publications, vol. 47, American Mathematical Society, Providence, RI, 1999. MATHGoogle Scholar
  111. Man01.
    M. A. Mandell, E ∞ algebras and p-adic homotopy theory, Topology 40 (2001), no. 1, 43–94. MathSciNetMATHGoogle Scholar
  112. Man06.
    —, Cochains and homotopy type, Publ. Math. Inst. Hautes Études Sci. (2006), no. 103, 213–246. Google Scholar
  113. Mar04.
    —, Homotopy algebras are homotopy algebras, Forum Math. 16 (2004), no. 1, 129–160. MathSciNetMATHGoogle Scholar
  114. Men09.
    Luc Menichi, Batalin-Vilkovisky algebra structures on Hochschild cohomology, Bulletin de la Société Mathématique de France 137 (2009), no. 2, 277–295. MathSciNetMATHGoogle Scholar
  115. Mer05.
    —, Nijenhuis infinity and contractible differential graded manifolds, Compos. Math. 141 (2005), no. 5, 1238–1254. MathSciNetGoogle Scholar
  116. Mer08.
    —, Exotic automorphisms of the Schouten algebra of polyvector fields, arXiv:0809.2385 (2008).
  117. Mer10b.
    —, Wheeled props in algebra, geometry and quantization, European Congress of Mathematics, Eur. Math. Soc., Zürich, 2010, pp. 83–114. Google Scholar
  118. Mil09.
    D. V. Millionshchikov, The algebra of formal vector fields on the line and Buchstaber’s conjecture, Funktsional. Anal. i Prilozhen. 43 (2009), no. 4, 26–44. MathSciNetGoogle Scholar
  119. Mnë09.
    P. Mnëv, Notes on simplicial BF theory, Mosc. Math. J. 9 (2009), no. 2, 371–410. MathSciNetMATHGoogle Scholar
  120. MR96.
    G. Melançon and C. Reutenauer, Free Lie superalgebras, trees and chains of partitions, J. Algebraic Combin. 5 (1996), no. 4, 337–351. MathSciNetMATHGoogle Scholar
  121. MR09.
    M. Markl and E. Remm, (non-)Koszulness of operads for n-ary algebras, gagalim and other curiosities, arXiv:0907.1505v2 (2009).
  122. 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. Google Scholar
  123. 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
  124. MV09b.
    —, Deformation theory of representations of prop(erad)s. II, J. Reine Angew. Math. 636 (2009), 123–174. MathSciNetMATHGoogle Scholar
  125. Nto94.
    Patricia Ntolo, Homologie de Leibniz d’algèbres de Lie semi-simples, C. R. Acad. Sci. Paris Sér. I Math. 318 (1994), no. 8, 707–710. MathSciNetMATHGoogle Scholar
  126. OG08.
    J.-M. Oudom and D. Guin, On the Lie enveloping algebra of a pre-Lie algebra, J. K-Theory 2 (2008), no. 1, 147–167. MathSciNetMATHGoogle Scholar
  127. Pir94.
    Teimuraz Pirashvili, On Leibniz homology, Ann. Inst. Fourier (Grenoble) 44 (1994), no. 2, 401–411. MathSciNetMATHGoogle Scholar
  128. Qui69.
    —, Rational homotopy theory, Ann. of Math. (2) 90 (1969), 205–295. MathSciNetMATHGoogle Scholar
  129. Ret93.
    Vladimir S. Retakh, Lie-Massey brackets and n-homotopically multiplicative maps of differential graded Lie algebras, J. Pure Appl. Algebra 89 (1993), no. 1-2, 217–229. MathSciNetMATHGoogle Scholar
  130. Reu93.
    Ch. Reutenauer, Free Lie algebras, xviii+269, Oxford Science Publications. Google Scholar
  131. Ron00.
    María Ronco, Primitive elements in a free dendriform algebra, New trends in Hopf algebra theory (La Falda, 1999), Contemp. Math., vol. 267, Amer. Math. Soc., Providence, RI, 2000, pp. 245–263. Google Scholar
  132. Ron02.
    —, Eulerian idempotents and Milnor-Moore theorem for certain non-cocommutative Hopf algebras, J. Algebra 254 (2002), no. 1, 152–172. MathSciNetMATHGoogle Scholar
  133. Ron11.
    —, Shuffle algebras, Annales Instit. Fourier 61 (2011), no. 1, 799–850. MathSciNetMATHGoogle Scholar
  134. RW02.
    A. Robinson and S. Whitehouse, Operads and Γ-homology of commutative rings, Math. Proc. Cambridge Philos. Soc. 132 (2002), no. 2, 197–234. MathSciNetMATHGoogle Scholar
  135. Sam53.
    Hans Samelson, A connection between the Whitehead and the Pontryagin product, Amer. J. Math. 75 (1953), 744–752. MathSciNetMATHGoogle Scholar
  136. Sch68.
    Michael Schlessinger, Functors of Artin rings, Trans. Amer. Math. Soc. 130 (1968), 208–222. MathSciNetMATHGoogle Scholar
  137. Sch93.
    Albert Schwarz, Geometry of Batalin-Vilkovisky quantization, Comm. Math. Phys. 155 (1993), no. 2, 249–260. MathSciNetMATHGoogle Scholar
  138. Seg04.
    G. Segal, The definition of conformal field theory, Topology, geometry and quantum field theory, London Math. Soc. Lecture Note Ser., vol. 308, Cambridge Univ. Press, Cambridge, 2004, pp. 421–577. Google Scholar
  139. Sei08.
    Paul Seidel, Fukaya categories and Picard-Lefschetz theory, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2008. MATHGoogle Scholar
  140. Ser06.
    Jean-Pierre Serre, Lie algebras and Lie groups, Lecture Notes in Mathematics, vol. 1500, Springer-Verlag, Berlin, 2006, 1964 lectures given at Harvard University, Corrected fifth printing of the second (1992) edition. Google Scholar
  141. Šev10.
    Pavol Ševera, Formality of the chain operad of framed little disks, Lett. Math. Phys. 93 (2010), no. 1, 29–35. MathSciNetMATHGoogle Scholar
  142. Smi89.
    J. H. Smith, Simplicial group models for Ωn S n X, Israel J. Math. 66 (1989), no. 1-3, 330–350. MathSciNetMATHGoogle Scholar
  143. 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. MathSciNetMATHGoogle Scholar
  144. ST09.
    P. Salvatore and R. Tauraso, The operad Lie is free, J. Pure Appl. Algebra 213 (2009), no. 2, 224–230. MathSciNetMATHGoogle Scholar
  145. Sta63.
    Jim Stasheff, Homotopy associativity of H-spaces. I, II, Trans. Amer. Math. Soc. 108 (1963), 275–292; ibid. 108 (1963), 293–312. MathSciNetMATHGoogle Scholar
  146. 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. MATHGoogle Scholar
  147. Sul77.
    Dennis Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. (1977), no. 47, 269–331 (1978). Google Scholar
  148. Tak94.
    L. A. Takhtajan, Higher order analog of Chevalley-Eilenberg complex and deformation theory of n-gebras, Algebra i Analiz 6 (1994), no. 2, 262–272. MathSciNetMATHGoogle Scholar
  149. Tam99.
    D. E. Tamarkin, Operadic proof of M. Kontsevich’s formality theorem, 51 p., Thesis (Ph.D.)–The Pennsylvania State University. Google Scholar
  150. Tam03.
    —, Formality of chain operad of little discs, Lett. Math. Phys. 66 (2003), no. 1-2, 65–72. MathSciNetGoogle Scholar
  151. Tam07.
    —, What do dg-categories form?, Compos. Math. 143 (2007), no. 5, 1335–1358. MathSciNetGoogle Scholar
  152. Tou06.
    Victor Tourtchine, Dyer-Lashof-Cohen operations in Hochschild cohomology, Algebr. Geom. Topol. 6 (2006), 875–894 (electronic). MathSciNetMATHGoogle Scholar
  153. Tra08.
    T. Tradler, The Batalin-Vilkovisky algebra on Hochschild cohomology induced by infinity inner products, Annales de l’institut Fourier 58 (2008), no. 7, 2351–2379. MathSciNetMATHGoogle Scholar
  154. TT00.
    D. Tamarkin and B. Tsygan, Noncommutative differential calculus, homotopy BV algebras and formality conjectures, Methods Funct. Anal. Topology 6 (2000), no. 2, 85–100. MathSciNetMATHGoogle Scholar
  155. TZ06.
    T. Tradler and M. Zeinalian, On the cyclic Deligne conjecture, J. Pure Appl. Algebra 204 (2006), no. 2, 280–299. MathSciNetMATHGoogle Scholar
  156. Uch10.
    Kyousuke Uchino, Derived bracket construction and Manin products, Lett. Math. Phys. 93 (2010), no. 1, 37–53. MathSciNetMATHGoogle Scholar
  157. Val07b.
    —, A Koszul duality for props, Trans. of Amer. Math. Soc. 359 (2007), 4865–4993. MathSciNetMATHGoogle Scholar
  158. Val08.
    —, Manin products, Koszul duality, Loday algebras and Deligne conjecture, J. Reine Angew. Math. 620 (2008), 105–164. MathSciNetMATHGoogle Scholar
  159. vdL03.
    —, Coloured Koszul duality and strongly homotopy operads, arXiv:math.QA/0312147 (2003).
  160. Vor00.
    Alexander A. Voronov, Homotopy Gerstenhaber algebras, Conférence Moshé Flato 1999, Vol. II (Dijon), Math. Phys. Stud., vol. 22, Kluwer Acad. Publ., Dordrecht, 2000, pp. 307–331. Google Scholar
  161. VV97.
    A. Vinogradov and M. Vinogradov, On multiple generalizations of Lie algebras and Poisson manifolds, Secondary calculus and cohomological physics (Moscow, 1997), Contemp. Math., vol. 219, 1997, pp. 273–287. Google Scholar
  162. Whi41.
    J. H. C. Whitehead, On adding relations to homotopy groups, Ann. of Math. (2) 42 (1941), 409–428. MathSciNetGoogle Scholar
  163. Zin12.
    G. W. Zinbiel, Encyclopedia of types of algebras 2010, Proc. Int. Conf., in Nankai Series in Pure, Applied Mathematics and Theoretical Physics, vol. 9, World Scientific, Singapore, 2012. Google Scholar
  164. ZSSS82.
    K. A. Zhevlakov, A. M. Slin’ko, I. P. Shestakov, and A. I. Shirshov, Rings that are nearly associative, Pure and Applied Mathematics, vol. 104, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1982, Translated from the Russian by Harry F. Smith. MATHGoogle Scholar
  165. Zwi93.
    Barton Zwiebach, Closed string field theory: quantum action and the Batalin-Vilkovisky master equation, Nuclear Phys. B 390 (1993), no. 1, 33–152. MathSciNetGoogle 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

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