Algebraic Operads pp 479-566 | Cite as
Examples of Algebraic Operads
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 AlgebraReferences
- AL04.M. Aguiar and J.-L. Loday, Quadri-algebras, J. Pure Appl. Algebra 191 (2004), no. 3, 205–221. MathSciNetMATHGoogle Scholar
- 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
- 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
- BD04.A. Beilinson and V. Drinfeld, Chiral algebras, American Mathematical Society Colloquium Publications, vol. 51, American Mathematical Society, Providence, RI, 2004. MATHGoogle Scholar
- 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
- Ber96.Clemens Berger, Opérades cellulaires et espaces de lacets itérés, Ann. Inst. Fourier (Grenoble) 46 (1996), no. 4, 1125–1157. MathSciNetMATHGoogle Scholar
- BF04.C. Berger and B. Fresse, Combinatorial operad actions on cochains, Math. Proc. Cambridge Philos. Soc. 137 (2004), no. 1, 135–174. MathSciNetMATHGoogle Scholar
- 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
- 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
- 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
- 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
- BV81.I. A. Batalin and G. A. Vilkovisky, Gauge algebra and quantization, Phys. Lett. B 102 (1981), no. 1, 27–31. MathSciNetGoogle Scholar
- CE48.C. Chevalley and S. Eilenberg, Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc. 63 (1948), 85–124. MathSciNetMATHGoogle Scholar
- CG08.X. Z. Cheng and E. Getzler, Transferring homotopy commutative algebraic structures, J. Pure Appl. Algebra 212 (2008), no. 11, 2535–2542. MathSciNetMATHGoogle Scholar
- 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
- Cha02.—, Opérades différentielles graduées sur les simplexes et les permutoèdres, Bull. Soc. Math. France 130 (2002), no. 2, 233–251. MathSciNetMATHGoogle Scholar
- CK98.A. Connes and D. Kreimer, Hopf algebras, renormalization and noncommutative geometry, Comm. Math. Phys. 199 (1998), no. 1, 203–242. MathSciNetMATHGoogle Scholar
- 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
- CLP02.J. M. Casas, J.-L. Loday, and T. Pirashvili, Leibniz n-algebras, Forum Math. 14 (2002), no. 2, 189–207. MathSciNetMATHGoogle Scholar
- 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
- Cos07.Kevin Costello, Topological conformal field theories and Calabi–Yau categories, Adv. Math. 210 (2007), no. 1, 165–214. MathSciNetMATHGoogle Scholar
- DCV11.G. Drummond-Cole and B. Vallette, Minimal model for the Batalin–Vilkovisky operad, ArXiv:1105.2008 (2011).
- Del93.Pierre Deligne, Letter to Stasheff, Gerstenhaber, May, Schechtman, Drinfeld. Google Scholar
- DK09.—, Free resolutions via Gröbner bases, arXiv:0912.4895 (2009).
- 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
- Dok09.Ioannis Dokas, Zinbiel algebras and commutative algebras with divided powers, Glasg. Math. J. 51 (2009), no. 10, 1–11. Google Scholar
- 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
- Dup76.Johan L. Dupont, Simplicial de Rham cohomology and characteristic classes of flat bundles, Topology 15 (1976), no. 3, 233–245. MathSciNetMATHGoogle Scholar
- Dup78.—, Curvature and characteristic classes, Lecture Notes in Mathematics, Vol. 640, Springer-Verlag, Berlin, 1978. MATHGoogle Scholar
- 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
- EFG07.—, Coherent unit actions on regular operads and Hopf algebras, Theory Appl. Categ. 18 (2007), 348–371. MathSciNetMATHGoogle Scholar
- 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
- 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
- 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
- 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
- Fre00.Benoit Fresse, On the homotopy of simplicial algebras over an operad, Trans. Amer. Math. Soc. 352 (2000), no. 9, 4113–4141. MathSciNetMATHGoogle Scholar
- Fre06.—, Théorie des opérades de Koszul et homologie des algèbres de Poisson, Ann. Math. Blaise Pascal 13 (2006), no. 2, 237–312. MathSciNetMATHGoogle Scholar
- Fre10.—, Koszul duality complexes for the cohomology of iterated loop spaces of spheres, ArXiv e-prints (2010). Google Scholar
- Fre11a.—, Iterated bar complexes of E-infinity algebras and homology theories, Algebr. Geom. Topol. 11 (2011), no. 2, 747–838. MathSciNetMATHGoogle Scholar
- Fre11b.—, Koszul duality of E n -operads, Selecta Math. (N.S.) 17 (2011), no. 2, 363–434. MR 2803847 MathSciNetMATHGoogle Scholar
- Gan03.Wee Liang Gan, Koszul duality for dioperads, Math. Res. Lett. 10 (2003), no. 1, 109–124. MathSciNetMATHGoogle Scholar
- GCTV09.I. Galvez-Carrillo, A. Tonks, and B. Vallette, Homotopy Batalin-Vilkovisky algebras, Journal Noncommutative Geometry (2009), arXiv:0907.2246, 49 pp.
- Ger63.M. Gerstenhaber, The cohomology structure of an associative ring, Ann. of Math. (2) 78 (1963), 267–288. MathSciNetMATHGoogle Scholar
- Get94.E. Getzler, Batalin-Vilkovisky algebras and two-dimensional topological field theories, Comm. Math. Phys. 159 (1994), no. 2, 265–285. MathSciNetMATHGoogle Scholar
- 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
- Get09.—, Lie theory for nilpotent L ∞ -algebras, Ann. of Math. (2) 170 (2009), no. 1, 271–301. MathSciNetMATHGoogle Scholar
- Gin06.V. Ginzburg, Calabi-Yau algebras, arXiv:math/0612139 (2006).
- GJ94.E. Getzler and J. D. S. Jones, Operads, homotopy algebra and iterated integrals for double loop spaces, hep-th/9403055 (1994).
- GK94.V. Ginzburg and M. Kapranov, Koszul duality for operads, Duke Math. J. 76 (1994), no. 1, 203–272. MathSciNetMATHGoogle Scholar
- 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
- GK98.E. Getzler and M. M. Kapranov, Modular operads, Compositio Math. 110 (1998), no. 1, 65–126. MathSciNetMATHGoogle Scholar
- GL89.R. Grossman and R. G. Larson, Hopf-algebraic structure of families of trees, J. Algebra 126 (1989), no. 1, 184–210. MathSciNetMATHGoogle Scholar
- 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
- 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
- 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
- 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
- Har62.D. K. Harrison, Commutative algebras and cohomology, Trans. Amer. Math. Soc. 104 (1962), 191–204. MathSciNetMATHGoogle Scholar
- Hin03.—, Tamarkin’s proof of Kontsevich formality theorem, Forum Math. 15 (2003), no. 4, 591–614. MathSciNetMATHGoogle Scholar
- HKR62.G. Hochschild, B. Kostant, and A. Rosenberg, Differential forms on regular affine algebras, Trans. Amer. Math. Soc. 102 (1962), 383–408. MathSciNetMATHGoogle Scholar
- HM10.J. Hirsch and J. Millès, Curved Koszul duality theory, arXiv:1008.5368 (2010).
- Hol06.Ralf Holtkamp, On Hopf algebra structures over free operads, Adv. Math. 207 (2006), no. 2, 544–565. MathSciNetMATHGoogle Scholar
- 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
- 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
- HW95.P. Hanlon and M. Wachs, On Lie k-algebras, Adv. Math. 113 (1995), no. 2, 206–236. MathSciNetMATHGoogle Scholar
- 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
- Jon99.Vaughan F. R. Jones, Planar algebras, i, arXiv:math/9909027 (1999).
- Jon10.—, Quadratic tangles in planar algebras, arXiv:1007.1158 (2010).
- Kac98.Victor Kac, Vertex algebras for beginners, second ed., University Lecture Series, vol. 10, American Mathematical Society, Providence, RI, 1998. MATHGoogle Scholar
- 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
- 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
- Kas93.Takuji Kashiwabara, On the homotopy type of configuration complexes, Algebraic topology (Oaxtepec, 1991), Contemp. Math., vol. 146, 1993, pp. 159–170. Google Scholar
- 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
- Kau08.—, A proof of a cyclic version of Deligne’s conjecture via Cacti, Mathematical Research Letters 15 (2008), no. 5, 901–921. MathSciNetMATHGoogle Scholar
- 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
- Klj74.A. A. Kljačko, Lie elements in a tensor algebra, Sibirsk. Mat. Ž. 15 (1974), 1296–1304, 1430. MR 0371961 (51 #8178) MathSciNetGoogle Scholar
- 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
- 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
- Kon03.—, Deformation quantization of Poisson manifolds, Lett. Math. Phys. 66 (2003), no. 3, 157–216. MathSciNetMATHGoogle Scholar
- 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
- 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
- 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
- KS96.Yvette Kosmann-Schwarzbach, From Poisson algebras to Gerstenhaber algebras, Ann. Inst. Fourier (Grenoble) 46 (1996), no. 5, 1243–1274. MathSciNetMATHGoogle Scholar
- 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
- KS04.Yvette Kosmann-Schwarzbach, Derived brackets, Lett. Math. Phys. 69 (2004), 61–87. MathSciNetMATHGoogle Scholar
- 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
- KS10.—, Deformation theory. I [Draft], http://www.math.ksu.edu/~soibel/Book-vol1.ps, 2010.
- Lis52.William G. Lister, A structure theory of Lie triple systems, Trans. Amer. Math. Soc. 72 (1952), 217–242. MathSciNetMATHGoogle Scholar
- LM05.—, Symmetric brace algebras, Appl. Categ. Structures 13 (2005), no. 4, 351–370. MathSciNetMATHGoogle Scholar
- 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
- Lod89.—, Opérations sur l’homologie cyclique des algèbres commutatives, Invent. Math. 96 (1989), no. 1, 205–230. MathSciNetGoogle Scholar
- 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
- Lod95.—, Cup-product for Leibniz cohomology and dual Leibniz algebras, Math. Scand. 77 (1995), no. 2, 189–196. MathSciNetGoogle Scholar
- 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
- 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
- Lod01.—, Dialgebras, Dialgebras and related operads, Lecture Notes in Math., vol. 1763, Springer, Berlin, 2001, pp. 7–66. Google Scholar
- Lod02.—, Arithmetree, J. Algebra 258 (2002), no. 1, 275–309, Special issue in celebration of Claudio Procesi’s 60th birthday. MathSciNetGoogle Scholar
- 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
- Lod06.—, Completing the operadic butterfly, Georgian Math. J. 13 (2006), no. 4, 741–749. MathSciNetGoogle Scholar
- Lod07.—, On the algebra of quasi-shuffles, Manuscripta Math. 123 (2007), no. 1, 79–93. MathSciNetGoogle Scholar
- Lod08.—, Generalized bialgebras and triples of operads, Astérisque (2008), no. 320, x+116. Google Scholar
- Lod10.—, On the operad of associative algebras with derivation, Georgian Math. J. 17 (2010), no. 2, 347–372. MathSciNetGoogle Scholar
- 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
- 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
- LR06.—, On the structure of cofree Hopf algebras, J. Reine Angew. Math. 592 (2006), 123–155. MathSciNetMATHGoogle Scholar
- LR10.—, Combinatorial Hopf algebras, Clay Mathematics Proceedings 12 (2010), 347–384. MathSciNetGoogle Scholar
- LR12.—, Permutads, Journal of Combinatorial Theory A (2012). Google Scholar
- 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
- 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
- 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
- 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
- Man01.M. A. Mandell, E ∞ algebras and p-adic homotopy theory, Topology 40 (2001), no. 1, 43–94. MathSciNetMATHGoogle Scholar
- Man06.—, Cochains and homotopy type, Publ. Math. Inst. Hautes Études Sci. (2006), no. 103, 213–246. Google Scholar
- Mar04.—, Homotopy algebras are homotopy algebras, Forum Math. 16 (2004), no. 1, 129–160. MathSciNetMATHGoogle Scholar
- 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
- Mer05.—, Nijenhuis infinity and contractible differential graded manifolds, Compos. Math. 141 (2005), no. 5, 1238–1254. MathSciNetGoogle Scholar
- Mer08.—, Exotic automorphisms of the Schouten algebra of polyvector fields, arXiv:0809.2385 (2008).
- Mer10b.—, Wheeled props in algebra, geometry and quantization, European Congress of Mathematics, Eur. Math. Soc., Zürich, 2010, pp. 83–114. Google Scholar
- 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
- Mnë09.P. Mnëv, Notes on simplicial BF theory, Mosc. Math. J. 9 (2009), no. 2, 371–410. MathSciNetMATHGoogle Scholar
- 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
- MR09.M. Markl and E. Remm, (non-)Koszulness of operads for n-ary algebras, gagalim and other curiosities, arXiv:0907.1505v2 (2009).
- 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
- 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
- MV09b.—, Deformation theory of representations of prop(erad)s. II, J. Reine Angew. Math. 636 (2009), 123–174. MathSciNetMATHGoogle Scholar
- 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
- 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
- Pir94.Teimuraz Pirashvili, On Leibniz homology, Ann. Inst. Fourier (Grenoble) 44 (1994), no. 2, 401–411. MathSciNetMATHGoogle Scholar
- Qui69.—, Rational homotopy theory, Ann. of Math. (2) 90 (1969), 205–295. MathSciNetMATHGoogle Scholar
- 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
- Reu93.Ch. Reutenauer, Free Lie algebras, xviii+269, Oxford Science Publications. Google Scholar
- 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
- Ron02.—, Eulerian idempotents and Milnor-Moore theorem for certain non-cocommutative Hopf algebras, J. Algebra 254 (2002), no. 1, 152–172. MathSciNetMATHGoogle Scholar
- Ron11.—, Shuffle algebras, Annales Instit. Fourier 61 (2011), no. 1, 799–850. MathSciNetMATHGoogle Scholar
- 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
- Sam53.Hans Samelson, A connection between the Whitehead and the Pontryagin product, Amer. J. Math. 75 (1953), 744–752. MathSciNetMATHGoogle Scholar
- Sch68.Michael Schlessinger, Functors of Artin rings, Trans. Amer. Math. Soc. 130 (1968), 208–222. MathSciNetMATHGoogle Scholar
- Sch93.Albert Schwarz, Geometry of Batalin-Vilkovisky quantization, Comm. Math. Phys. 155 (1993), no. 2, 249–260. MathSciNetMATHGoogle Scholar
- 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
- Sei08.Paul Seidel, Fukaya categories and Picard-Lefschetz theory, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2008. MATHGoogle Scholar
- 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
- Šev10.Pavol Ševera, Formality of the chain operad of framed little disks, Lett. Math. Phys. 93 (2010), no. 1, 29–35. MathSciNetMATHGoogle Scholar
- Smi89.J. H. Smith, Simplicial group models for Ωn S n X, Israel J. Math. 66 (1989), no. 1-3, 330–350. MathSciNetMATHGoogle Scholar
- 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
- ST09.P. Salvatore and R. Tauraso, The operad Lie is free, J. Pure Appl. Algebra 213 (2009), no. 2, 224–230. MathSciNetMATHGoogle Scholar
- 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
- 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
- Sul77.Dennis Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. (1977), no. 47, 269–331 (1978). Google Scholar
- 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
- Tam99.D. E. Tamarkin, Operadic proof of M. Kontsevich’s formality theorem, 51 p., Thesis (Ph.D.)–The Pennsylvania State University. Google Scholar
- Tam03.—, Formality of chain operad of little discs, Lett. Math. Phys. 66 (2003), no. 1-2, 65–72. MathSciNetGoogle Scholar
- Tam07.—, What do dg-categories form?, Compos. Math. 143 (2007), no. 5, 1335–1358. MathSciNetGoogle Scholar
- Tou06.Victor Tourtchine, Dyer-Lashof-Cohen operations in Hochschild cohomology, Algebr. Geom. Topol. 6 (2006), 875–894 (electronic). MathSciNetMATHGoogle Scholar
- 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
- 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
- TZ06.T. Tradler and M. Zeinalian, On the cyclic Deligne conjecture, J. Pure Appl. Algebra 204 (2006), no. 2, 280–299. MathSciNetMATHGoogle Scholar
- Uch10.Kyousuke Uchino, Derived bracket construction and Manin products, Lett. Math. Phys. 93 (2010), no. 1, 37–53. MathSciNetMATHGoogle Scholar
- Val07b.—, A Koszul duality for props, Trans. of Amer. Math. Soc. 359 (2007), 4865–4993. MathSciNetMATHGoogle Scholar
- Val08.—, Manin products, Koszul duality, Loday algebras and Deligne conjecture, J. Reine Angew. Math. 620 (2008), 105–164. MathSciNetMATHGoogle Scholar
- vdL03.—, Coloured Koszul duality and strongly homotopy operads, arXiv:math.QA/0312147 (2003).
- 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
- 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
- Whi41.J. H. C. Whitehead, On adding relations to homotopy groups, Ann. of Math. (2) 42 (1941), 409–428. MathSciNetGoogle Scholar
- 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
- 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
- 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