Artículo
Abstract:
The dg operad C of cellular chains on the operad of spineless cacti of Kaufmann (Topology 46(1):39–88, 2007) is isomorphic to the Gerstenhaber–Voronov dg operad codifying the cup product and brace operations on the Hochschild cochains of an associative algebra, and to the suboperad F2X of the surjection operad of Berger and Fresse (Math Proc Camb Philos Soc 137(1):135–174, 2004), McClure and Smith (Recent progress in homotopy theory (Baltimore, MD, 2000). Contemp Math., Amer. Math. Soc., Providence 293:153–193, 2002) and McClure and Smith (J Am Math Soc 16(3):681–704, 2003). Its homology is the Gerstenhaber dg operad G. We construct a map of dg operads ψ: A∞ ⟶ C such that ψ(m2) is commutative and H∗(ψ) is the canonical map A → Com → G. This formalises the idea that, since the cup product is commutative in homology, its symmetrisation is a homotopy associative operation. Our explicit A∞ structure does not vanish on non-trivial shuffles in higher degrees, so does not give a map Com∞ → C. If such a map could be written down explicitly, it would immediately lead to a G∞ structure on C and on Hochschild cochains, that is, to an explicit and direct proof of the Deligne conjecture. © 2015, Springer Basel.
Registro:
| Documento: | Artículo |
| Título: | An A∞ Operad in Spineless Cacti |
| Autor: | Gálvez-Carrillo, I.; Lombardi, L.; Tonks, A. |
| Filiación: | Departament de Matemàtica Aplicada III, Universitat Politècnica de Catalunya, Escola d’Enginyeria de Terrassa, Carrer Colom 1, Terrassa, Barcelona 08222, Spain Departamento de Matemática, Universidad de Buenos Aires, Ciudad Universitaria, Buenos Aires, Argentina Department of Mathematics, University of Leicester, University Road, Leicester, LE1 7RH, United Kingdom |
| Año: | 2015 |
| Volumen: | 12 |
| Número: | 4 |
| Página de inicio: | 1215 |
| Página de fin: | 1226 |
| DOI: | http://dx.doi.org/10.1007/s00009-015-0577-4 |
| Título revista: | Mediterranean Journal of Mathematics |
| Título revista abreviado: | Mediterr. J. Math. |
| ISSN: | 16605446 |
| Registro: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_16605446_v12_n4_p1215_GalvezCarrillo |
Referencias:
- Berger, C., Fresse, B., Combinatorial operad actions on cochains (2004) Math. Proc. Camb. Philos. Soc., 137 (1), pp. 135-174
- Cohen, R.L., Jones, J.D.S., A homotopy theoretic realization of string topology (2002) Math. Ann, 324 (4), pp. 773-798
- Gerstenhaber, M., Voronov, A.A.: Homotopy G-algebras and moduli space operad. Int. Math. Res. Not., 3, 141–153 (1995) (electronic); http://arxiv.org/abs/hep-th/9403055, Getzler, E., Jones, J.D.S.: Operads, homotopy algebra and iterated integrals for double loop spaces. (1994); Kaufmann, R.M.: On several varieties of cacti and their relations. Algebr. Geom. Topol., 5, 237–300 (2005) (electronic); Kaufmann, R.M., On spineless cacti, Deligne’s conjecture and Connes–Kreimer’s Hopf algebra (2007) Topology, 46 (1), pp. 39-88
- Kaufmann, R.M., A proof of a cyclic version of Deligne’s conjecture via cacti (2008) Math. Res. Lett., 15 (5), pp. 901-921
- Kaufmann, R.M., Livernet, M., Penner, R.C.: Arc operads and arc algebras. Geom. Topol., 7, 511–568 (2003) (electronic); Kaufmann, R.M., Schwell, R., Associahedra, cyclohedra and a topological solution to the (Formula Presented.) Deligne conjecture (2010) Adv. Math, 223 (6), pp. 2166-2199
- Kontsevich, M., Soibelman, Y., Deformations of algebras over operads and the Deligne conjecture (2000) Conférence Moshé Flato 1999, Vol. I (Dijon), Math. Phys. Stud., vol. 21, pp. 255-307. , Kluwer Acad. Publ., Dordrecht
- Loday, J.-L., Vallette, B., Algebraic operads (2012) Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol, p. 346. , Springer, Heidelberg
- McClure, J.E., Smith, J.H., A solution of Deligne’s Hochschild cohomology conjecture (2002) Recent progress in homotopy theory (Baltimore, MD, 2000), Contemp. Math., vol. 293, pp. 153-193. , Amer. Math. Soc., Providence
- McClure, J.E., Smith, J.H.: Multivariable cochain operations and little n-cubes. J. Am. Math. Soc., 16(3), 681–704 (2003) (electronic); Salvatore, P., The topological cyclic Deligne conjecture (2009) Algebr. Geom. Topol, 9 (1), pp. 237-264
- http://arxiv.org/abs/math/9803025, Tamarkin, D.E.: Another proof of M. Kontsevich formality theorem. (1998); Voronov, A.A., Notes on universal algebra (2005) Graphs and Patterns in Mathematics and Theoretical Physics, Proc. Sympos. Pure Math., vol. 73, pp. 81-103. , Amer. Math. Soc., Providence
- Zhang, Y., Bai, C., Guo, L., (2012) The category and operad of matching dialgebras. Appl, , Categor, Struct
- Zinbiel, G.W., Guo, L., Bai, C., Loday, J.-L., Encyclopedia of types of algebras (2010) Operads and universal algebra. Nankai Series in Pure, Applied Mathematics and Theoretical Physics, vol, p. 9. , World Sci. Publ., Hackensack
Citas:
---------- APA ----------
Gálvez-Carrillo, I., Lombardi, L. & Tonks, A. (2015) . An A∞ Operad in Spineless Cacti. Mediterranean Journal of Mathematics, 12(4), 1215-1226.http://dx.doi.org/10.1007/s00009-015-0577-4
---------- CHICAGO ----------
Gálvez-Carrillo, I., Lombardi, L., Tonks, A. "An A∞ Operad in Spineless Cacti" . Mediterranean Journal of Mathematics 12, no. 4 (2015) : 1215-1226.http://dx.doi.org/10.1007/s00009-015-0577-4
---------- MLA ----------
Gálvez-Carrillo, I., Lombardi, L., Tonks, A. "An A∞ Operad in Spineless Cacti" . Mediterranean Journal of Mathematics, vol. 12, no. 4, 2015, pp. 1215-1226.http://dx.doi.org/10.1007/s00009-015-0577-4
---------- VANCOUVER ----------
Gálvez-Carrillo, I., Lombardi, L., Tonks, A. An A∞ Operad in Spineless Cacti. Mediterr. J. Math. 2015;12(4):1215-1226.http://dx.doi.org/10.1007/s00009-015-0577-4
