With cedram.org
Annales Mathématiques
Blaise Pascal
Search for an article
Search within the site
Table of contents for this issue | Previous article | Next article
Walid Aloulou
Les $(a,b)$-algèbres à homotopie près
Annales mathématiques Blaise Pascal, 17 no. 1 (2010), p. 97-151, doi: 10.5802/ambp.279
Article PDF | Reviews MR 2674655 | Zbl 1204.18007
Class. Math.: 18G55, 16W30, 17B63, 16E45
Keywords: Algèbres homotopiques, cogèbres, algèbres de Poisson, algèbres différentielles graduées

Résumé - Abstract

We study in this article the concepts of algebra up to homotopy for a structure defined by two operations $ \hbox{.}$ and $[~,~]$. Having determined the structure of $ G_\infty $ algebras and $ P_\infty $ algebras, we generalize this construction and we define a structure of $ (a, b)$-algebra up to homotopy. Given a structure of commutative and differential graded Lie algebra for two shifts degree given by $a$ and $b$, we will give an explicit construction of the associate algebra up to homotopy and we clarify the relationship between $(a, b)$-algebra and algebra over the operad of little $n+1$-dimensional cubes.

Bibliography

[1] W. Aloulou, D. Arnal and R. Chatbouri. Cohomologie de Chevalley des graphes vectoriels. Pacific J of Math, 229(2):257-292, 2007. Article |  MR 2276511 |  Zbl pre05366194
[2] W. Aloulou, D. Arnal and R. Chatbouri. Algèbres et cogèbres de Gerstenhaber et cohomologies de Chevalley-Harrison. Bulletin des Sciences Mathématiques, 133(1):1-50, 2009. Article |  MR 2483521 |  Zbl 1159.18006
[3] D. Arnal, D. Manchon and M. Masmoudi. Choix des signes pour la formalité de M. Kontsevich. Pacific J of Math, 203(1):23-66, 2002. Article |  MR 1895924 |  Zbl 1055.53066
[4] J. A.de Azcárraga, J. M. Izquierdo, A. M. Perelemov and J. C. Pérez-Bueno. The $\mathbb{Z}_2$-graded Schouten-Nijenhuis bracket and generalized super-Poisson structures. preprint arXiv :hep-th/9612186v2, 1997 arXiv |  Zbl 0883.58007
[5] I. Basdouri and M. Ben Ammar. Cohomology of $\mathfrak{osp}(1|2)$ acting on linear differential operators on the supercircle $S^{1|1}$. Preprint arXiv :0709.1768v1 [math.RT] 12, 2007 arXiv |  Zbl 1138.53066
[6] J. M. Boardman and R. M. Vogt. Homotopy invariant algebraic structures on topological spaces. Springer-Verlag, 1973  MR 420609 |  Zbl 0285.55012
[7] M. Bordemann, G. Ginot, G. Halbout, H.C. Herbig and S. Waldmann. Formalité $G_{\infty }$ adaptée et star-représentations sur des sous variétés coïsotropes. Preprint arXiv :math.QA/0504276 v 1, 2005 arXiv
[8] A. S. Cattaneo and G. Felder. Relative formality theorem and quantisation of coisotropic submanifolds. Adv. Math., 208(2):521-548, 2007. Article |  MR 2304327 |  Zbl 1106.53060
[9] F. Cohen. Homology of $\Omega ^{n+1}\Sigma ^{n+1}X$ and $C_{n+1}X$, $n>0$. Bull. Amer. Math. Soc., 79(6):1236-1241, 1973. Article |  MR 339176 |  Zbl 0281.55004
[10] R. L. Cohen and A. A. Voronov. Notes on string topology, string topology and cyclic homology. Adv. courses Math. CRM Barcelona, Birkhäuser, Basel:1-95, 2006.  MR 2240287
[11] B. Fresse. Théorie des opérades de Koszul et homologie des algèbres de Poisson. Annales mathématiques Blaise Pascal, 13(2):237-312, 2006. Cedram |  MR 2275449 |  Zbl 1141.55006
[12] E. Getzler. Batalin-Vilkovisky algebras and two-dimensional topological field theories. Comm. Math. Phys, 159(2):265-285, 1994. Article |  MR 1256989 |  Zbl 0807.17026
[13] E. Getzler and J.D.S. Jones. Operads, homotopy algebra and iterated integrals for double loop spaces. Preprint arXiv :hep-th/9403055, 1994 arXiv
[14] G. Ginot. Homologie et modèle minimal des algèbres de Gerstenhaber. Annales mathématiques Blaise Pascal, 11(1):95-126, 2004. Cedram |  MR 2077240 |  Zbl 1139.16301
[15] G. Ginot and G. Halbout. A formality theorem for Poisson manifolds. Lett. Math. Phys., 66:37-64, 2003. Article |  MR 2064591 |  Zbl 1066.53145
[16] M. Kontsevich and Y. Soibelman. Deformations of algebras over operads and the Deligne conjecture. Conférence Moshé Flato 1999, vol I (Dijon), Math. Phys. Stud., 21, Kluwer Acad. Publ., Dordrecht:255-307, 2000.  MR 1805894 |  Zbl 0972.18005
[17] B. Kupershmidt. Odd and even Poisson brackets in dynamical systems. Lett. Math. Phys., 9:323-330, 1985. Article |  MR 796633 |  Zbl 0585.58020
[18] D. A. Leites. New superalgebras and mechanics. Sov. Math. Dokl., 18:1277-1280, 1977.  Zbl 0403.17002
[19] J.L. Loday. Cyclic Homology. Second Edition Grundlerhren der Mathematischen Wissenschaften A series of comprehensive studies in mathematics Springer-Verlag, 1992  MR 1217970 |  Zbl 0780.18009
[20] S. MacLane. Homology. Grundlerhren der Mathematischen Wissenschaften Springer-Verlag, Berlin, 1963  MR 349792 |  Zbl 0133.26502
[21] J.P. May. The geometry of iterated loop spaces. Springer-Verlag, Berlin-New York, 1972  MR 420610 |  Zbl 0244.55009
[22] D. Tamarkin. Another proof of M. Kontsevich formality theorem. Preprint arXiv :math.QA/9803025 v 4, 1998 arXiv
[23] D. Tamarkin and B. Tsygan. Noncommutative differential calculus, homotopy BV algebras and formality conjectures. Methods Funct. Anal. Topology, 6(2):85-100, 2000.  MR 1783778 |  Zbl 0965.58010