Let be a differential manifold. Let be a Drinfeld associator. In this paper we explain how to construct a global formality morphism starting from . More precisely, following Tamarkin’s proof, we construct a Lie homomorphism “up to homotopy" between the Lie algebra of Hochschild cochains on and its cohomology ). This paper is an extended version of a course given 8 - 12 March 2004 on Tamarkin’s works. The reader will find explicit examples, recollections on -structures, explanation of the Etingof-Kazhdan quantization-dequantization theorem, of Tamarkin’s cohomological obstruction and of globalization process needed to get the formality theorem. Finally, we prove here that Tamarkin’s formality maps can be globalized.
@article{AMBP_2006__13_2_313_0, author = {Halbout, Gilles}, title = {Formality theorems: from associators to a global formulation}, journal = {Annales math\'ematiques Blaise Pascal}, volume = {13}, year = {2006}, pages = {313-348}, doi = {10.5802/ambp.220}, zbl = {1112.53067}, mrnumber = {2275450}, language = {en}, url = {http://dml.mathdoc.fr/item/AMBP_2006__13_2_313_0} }
Halbout, Gilles. Formality theorems: from associators to a global formulation. Annales mathématiques Blaise Pascal, Tome 13 (2006) pp. 313-348. doi : 10.5802/ambp.220. http://gdmltest.u-ga.fr/item/AMBP_2006__13_2_313_0/
[1] The double bar and cobar constructions, Compos. Math, Tome 43 (1981), pp. 331-341 | Numdam | MR 632433 | Zbl 0478.57027
[2] Covariant and equivariant formality theorems, Adv. Math., Tome 191 (2005), pp. 147-177 | Article | MR 2102846 | Zbl 02134411
[3] Quasi-Hopf algebras, Leningrad Math. J., Tome 1 (1990), pp. 1419-1457 | MR 1047964
[4] Quantum groups, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), Amer. Math. Soc., Providence, RI (1993), pp. 798-820 | MR 934283
[5] A cohomological construction of quantization functors of Lie bialgebras, Adv. Math., Tome 197 (2005), pp. 430-479 | Article | MR 2173841 | Zbl 02231207
[6] Quantization of Lie bialgebras. I, Selecta Math. (N.S.), Tome 2 (1996), pp. 1-41 | Article | MR 1403351 | Zbl 0863.17008
[7] Quantization of Lie bialgebras. II, III, Selecta Math. (N.S.), Tome 4 (1998), p. 213-231, 233-269 | Article | MR 1669953 | Zbl 0915.17009
[8] A simple geometrical construction of deformation quantization, J. Diff. Geom., Tome 40 (1994), pp. 213-238 | MR 1293654 | Zbl 0812.53034
[9] Homotopy G-algebras and moduli space operad, Internat. Math. Res. Notices, Tome 3 (1995), pp. 141-153 | Article | MR 1321701 | Zbl 0827.18004
[10] Homologie et modèle minimal des algèbres de Gerstenhaber, Ann. Math. Blaise Pascal, Tome 11 (2004), pp. 95-127 | Article | Numdam | MR 2077240 | Zbl 02207860
[11] A formality theorem for Poisson manifold, Lett. Math. Phys., Tome 66 (2003), pp. 37-64 | Article | MR 2064591 | Zbl 1066.53145
[12] Koszul duality for operads, Duke Math. J., Tome 76 (1994), pp. 203-272 | Article | MR 1301191 | Zbl 0855.18006
[13] Formule d’homotopie entre les complexes de Hochschild et de de Rham, Compositio Math., Tome 126 (2001), pp. 123-145 | Article | MR 1827641 | Zbl 1007.16008
[14] Tamarkin’s proof of Kontsevich’s formality theorem, Forum Math., Tome 15 (2003), pp. 591-614 | Article | MR 1978336 | Zbl 01916218
[15] Differential forms on regular affine algebras, Transactions AMS, Tome 102 (1962), pp. 383-408 | Article | MR 142598 | Zbl 0102.27701
[16] Homologie cyclique, caractère de Chern et lemme de perturbation, J. Reine Angew. Math., Tome 408 (1990), pp. 159-180 | Article | MR 1058987 | Zbl 0691.18002
[17] Operations on cyclic homology, the X complex, and a conjecture of Deligne, Comm. Math. Phys., Tome 202 (1999), pp. 309-323 | Article | MR 1689975 | Zbl 0952.16008
[18] Formality conjecture. Deformation theory and symplectic geometry, Math. Phys. Stud., Tome 20 (1996), pp. 139-156 | MR 1480721 | Zbl 1149.53325
[19] Deformation quantization of Poisson manifolds, I, Lett. Math. Phys., Tome 66 (2003), pp. 157-216 | Article | MR 2062626 | Zbl 1058.53065
[20] Deformations of algebras over operads and the Deligne conjecture (2000), pp. 255-307 | MR 1805894 | Zbl 0972.18005
[21] A homotopy formula for the Hochschild cohomology, Compositio Math., Tome 96 (1995), pp. 99-109 | Numdam | MR 1323727 | Zbl 0842.16006
[22] Another proof of M. Kontsevich’s formality theorem (1998) (math.QA/9803025)
[23] Homotopy Gerstenhaber algebras, Conférence Moshé Flato 1999, Vol. II (Dijon), Math. Phys. Stud., 22 (2000), pp. 307-331 | MR 1805923 | Zbl 0974.16005