Cohomologie tangente et cup-produit pour la quantification de Kontsevich
Dominique Manchon; Charles Torossian
Annales Mathématiques Blaise Pascal, Volume 10 (2003) no. 1, p. 75-106

On a flat manifold M= d , M. Kontsevich’s formality quasi-isomorphism is compatible with cup-products on tangent cohomology spaces, in the sense that for any formal Poisson 2-tensor γ the derivative at γ of the quasi-isomorphism induces an isomorphism of graded commutative algebras from Poisson cohomology space to Hochschild cohomology space relative to the deformed multiplication built from γ via the quasi-isomorphism. We give here a detailed proof of this result, with signs and orientations precised.

@article{AMBP_2003__10_1_75_0,
     author = {Dominique Manchon and Charles Torossian},
     title = {Cohomologie tangente et cup-produit pour la quantification de Kontsevich},
     journal = {Annales Math\'ematiques Blaise Pascal},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {10},
     number = {1},
     year = {2003},
     pages = {75-106},
     doi = {10.5802/ambp.168},
     mrnumber = {1990011},
     zbl = {02068411},
     language = {fr},
     url = {https://ambp.centre-mersenne.org/item/AMBP_2003__10_1_75_0}
}
Manchon, Dominique; Torossian, Charles. Cohomologie tangente et cup-produit pour la quantification de Kontsevich. Annales Mathématiques Blaise Pascal, Volume 10 (2003) no. 1, pp. 75-106. doi : 10.5802/ambp.168. ambp.centre-mersenne.org/item/AMBP_2003__10_1_75_0/

[1] M. Andler; A. Dvorsky; S. Sahi Kontsevich quantization and invariant distributions on Lie groups, Ann. Sci. Ec.Normale Sup. (4), Tome 35, no.3  (2002), pp. 371-390 | Numdam | MR 1914002 | Zbl 1009.22020

[2] D. Arnal; D. Manchon; M. Masmoudi Choix des signes pour la formalité de Kontsevich, Pacific J. Math., Tome 203 (2002), pp. 23-66 | Article | MR 1895924 | Zbl 01818958

[3] F. Bayen; M. Flato; C. Frønsdal; A. Lichnerowicz; D. Sternheimer Deformation theory and quantization I. Deformations of symplectic structures, Ann. Phys., Tome 111 (1978), pp. 61-110 | Article | MR 496157 | Zbl 0377.53024

[4] A. Cattaneo; G. Felder; L. Tomassini From local to global deformation quantization of Poisson manifolds (2000) (arXiv : math/QA/0012228) | Zbl 1037.53063

[5] W. Fulton; R. MacPherson Compactification of configuration spaces, Ann. Math., Tome 139 (1994), pp. 183-225 | Article | MR 1259368 | Zbl 0820.14037

[6] G. Ginot; G. Halbout A deformed version of Tamarkin’s formality theorem (2002) (Prépublication, IRMA Strasbourg)

[7] M. Kontsevich Deformation quantization of Poisson manifolds I (1997) (arXiv : math/QA/9709040)

[8] T. Mochizuki On the morphism of Duflo-Kirillov type, Journal of Geometry and Physics, Tome 41 (2002), pp. 73-113 | Article | MR 1872382 | Zbl 01758080

[9] D. Tamarkin Another proof of M. Kontsevich Formality theorem for R-n (1998) (arXiv : math/QA/9803025)