Cohomologie tangente et cup-produit pour la quantification de Kontsevich

Dominique Manchon[1]; Charles Torossian[2]

  • [1] CNRS - UMR 6620 Université Blaise Pascal 24 avenue des Landais 63177 Aubière cedex France
  • [2] CNRS - UMR 8553 Ecole Normale Supérieure 45 rue d’Ulm 75230 Paris Cedex 05 France

Annales mathématiques Blaise Pascal (2003)

  • Volume: 10, Issue: 1, page 75-106
  • ISSN: 1259-1734

Abstract

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On a flat manifold M = ℝ 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 ℏ γ the derivative at ℏ 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 ℏ γ via the quasi-isomorphism. We give here a detailed proof of this result, with signs and orientations precised.

How to cite

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Manchon, Dominique, and Torossian, Charles. "Cohomologie tangente et cup-produit pour la quantification de Kontsevich." Annales mathématiques Blaise Pascal 10.1 (2003): 75-106. <http://eudml.org/doc/10485>.

@article{Manchon2003,
affiliation = {CNRS - UMR 6620 Université Blaise Pascal 24 avenue des Landais 63177 Aubière cedex France; CNRS - UMR 8553 Ecole Normale Supérieure 45 rue d’Ulm 75230 Paris Cedex 05 France},
author = {Manchon, Dominique, Torossian, Charles},
journal = {Annales mathématiques Blaise Pascal},
keywords = {Kontsevich quantization; formality quasi-isomorphism; Poisson cohomology; Hochschild cohomology},
language = {fre},
month = {1},
number = {1},
pages = {75-106},
publisher = {Annales mathématiques Blaise Pascal},
title = {Cohomologie tangente et cup-produit pour la quantification de Kontsevich},
url = {http://eudml.org/doc/10485},
volume = {10},
year = {2003},
}

TY - JOUR
AU - Manchon, Dominique
AU - Torossian, Charles
TI - Cohomologie tangente et cup-produit pour la quantification de Kontsevich
JO - Annales mathématiques Blaise Pascal
DA - 2003/1//
PB - Annales mathématiques Blaise Pascal
VL - 10
IS - 1
SP - 75
EP - 106
LA - fre
KW - Kontsevich quantization; formality quasi-isomorphism; Poisson cohomology; Hochschild cohomology
UR - http://eudml.org/doc/10485
ER -

References

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  1. M. Andler, A. Dvorsky, S. Sahi, Kontsevich quantization and invariant distributions on Lie groups, Ann. Sci. Ec.Normale Sup. (4) 35, no.3  (2002), 371-390 Zbl1009.22020MR1914002
  2. D. Arnal, D. Manchon, M. Masmoudi, Choix des signes pour la formalité de Kontsevich, Pacific J. Math. 203 (2002), 23-66 Zbl1055.53066MR1895924
  3. F. Bayen, M. Flato, C. Frønsdal, A. Lichnerowicz, D. Sternheimer, Deformation theory and quantization I. Deformations of symplectic structures, Ann. Phys. 111 (1978), 61-110 Zbl0377.53024MR496157
  4. A. Cattaneo, G. Felder, L. Tomassini, From local to global deformation quantization of Poisson manifolds, (2000) Zbl1037.53063
  5. W. Fulton, R. MacPherson, Compactification of configuration spaces, Ann. Math. 139 (1994), 183-225 Zbl0820.14037MR1259368
  6. G. Ginot, G. Halbout, A deformed version of Tamarkin’s formality theorem, (2002) 
  7. M. Kontsevich, Deformation quantization of Poisson manifolds I, (1997) Zbl1058.53065
  8. T. Mochizuki, On the morphism of Duflo-Kirillov type, Journal of Geometry and Physics 41 (2002), 73-113 Zbl1134.53304MR1872382
  9. D. Tamarkin, Another proof of M. Kontsevich Formality theorem for R-n, (1998) 

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