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Deformation, quantization, Lie theory. (Déformation, quantification, théorie de Lie.) (English, French) Zbl 1093.53095
Panoramas et Synthèses 20. Paris: Société Mathématique de France (ISBN 2-85629-183-X/pbk). vii, 186 p. (2005).
In 1997, M. Kontsevich proved that every Poisson manifold admits a formal quantization, canonical up to equivalence. In doing so he solved a longstanding problem in mathematical physics and he also opened up new research avenues in Lie theory, quantum group theory, deformation theory and the study of operads. In this volume, the authors present the main results of Kontsevich’s 1997 preprint [M. Kontsevich, Deformation quantization of Poisson manifolds I, math.QA/9709040; see also Lett. Math. Phys. 66, 157–216 (2003; Zbl 1058.53065)], show the relevance of Kontsevich’s theorem for Lie theory and explain the idea from topological string theory which inspired Kontsevich’s proof.
In the first part of this volume, B. Keller first describes the explicit formula which Kontsevich gave for the star product associated with a Poisson bracket on an open subset $$M$$ of $${\mathbb R}^{2n}$$ and gives a more precise form of Kontsevich’s theorem, which establishes a parametrization of the equivalence classes of star products on a smooth manifold in terms of the equivalence classes of formal deformations of the vanishing Poisson bracket, second presents the deformation-theoretic framework of Kontsevich’s formality theorem, and third sketches Tamarkin’s approach to a purely algebraic version of Kontsevich’s theorem for $${\mathbb R}^{2n}$$.
In the second part, C. Torossian describes some links between the Kontsevich construction and the Duflo isomorphism.
In the third part, A. Cattaneo explains how to obtain Kontsevich’s formula from the perturbative computation of the functional integral of a topological field theory known as the Poisson sigma model.
An appendix (by A. Bruguières) is devoted to the geometry of configurations spaces.

##### MSC:
 53D55 Deformation quantization, star products 16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) 53D17 Poisson manifolds; Poisson groupoids and algebroids 81S10 Geometry and quantization, symplectic methods 22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods