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Déformation, Quantification, Théorie de Lie
Alberto Cattaneo, University of Zurich, Switzerland, Bernhard Keller, University of Paris VII, France, Charles Torossian, DMA-ENS, Paris, France, and Alain Bruguières, Université Montpellier II, France
A publication of the Société Mathématique de France.
cover
Panoramas et Synthèses
2006; 186 pp; softcover
Number: 20
ISBN-10: 2-85629-183-X
ISBN-13: 978-2-85629-183-2
List Price: US$38
Member Price: US$30.40
Order Code: PASY/20
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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. Through his proof and his interpretation of a later proof given by Tamarkin, he also opened up new research avenues in Lie theory, quantum group theory, deformation theory and the study of operads ... and uncovered fascinating links of these topics with number theory, knot theory and the theory of motives. Without doubt, his work on deformation quantization will continue to influence these fields for many years to come. In the three parts of this volume, we will 1) present the main results of Kontsevich's 1997 preprint and sketch his interpretation of Tamarkin's approach, 2) show the relevance of Kontsevich's theorem for Lie theory and 3) explain the idea from topological string theory which inspired Kontsevich's proof. An appendix is devoted to the geometry of configuration spaces.

A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.

Readership

Graduate students and research mathematicians interested in geometry and topology.

Table of Contents

  • Introduction
  • Introduction (English translation)
Part I. Deformation quantization after Kontsevich and Tamarkin (B. Keller)
  • Presentation of the main results
  • Deformation theory
  • On Tamarkin's approach
Part II. Application à la théorie de Lie (C. Torossian)
  • Introduction
  • La formule de Kontsevich pour \(\mathbb{R}^n\)
  • Exemples de calculs de graphes
  • Application au cas des algèbres de Lie
  • Formalité dans le cas \(\mathbb{R}^n\)
Part III. Deformation quantization from functional integrals (A. Cattaneo)
  • Introduction
  • Functional integrals
  • Symmetries and the BRST formalism
  • The Poisson sigma model
  • Deformation quantization of affine Poisson structures
Appendice (A. Bruguières)
  • Espaces de configurations
  • Bibliographie
  • Index
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