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.

Cattaneo, A S; Keller, B; Torossian, C; Bruguières, A (2005). *Déformation, quantification, théorie de Lie.* Paris: Société Mathématique de France.

## Abstract

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.

## Altmetrics

## Additional indexing

Item Type: | Monograph |
---|---|

Communities & Collections: | 07 Faculty of Science > Institute of Mathematics |

Dewey Decimal Classification: | 510 Mathematics |

Uncontrolled Keywords: | Deformation theory, deformation quantization, mathematical physics, Poisson manifold, Hochschild cohomology, Lie algebra, Duflo isomorphism, Campbell-Baker-Hausdorff formula, string theory, configuration space. |

Language: | English |

Date: | 2005 |

Deposited On: | 16 Dec 2009 12:42 |

Last Modified: | 05 Apr 2016 13:24 |

Publisher: | Société Mathématique de France |

Series Name: | Panoramas et Synthèses [Panoramas and Syntheses] |

Volume: | 20 |

Number of Pages: | 186 |

ISBN: | 2-85629-183-X |

Official URL: | http://smf.emath.fr/Publications/PanoramasSyntheses/2005/20/html/smf_pano-synth_20.html |

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