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© 2013

Poisson Structures

Benefits

  • First book about Poisson structures giving a comprehensive introduction as well as solid foundations to the theory

  • Unique structure of the volume tailored to graduate students or advanced researchers

  • Provides examples and exercises?

Book
  • 30k Downloads

Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 347)

Table of contents

  1. Front Matter
    Pages I-XXIV
  2. Theoretical Background

    1. Front Matter
      Pages 1-1
    2. Camille Laurent-Gengoux, Anne Pichereau, Pol Vanhaecke
      Pages 3-35
    3. Camille Laurent-Gengoux, Anne Pichereau, Pol Vanhaecke
      Pages 37-61
    4. Camille Laurent-Gengoux, Anne Pichereau, Pol Vanhaecke
      Pages 63-89
    5. Camille Laurent-Gengoux, Anne Pichereau, Pol Vanhaecke
      Pages 91-112
    6. Camille Laurent-Gengoux, Anne Pichereau, Pol Vanhaecke
      Pages 113-157
  3. Examples

    1. Front Matter
      Pages 159-159
    2. Camille Laurent-Gengoux, Anne Pichereau, Pol Vanhaecke
      Pages 161-178
    3. Camille Laurent-Gengoux, Anne Pichereau, Pol Vanhaecke
      Pages 179-203
    4. Camille Laurent-Gengoux, Anne Pichereau, Pol Vanhaecke
      Pages 205-232
    5. Camille Laurent-Gengoux, Anne Pichereau, Pol Vanhaecke
      Pages 233-267
    6. Camille Laurent-Gengoux, Anne Pichereau, Pol Vanhaecke
      Pages 269-290
    7. Camille Laurent-Gengoux, Anne Pichereau, Pol Vanhaecke
      Pages 291-325
  4. Applications

    1. Front Matter
      Pages 327-327
    2. Camille Laurent-Gengoux, Anne Pichereau, Pol Vanhaecke
      Pages 329-351
    3. Camille Laurent-Gengoux, Anne Pichereau, Pol Vanhaecke
      Pages 353-409
  5. Back Matter
    Pages 411-461

About this book

Introduction

Poisson structures appear in a large variety of contexts, ranging from string theory, classical/quantum mechanics and differential geometry to abstract algebra, algebraic geometry and representation theory. In each one of these contexts, it turns out that the Poisson structure is not a theoretical artifact, but a key element which, unsolicited, comes along with the problem that is investigated, and its delicate properties are decisive for the solution to the problem in nearly all cases. Poisson Structures is the first book that offers a comprehensive introduction to the theory, as well as an overview of the different aspects of Poisson structures. The first part covers solid foundations, the central part consists of a detailed exposition of the different known types of Poisson structures and of the (usually mathematical) contexts in which they appear, and the final part is devoted to the two main applications of Poisson structures (integrable systems and deformation quantization).
The clear structure of the book makes it adequate for readers who come across Poisson structures in their research or for graduate students or advanced researchers who are interested in an introduction to the many facets and applications of Poisson structures.​

Keywords

17B63, 53D17, 17B80, 53D55, 17B62 Poisson Algebras Poisson geometry Poisson structures deformation quantization integrable systems

Authors and affiliations

  1. 1.CNRS UMR 7122, Laboratoire de MathématiquesUniversité de LorraineMetzFrance
  2. 2.CNRS UMR 5208, Université Jean MonnetUniversité de LyonSaint EtienneFrance
  3. 3.CNRS UMR 7348, Lab. Mathématiques et ApplicationsUniversité de PoitiersFuturoscope ChasseneuilFrance

About the authors

C. Laurent-Gengoux research focus lies on Poisson geometry, Lie-groups and integrable systems. He is the author of 14 research articles. Furthermore, he is committed to teaching and set up several mathematics projects with local high schools. In 2002 he earned his doctorate in mathematics with a dissertation on " Quelques problèmes analytiques et géométriques sur les algèbres et superalgèbres de champs et superchamps de vecteurs”.

A. Pichereau earned her doctorate in mathematics with a dissertation on “Poisson (co)homology and isolated singularities in low dimensions, with an application in the theory of deformations” under the supervision of P. Vanheacke in 2006. She has since published four journal articles on Poisson structures and contributed to the Proceedings of "Algebraic and Geometric Deformation Spaces”.

P. Vanheacke’s research focus lies on integrable systems, Abelian varieties, Poisson algebra/geometry and deformation theory. In 1991 he earned his doctorate in mathematics with a dissertation on “Explicit techniques for studying two-dimensional integrable systems” and has published numerous research articles since.

Bibliographic information

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Reviews

From the book reviews:

“Each chapter contains a series of exercises as well as a number of notes aimed at giving further hints as to how the various items in the book are interrelated and, furthermore, at placing the material in the literature. … The book is a timely and courageous attempt to make accessible a flourishing research area to a wider audience in the form of a research monograph/textbook and as such it is very welcome.” (Johannes Huebschmann, Jahresbericht der Deutschen Mathematiker-Vereinigung, Vol. 116, 2014)

“This book is an excellent presentation of Poisson geometry, its applications and related topics. … This book is suitable for those who have a solid foundation of differential geometry and Lie algebras. The reader will understand why Poisson geometry is such an interesting and important subject.” (Zhuo Chen, zbMATH, Vol. 1284, 2014)

“This book provides a comprehensive introduction to Poisson structures. … Exercises are given at the end of each chapter … to help readers understand the basic theory. … This is a nice introductory book for both entry level graduate students and advanced researchers who are interested in the subject.” (Xiang Tang, Mathematical Reviews, August, 2013)

“The book under review deals with very exciting (and current) material presented from a fascinating vantage point and should be welcomed by any scholar whose work touches upon the matters … . its thirteen chapters are peppered with sets of exercises and each chapter comes equipped with supplemental notes that go a bit beyond the text, introduce some historical material, and point to other relevant sources.” (Michael Berg, MAA Reviews, November, 2012)