Poisson Geometry, Deformation Quantisation and Group Representations

doi:10.1017/CBO9780511734878

Poisson geometry lies at the cusp of noncommutative algebra and differential geometry, with natural and important links to classical physics and quantum mechanics. This book presents an introduction to the subject from a small group of leading researchers, and the result is a volume accessible to graduate students or experts from other fields. The contributions are: Poisson Geometry and Morita Equivalence by Bursztyn and Weinstein; Formality and Star Products by Cattaneo; Lie Groupoids, Sheaves and Cohomology by Moerdijk and Mrcun; Geometric Methods in Representation Theory by Schmid; Deformation Theory: A Powerful Tool in Physics Modelling by Sternheimer.

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Poisson Geometry, Deformation Quantisation and Group Representations

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Contents

Frontmatter
pp i-iv

Contents
pp v-viii

Preface
pp ix-x

Part One - Poisson geometry and morita equivalence
pp 1-2

1 - Introduction
pp 3-4

2 - Poisson geometry and some generalizations
pp 5-24

3 - Algebraic Morita equivalence
pp 25-36

4 - Geometric Morita equivalence
pp 37-66

5 - Geometric representation equivalence
pp 67-71

Bibliography
pp 72-78

Part Two - Formality and star products
pp 79-80

1 - Introduction
pp 81-86

2 - The star product
pp 87-92

3 - Rephrasing the main problem: the formality
pp 93-112

4 - Digression: what happens in the dual
pp 113-119

5 - The Kontsevich formula
pp 120-133

6 - From local to global deformation quantization
pp 134-140

Bibliography
pp 141-144

Part Three - Lie groupoids, sheaves and cohomology
pp 145-146

1 - Introduction
pp 147-148

2 - Lie groupoids
pp 149-174

3 - Sheaves on Lie groupoids
pp 175-209

4 - Sheaf cohomology
pp 210-241

5 - Compactly supported cohomology
pp 242-268

Bibliography
pp 269-272

Part Four - Geometric methods in representation theory
pp 273-274

1 - Reductive Lie Groups: Definitions and Basic Properties
pp 275-281

2 - Compact Lie Groups
pp 282-293

3 - Representations of Reductive Lie Groups
pp 294-304

4 - Geometric Constructions of Representations
pp 305-320

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Poisson Geometry, Deformation Quantisation and Group Representations

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