Poisson Geometry, Deformation Quantisation

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.

Preface

ix

Part One: Poisson geometry and morita equivalence

1

(78)

1 Introduction

3

(2)

2 Poisson geometry and some generalizations

5

(20)

2.1 Poisson manifolds

5

(2)

2.2 Dirac structures

7

(4)

2.3 Twisted structures

11

(2)

2.4 Symplectic leaves and local structure of Poisson manifolds

13

(2)

2.5 Presymplectic leaves and Dirac manifolds

15

(3)

2.6 Poisson maps

18

(2)

2.7 Dirac maps

20

(5)

3 Algebraic Morita equivalence

25

(12)

3.1 Ring-theoretic Morita equivalence of algebras

25

(4)

3.2 Strong Morita equivalence of C*-algebras

29

(4)

3.3 Morita equivalence of deformed algebras

33

(4)

4 Geometric Morita equivalence

37

(30)

4.1 Representations and tensor product

37

(3)

4.2 Symplectic groupoids

40

(7)

4.3 Morita equivalence for groups and groupoids

47

(2)

4.4 Modules over Poisson manifolds and groupoid actions

49

(3)

4.5 Morita equivalence and symplectic groupoids

52

(6)

4.6 Picard groups

58

(3)

4.7 Fibrating Poisson manifolds and Morita invariants

61

(3)

4.8 Gauge equivalence of Poisson structures

64

(3)

5 Geometric representation equivalence

67

(5)

5.1 Symplectic torsors

67

(2)

5.2 Symplectic categories

69

(1)

5.3 Symplectic categories of representations

70

(2)

Bibliography

72

(7)

Part Two: Formality and star products

79

(66)

1 Introduction

81

(6)

1.1 Physical motivation

81

(2)

1.2 Historical review of deformation quantization

83

(2)

1.3 Plan of the work

85

(2)

2 The star product

87

(6)

3 Rephrasing the main problem: the formality

93

(20)

3.1 DGLA's, Linfinity algebras and deformation functors

94

(8)

3.2 Multivector fields and multidifferential operators

102

(9)

3.2.1 The DGLA V

103

(3)

3.2.2 The DGLA D

106

(5)

3.3 The first term: U1

111

(2)

4 Digression: what happens in the dual

113

(7)

5 The Kontsevich formula

120

(14)

5.1 Admissible graphs, weights and BΓs

121

(4)

5.2 The proof: Stokes' theorem & Vanishing theorems

125

(9)

6 From local to global deformation quantization

134

(7)

Bibliography

141

(4)

Part Three: Lie groupoids, sheaves and cohomology

145

(128)

1 Introduction

147

(2)

2 Lie groupoids

149

(26)

2.1 Lie groupoids and weak equivalences

151

(3)

2.2 The monodromy and holonomy groupoids of a foliation

154

(2)

2.3 Etale groupoids and foliation groupoids

156

(3)

2.4 Sonic general constructions

159

(5)

2.5 Principal bundles as morphisms

164

(4)

2.6 The principal bundles category

168

(7)

3 Sheaves on Lie groupoids

175

(35)

3.1 Sheaves on groupoids

176

(6)

3.2 Functoriality and Morita equivalence

182

(5)

3.3 The fundamental group and locally constant sheaves

187

(14)

3.4 G-sheaves of R-modules

201

(4)

3.5 Derived categories

205

(5)

4 Sheaf cohomology

210

(32)

4.1 Sheaf cohomology of foliation groupoids

211

(3)

4.2 The bar resolution for kale groupoids

214

(7)

4.3 Proper maps and orbifolds

221

(6)

4.4 A comparison theorem for foliations

227

(5)

4.5 The embedding category of an étale groupoid

232

(6)

4.6 Degree one cohomology and the fundamental group

238

(4)

5 Compactly supported cohomology

242

(27)

5.1 Sheaves over non-Hausdorff manifolds

243

(6)

5.2 Compactly supported cohomology of etale groupoids

249

(5)

5.3 The operation φ!

254

(4)

5.4 Leray spectral sequence, and change-of-base

258

(6)

5.5 Homology of the embedding category

264

(5)

Bibliography

269

(4)

Part Four: Geometric methods in representation theory

273

(52)

1 Reductive Lie Groups: Definitions and Basic Properties

275

(7)

1.1 Basic Definitions and Examples

275

(1)

1.2 The Cartan Decomposition

276

(3)

1.3 Complexifications of Linear Groups

279

(3)

2 Compact Lie Groups

282

(12)

2.1 Maximal Tori, the Unit Lattice, and the Weight Lattice

282

(2)

2.2 Weights, Roots, and the Weyl Group

284

(2)

2.3 The Theorem of the Highest Weight

286

(3)

2.4 Borel Subalgebras and the Flag Variety

289

(2)

2.5 The Borel-Weil-Bott Theorem

291

(3)

3 Representations of Reductive Lie Groups

294

(11)

3.1 Continuity, Admissibility, KR-finite and Cinfinity Vectors

294

(4)

3.2 Harish-Chandra Modules

298

(7)

4 Geometric Constructions of Representations

305

(16)

4.1 The Universal Cartan Algebra and Infinitesimal Characters

306

(1)

4.2 Twisted D-modules

307

(4)

4.3 Construction of Harish-Chandra Modules

311

(3)

4.4 Construction of GR-representations

314

(3)

4.5 Matsuki Correspondence

317

(4)

Bibliography

321

(4)

Part Five: Deformation theory: a powerful tool in physics modelling

325

(30)

1 Introduction

327

(5)

1.1 It ain't necessarily so

327

(1)

1.2 Epistemological importance of deformation theory

328

(4)

2 Composite elementary particles in AdS microworld

332

(6)

2.1 A qualitative overview

333

(2)

2.2 A brief overview of singleton symmetry & field theory

335

(3)

3 Nonlinear covariant field equations

338

(2)

4 Quantisation is a deformation

340

(8)

4.1 The Gerstenhaber theory of deformations of algebras

340

(2)

4.2 The invention of deformation quantisation

342

(3)

4.3 Deformation quantisation and its developments

345

(3)

Bibliography

348

(7)

Index

355

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Poisson Geometry, Deformation Quantisation and Group Representations

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