Grundlehren dermathematischen Wissenschaften 347A Series of Comprehensive Studies in Mathematics

Series editors

M. Berger P. de la Harpe F. HirzebruchN.J. Hitchin L. Hörmander A. KupiainenG. Lebeau F.-H. Lin S. MoriB.C. Ngô M. Ratner D. SerreN.J.A. Sloane A.M. Vershik M. Waldschmidt

Editor-in-Chief

A. Chenciner J. Coates S.R.S. Varadhan

For further volumes:www.springer.com/series/138

Camille Laurent-Gengoux � Anne Pichereau �

Pol Vanhaecke

Poisson Structures

Camille Laurent-GengouxCNRS UMR 7122, Laboratoire

de MathématiquesUniversité de LorraineMetz, France

Anne PichereauCNRS UMR 5208, Institut

Camille JordanUniversité Jean MonnetSaint-Etienne, France

Pol VanhaeckeCNRS UMR 7348, Lab. Mathématiques

et ApplicationsUniversité de PoitiersFuturoscope Chasseneuil, France

ISSN 0072-7830 Grundlehren der mathematischen WissenschaftenISBN 978-3-642-31089-8 ISBN 978-3-642-31090-4 (eBook)DOI 10.1007/978-3-642-31090-4Springer Heidelberg New York Dordrecht London

Library of Congress Control Number: 2012947315

Mathematics Subject Classification: 17B63, 53D17, 17B80, 53D55, 17B62

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For Clio

Preface

Poisson structures naturally appear in very different forms and contexts. Symplec-tic manifolds, Lie algebras, singularity theory, r-matrices, for example, all lead to acertain type of Poisson structures, sharing several features, despite the distances be-tween the mathematics they originate from. This observation motivated us to bringthe different worlds in which Poisson structures live together in a single volume,providing a multitude of entrances to the book, and hence to the subject. Thus, theidea of the body of the book, Part II, was born.

It is well known, i.e., it is commonly agreed upon by the experts, that results andtechniques which are valid for one type of Poisson structure ought to apply, mutatismutandis, to other types of Poisson structures. When starting to write Part I of thebook, we soon realized that not everything could be derived from the general con-cept of a Poisson algebra, so we faced the challenge of presenting the concepts andthe results in detail, for the algebraic, algebraic-geometric and geometric contexts,without copy-pasting large parts of the text two or three times. But as the writingmoved on, both the algebraic and geometric contexts kept imposing themselves; fi-nally, each found its proper place, appearing as being complementary and dual tothe other, rather than a consequence or rephrasing, one of the other. It added, unex-pectedly, an extra dimension to the book.

It was pointed out by one of the referees that the main applications of Poissonstructures should be present in a book which has Poisson structures as its mainsubject. This was quite another challenge, giving rise to the third and final part ofthe book, Part III, undoubtedly an important addition.

Several years were necessary for this project, a big part was done at Poitiers,when the three of us were appointed to the “Laboratoire de Mathematiques et Ap-plications”. We were spending long hours together in what our colleagues calledThe Aquarium (for an explanation of the name, look up “poisson” in a French dic-tionary). When two of us moved away from Poitiers, we sometimes worked to-gether in other places, which always offered us a pleasant and stimulating workingatmosphere. Thus, we are happy to acknowledge the hospitality of our colleaguesfrom the mathematics departments at the universities of Poitiers, Antwerp, Coimbra,

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Saint-Etienne and at the CRM in Barcelona and the Max-Planck Institute in Bonn.Each one of us was partially supported by an ANR contract (TcChAm, DPSing andGIMP, respectively), which made it possible to keep working on a blackboard, ratherthan seeing each other on a computer screen.

We learned Poisson structures from our teachers, friends and collaborators: MarkAdler, Paulo Antunes, Pantelis Damianou, Rui Fernandes, Benoit Fresse, Eva Mi-randa, Joana Nunes da Costa, Marco Pedroni, Michael Penkava, Claude Roger,Pierre van Moerbeke, Yvette Kosmann-Schwarzbach, Herve Sabourin, MathieuStienon, Friedrich Wagemann, Alan Weinstein, Ping Xu and Marco Zambon. Themultitude of different points of view on the subject, which we learned from them,have given a quite specific flavor to the book. Even if a book cannot replace what onelearns through lectures, discussions and collaborations, it is our hope that this bookmay further transfer what we learned from them and that it invites other researchersto explore the subject of Poisson structures, which turns out to be as diverse and richas the colorful fauna and flora which inhabit the bottom of our oceans.

April 1, 2012 Camille Laurent-Gengoux, Anne Pichereau and Pol Vanhaecke

Contents

Part I Theoretical Background

1 Poisson Structures: Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1 Poisson Algebras and Their Morphisms . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.1 Poisson Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.2 Morphisms of Poisson Algebras . . . . . . . . . . . . . . . . . . . . . . . . 51.1.3 Hamiltonian Derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 Poisson Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2.1 Affine Poisson Varieties and Their Morphisms . . . . . . . . . . . . 81.2.2 The Poisson Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.2.3 The Rank of a Poisson Structure . . . . . . . . . . . . . . . . . . . . . . . . 13

1.3 Poisson Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.3.1 Bivector Fields on Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 161.3.2 Poisson Manifolds and Poisson Maps . . . . . . . . . . . . . . . . . . . 191.3.3 Local Structure: Weinstein’s Splitting Theorem . . . . . . . . . . . 231.3.4 Global Structure: The Symplectic Foliation . . . . . . . . . . . . . . 26

1.4 Poisson Structures on a Vector Space . . . . . . . . . . . . . . . . . . . . . . . . . . 321.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2 Poisson Structures: Basic Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . 372.1 The Tensor Product of Poisson Algebras and the Product

of Poisson Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.1.1 The Tensor Product of Poisson Algebras . . . . . . . . . . . . . . . . . 382.1.2 The Product of Poisson Manifolds . . . . . . . . . . . . . . . . . . . . . . 42

2.2 Poisson Ideals and Poisson Submanifolds . . . . . . . . . . . . . . . . . . . . . . 462.2.1 Poisson Ideals and Poisson Subvarieties . . . . . . . . . . . . . . . . . 462.2.2 Poisson Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.3 Real and Holomorphic Poisson Structures . . . . . . . . . . . . . . . . . . . . . . 52

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2.3.1 Real Poisson Structures Associated to HolomorphicPoisson Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.3.2 Holomorphic Poisson Structures on Smooth AffinePoisson Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.4 Other Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562.4.1 Field Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562.4.2 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572.4.3 Germification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3 Multi-Derivations and Kahler Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.1 Multi-Derivations and Multivector Fields . . . . . . . . . . . . . . . . . . . . . . . 64

3.1.1 Multi-Derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.1.2 Basic Operations on Multi-Derivations . . . . . . . . . . . . . . . . . . 653.1.3 Multivector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.2 Kahler Forms and Differential Forms . . . . . . . . . . . . . . . . . . . . . . . . . . 683.2.1 Kahler Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.2.2 Kahler Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703.2.3 Algebra and Coalgebra Structure . . . . . . . . . . . . . . . . . . . . . . . 713.2.4 The de Rham Differential and Cohomology . . . . . . . . . . . . . . 723.2.5 Differential Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.3 The Schouten Bracket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753.3.1 Internal Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763.3.2 The Schouten Bracket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793.3.3 The Algebraic Schouten Bracket . . . . . . . . . . . . . . . . . . . . . . . 843.3.4 The (Generalized) Lie Derivative . . . . . . . . . . . . . . . . . . . . . . . 85

3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4 Poisson (Co)Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.1 Lie Algebra and Poisson Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.1.1 Lie Algebra Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924.1.2 Poisson Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.2 Lie Algebra and Poisson Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . 974.2.1 Lie Algebra Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 984.2.2 Poisson Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.3 Operations in Homology and Cohomology . . . . . . . . . . . . . . . . . . . . . 1014.4 The Modular Class of a Poisson Manifold . . . . . . . . . . . . . . . . . . . . . . 102

4.4.1 The Modular Vector Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034.4.2 The Modular Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1044.4.3 The Divergence of the Poisson Bracket . . . . . . . . . . . . . . . . . . 1074.4.4 Unimodular Poisson Manifolds . . . . . . . . . . . . . . . . . . . . . . . . 109

4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1104.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

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5 Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135.1 Lie Groups and (Their) Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.1.1 Lie Groups and Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 1145.1.2 Lie Group Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165.1.3 Adjoint and Coadjoint Action . . . . . . . . . . . . . . . . . . . . . . . . . . 1195.1.4 Invariant Bilinear Forms and Invariant Functions . . . . . . . . . . 121

5.2 Poisson Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1235.2.1 Algebraic Poisson Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1235.2.2 Poisson Reduction for Poisson Manifolds . . . . . . . . . . . . . . . . 127

5.3 Poisson–Dirac Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1345.3.1 Algebraic Poisson–Dirac Reduction . . . . . . . . . . . . . . . . . . . . . 1345.3.2 Poisson–Dirac Reduction for Poisson Manifolds . . . . . . . . . . 1375.3.3 The Transverse Poisson Structure . . . . . . . . . . . . . . . . . . . . . . . 143

5.4 Poisson Structures and Group Actions . . . . . . . . . . . . . . . . . . . . . . . . . 1485.4.1 Poisson Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1485.4.2 Poisson Actions and Quotient Spaces . . . . . . . . . . . . . . . . . . . 1495.4.3 Fixed Point Sets as Poisson–Dirac Submanifolds . . . . . . . . . . 1505.4.4 Reduction with Respect to a Momentum Map . . . . . . . . . . . . 153

5.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1555.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

Part II Examples

6 Constant Poisson Structures, Regular and Symplectic Manifolds . . . . 1616.1 Constant Poisson Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1626.2 Regular Poisson Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1656.3 Symplectic Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

6.3.1 Symplectic Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1676.3.2 Symplectic Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1686.3.3 Symplectic Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1726.3.4 Quotients by Finite Groups of Symplectomorphisms . . . . . . . 1746.3.5 Example 1: Cotangent Bundles . . . . . . . . . . . . . . . . . . . . . . . . . 1756.3.6 Example 2: Kahler Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 177

6.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

7 Linear Poisson Structures and Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . 1797.1 The Lie–Poisson Structure on g∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1807.2 The Lie–Poisson Structure on g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1827.3 Properties of the Lie–Poisson Structure . . . . . . . . . . . . . . . . . . . . . . . . 185

7.3.1 The Symplectic Foliation: Coadjoint Orbits . . . . . . . . . . . . . . 1857.3.2 The Cohomology of Lie–Poisson Structures . . . . . . . . . . . . . . 1907.3.3 The Modular Class of a Lie–Poisson Structure . . . . . . . . . . . . 192

7.4 Affine Poisson Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1937.5 The Linearization of Poisson Structures . . . . . . . . . . . . . . . . . . . . . . . . 1967.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

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8 Higher Degree Poisson Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2058.1 Polynomial and (Weight) Homogeneous Poisson Structures . . . . . . . 206

8.1.1 Polynomial Poisson Structures . . . . . . . . . . . . . . . . . . . . . . . . . 2068.1.2 Homogeneous Poisson Structures . . . . . . . . . . . . . . . . . . . . . . . 2078.1.3 Weight Homogeneous Poisson Structures . . . . . . . . . . . . . . . . 211

8.2 Quadratic Poisson Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2138.2.1 Multiplicative Poisson Structures . . . . . . . . . . . . . . . . . . . . . . . 2158.2.2 The Modular Vector Field of a Quadratic Poisson Structure . 218

8.3 Nambu–Poisson Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2228.4 Transverse Poisson Structures to Adjoint Orbits . . . . . . . . . . . . . . . . . 227

8.4.1 S-triples in Semi-Simple Lie Algebras . . . . . . . . . . . . . . . . . . . 2278.4.2 Weights Associated to an S-Triple . . . . . . . . . . . . . . . . . . . . . . 2288.4.3 Transverse Poisson Structures to Nilpotent Orbits . . . . . . . . . 229

8.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

9 Poisson Structures in Dimensions Two and Three . . . . . . . . . . . . . . . . . . 2339.1 Poisson Structures in Dimension Two . . . . . . . . . . . . . . . . . . . . . . . . . . 234

9.1.1 Global Point of View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2349.1.2 Local Point of View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2359.1.3 Classification in Dimension Two . . . . . . . . . . . . . . . . . . . . . . . 237

9.2 Poisson Structures in Dimension Three . . . . . . . . . . . . . . . . . . . . . . . . 2509.2.1 Poisson Structures on F

3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2509.2.2 Poisson Manifolds of Dimension Three . . . . . . . . . . . . . . . . . . 2549.2.3 Quadratic Poisson Structures on F

3 . . . . . . . . . . . . . . . . . . . . . 2589.2.4 Poisson Surfaces in C

3 and Du Val Singularities . . . . . . . . . . 2639.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

10 R-Brackets and r-Brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26910.1 Linear R-Brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

10.1.1 R-Matrices and the Yang–Baxter Equation . . . . . . . . . . . . . . . 27010.1.2 R-Matrices and Lie Algebra Splittings . . . . . . . . . . . . . . . . . . . 27210.1.3 Linear R-Brackets on g∗ and on g . . . . . . . . . . . . . . . . . . . . . . . 273

10.2 Linear r-Brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27510.2.1 Coboundary Lie Bialgebras and r-Matrices . . . . . . . . . . . . . . . 27510.2.2 Linear r-Brackets on g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27810.2.3 From r-Matrices to R-Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 280

10.3 R-Brackets and r-Brackets of Higher Degree . . . . . . . . . . . . . . . . . . . . 28310.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

11 Poisson–Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29111.1 Multiplicative Poisson Structures and Poisson–Lie Groups . . . . . . . . 292

11.1.1 The Condition of Multiplicativity . . . . . . . . . . . . . . . . . . . . . . . 29211.1.2 Basic Properties of Poisson–Lie Groups . . . . . . . . . . . . . . . . . 29411.1.3 Poisson–Lie Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29611.1.4 Linear Multiplicative Poisson Structures . . . . . . . . . . . . . . . . . 297

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11.1.5 Coboundary Poisson–Lie Groups . . . . . . . . . . . . . . . . . . . . . . . 29811.1.6 Multiplicative Poisson Structures on Vector Spaces . . . . . . . . 299

11.2 Lie Bialgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30211.2.1 Lie Bialgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30311.2.2 Lie Sub-bialgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30611.2.3 Duality for Lie Bialgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30711.2.4 Coboundary Lie Bialgebras and r-Matrices . . . . . . . . . . . . . . . 30811.2.5 Manin Triples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31011.2.6 Lie Bialgebras and Poisson Structures . . . . . . . . . . . . . . . . . . . 313

11.3 Poisson–Lie Groups and Lie Bialgebras . . . . . . . . . . . . . . . . . . . . . . . . 31411.3.1 The Lie Bialgebra of a Poisson–Lie Group . . . . . . . . . . . . . . . 31411.3.2 Coboundary Poisson–Lie Groups and Lie Bialgebras . . . . . . 31611.3.3 The Integration of Lie Bialgebras . . . . . . . . . . . . . . . . . . . . . . . 318

11.4 Dressing Actions and Symplectic Leaves . . . . . . . . . . . . . . . . . . . . . . . 32111.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

Part III Applications

12 Liouville Integrable Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32912.1 Functions in Involution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33012.2 Constructions of Functions in Involution . . . . . . . . . . . . . . . . . . . . . . . 332

12.2.1 Poisson’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33212.2.2 Noether’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33312.2.3 Bi-Hamiltonian Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 33312.2.4 Poisson Maps and Thimm’s Method . . . . . . . . . . . . . . . . . . . . 33412.2.5 Lax Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33612.2.6 The Adler–Kostant–Symes Theorem . . . . . . . . . . . . . . . . . . . . 337

12.3 The Liouville Theorem and the Action-Angle Theorem . . . . . . . . . . . 34112.3.1 Liouville Integrable Systems and Liouville’s Theorem . . . . . 34112.3.2 Foliation by Standard Liouville Tori . . . . . . . . . . . . . . . . . . . . 34312.3.3 Period Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34512.3.4 The Existence of Action-Angle Coordinates . . . . . . . . . . . . . . 346

12.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350

13 Deformation Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35313.1 Deformations of Associative Products . . . . . . . . . . . . . . . . . . . . . . . . . 354

13.1.1 Formal and k-th Order Deformations . . . . . . . . . . . . . . . . . . . . 35413.1.2 Hochschild Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35713.1.3 Deformations and Cohomology . . . . . . . . . . . . . . . . . . . . . . . . 35913.1.4 The Maurer–Cartan Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 36113.1.5 Star Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36213.1.6 A Star Product for Constant Poisson Structures . . . . . . . . . . . 36313.1.7 A Star Product for Symplectic Manifolds . . . . . . . . . . . . . . . . 365

13.2 Deformations of Poisson Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 36713.2.1 Formal and k-th Order Deformations . . . . . . . . . . . . . . . . . . . . 367

xiv Contents

13.2.2 Deformations and Cohomology . . . . . . . . . . . . . . . . . . . . . . . . 36813.2.3 The Maurer–Cartan Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 371

13.3 Differential Graded Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37113.3.1 The Symmetric Algebra of a Graded Vector Space . . . . . . . . 37113.3.2 Differential Graded Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . 37213.3.3 The Maurer–Cartan Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 37613.3.4 Gauge Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37813.3.5 Path Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38013.3.6 Applications to Deformation Theory . . . . . . . . . . . . . . . . . . . . 385

13.4 L∞-Morphisms of Differential Graded Lie Algebras . . . . . . . . . . . . . . 38613.4.1 ΩΦ is Well-Defined . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38913.4.2 Surjectivity of ΩΦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39213.4.3 Injectivity of ΩΦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395

13.5 Kontsevich’s Formality Theorem and Its Consequences . . . . . . . . . . . 39713.5.1 Kontsevich’s Formality Theorem . . . . . . . . . . . . . . . . . . . . . . . 39813.5.2 Kontsevich’s Formality Theorem for Rd . . . . . . . . . . . . . . . . . 39913.5.3 A Few Consequences of Kontsevich’s Formality Theorem . . 406

13.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408

A Multilinear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411A.1 Tensor Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411A.2 Exterior and Symmetric Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415A.3 Algebras and Graded Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418A.4 Coalgebras and Graded Coalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422A.5 Graded Derivations and Coderivations . . . . . . . . . . . . . . . . . . . . . . . . . 425

B Real and Complex Differential Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 427B.1 Real and Complex Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427B.2 The Tangent Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429B.3 Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433B.4 The Flow of a Vector Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435B.5 The Frobenius Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449

List of Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459

Introduction

Poisson structures appear in a large variety of different contexts, ranging from stringtheory, classical/quantum mechanics and differential geometry to abstract algebra,algebraic geometry and representation theory. In each one of these contexts, it turnsout that the Poisson structure is not a theoretical artifact, but a key element which,unsolicited, comes along with the problem which is investigated and its delicateproperties are in basically all cases decisive for the solution to the problem.

Hamiltonian mechanics and integrable systems. A first striking example of thisphenomenon appears in classical mechanics. Poisson’s classical bracket, which isgiven for smooth functions F and G on R

2r by

{F,G} :=r

∑i=1

(∂F∂qi

∂G∂ pi

− ∂G∂qi

∂F∂ pi

), (0.1)

allows one to write Hamilton’s equations of motion

qi =∂H∂ pi

, pi =−∂H∂qi

, i = 1, . . . ,r , (0.2)

associated to a Hamiltonian H = H(q, p), in a coordinate-free manner, treating po-sitions qi and momenta pi on an equal footing. Indeed, using (0.1), the equations ofmotion (0.2) can be written as

qi = {qi,H} , pi = {pi,H} , i = 1, . . . ,r . (0.3)

For every solution t �→ (q(t), p(t)) to these differential equations, it follows from(0.1) and (0.3) that

ddt

F(q(t), p(t)) = {F,H}(q(t), p(t)) , (0.4)

for all smooth functions F on R2r.

xv

xvi Introduction

In differential geometrical terms, the Poisson bracket defines an operator whichassociates to every smooth function H on R

2r a vector field XH on R2r. In terms

of local coordinates, the vector field XH is given by (0.3) and the variation of asmooth function F on R

2r along any integral curve (q(t), p(t)) of the vector fieldXH is given by (0.4). In fact, when the function H is the Hamiltonian (the energy)of the system, the integral curves of XH describe the time evolution of the system.Thus, the equations of motion of classical mechanics come along with an associatedPoisson bracket; for simple systems, such as systems of particles with a classicalinteraction, the Poisson structure is the above one, but for constrained and reducedsystems, the Poisson bracket takes a more complicated form.

The role which the Poisson structure plays in solving the problems of classi-cal mechanics was known partly to Poisson, who pointed out that the vanishing of{F,H} and {G,H} implies the vanishing of {{F,G} ,H}. In mechanical terms, thismeans in view of (0.4) that the Poisson bracket of two constants of motion is again aconstant of motion. Since constants of motion are a great help in the explicit integra-tion of Hamilton’s equations, Poisson’s theorem brings to light the Poisson bracketas a tool for integration.

A strong amplification of this fact is the Liouville theorem, which states that rindependent constants of motion H1 = H,H2, . . . ,Hr of (0.3) suffice for integrat-ing Hamilton’s equations by quadratures, under the hypothesis that these functionscommute for the Poisson bracket (are in involution). A yet further amplification isthe action-angle theorem, which states that in the neighborhood of a compact sub-manifold, traced out by the flows of the (commuting!) vector fields XHi , there exist(S1)r ×R

r-valued coordinates (θ1, . . . ,θr,ρ1, . . . ,ρr) in which the Poisson brackettakes the canonical form (0.1), namely

{F,G}=r

∑i=1

(∂F∂θi

∂G∂ρi

− ∂G∂θi

∂F∂ρi

),

and such that the functions H1, . . . ,Hr (including the Hamiltonian H) depend on thecoordinates ρ1, . . . ,ρr only. According to (0.2), Hamilton’s equations take in thesecoordinates the simple form

θi =∂H∂ρi

, ρi =−∂H∂θi

= 0 , i = 1, . . . ,r ,

which entails that the ρi are constant and hence that the θi are affine functions oftime (since H does not depend on θ1, . . . ,θr).

Thus, in a nutshell, the equations of motion of classical mechanics come with aPoisson bracket, the Poisson bracket is decisive for their integrability and it permitsus to construct coordinates in which the problem (including the Poisson bracket)takes a very simple form, from which the solutions and their characteristics can beread off at once.

Introduction xvii

Abstraction and generalization. Before giving another example of the key roleplayed by Poisson brackets, we explain the algebraic and geometric abstraction ofthe classical Poisson bracket (0.1). It is well known that Poisson’s theorem is ex-plained by the Jacobi identity

{{F,G} ,H}+{{G,H} ,F}+{{H,F} ,G}= 0 , (0.5)

valid for arbitrary smooth functions F,G and H on R2r. It is however noteworthy

to point out that Jacobi discovered this identity only thirty years after Poisson an-nounced his theorem. Taking arbitrary smooth functions xi j = −x ji defined on anon-empty open subset of Rd , with coordinates x1, . . . ,xd , the skew-symmetric bi-linear operation {· , ·}, defined on smooth functions by

{F,G} :=d

∑i, j=1

xi j∂F∂xi

∂G∂x j

, (0.6)

satisfies the Jacobi identity if and only if the functions xi j satisfy the identity

d

∑�=1

(x�k

∂xi j

∂x�+ x�i

∂x jk

∂x�+ x� j

∂xki

∂x�

)= 0 .

This fact was observed by Lie, who also pointed out that under the assumption ofconstancy of the rank (which he assumes implicitly), there exist local coordinatesq1, . . . ,qr, p1, . . . , pr,z1, . . . ,zs, where d = 2r+ s, such that (0.6) takes the form

{F,G}=r

∑i=1

(∂F∂qi

∂G∂ pi

− ∂G∂qi

∂F∂ pi

),

which is formally the same as (0.1), but on a d = 2r+s dimensional space, with s ofthe variables being totally absent from the bracket. An important class of examplesof brackets (0.6), satisfying (0.5), is defined on the dual g∗ of a (finite-dimensional)Lie algebra g, where the bracket is directly inherited by the Lie bracket. Many im-portant facts about the Poisson bracket on g∗, such as the relation to the coadjointrepresentation of the adjoint group G of g, were observed by Lie and were rediscov-ered and further expanded by Berezin [21], Kirillov [105, 106], Kostant [117] andSouriau [185].

It was pointed out by Lichnerowicz in [126] that, in abstract differential geomet-rical terms, a Poisson structure on a smooth manifold M is a smooth bivector field πon M, satisfying [π,π]S = 0, where [· , ·]S stands for the Schouten bracket (a naturalextension of the Lie bracket of vector fields to arbitrary multivector fields). It im-plies that for every open subset U of M, the bilinear operation {· , ·}U , defined forF,G ∈C∞(U) by

{F,G}U (m) := 〈dmF ∧dmG,πm〉 , (0.7)

for all m ∈ M, satisfies the Jacobi identity, hence defines a Lie algebra structureon C∞(U).

xviii Introduction

Moreover, this Lie algebra structure on C∞(U) and the associative structureon C∞(U) (pointwise multiplication of functions) are compatible in the sense that

{FG,H}U = F {G,H}U +G{F,H}U ,

for all F,G,H ∈C∞(U).

It is from here easy to make the algebraic abstraction of the notion of a Poissonstructure, leading to the notion of a Poisson algebra. For concreteness, we give thedefinition in the case of a vector space A over a field F of characteristic zero (onemay think of R or C). Two products (bilinear maps from A to itself) are assumedto be given on A , one denoted by “·”, which is assumed to be associative and com-mutative, while the other one, denoted by {· , ·}, is assumed to be a Lie bracket. Thetriple (A , ·,{· , ·}) is called a Poisson algebra if the two products are compatible inthe sense that

{F ·G,H}= F · {G,H}+G · {F,H} , (0.8)

for all F,G,H ∈ A . It is clear that in the case of a Poisson manifold (M,π), foreach open subset U of M, the triple (C∞(U), ·,{· , ·}U ) is a Poisson algebra, where ·stands for the pointwise product of functions and {· , ·}U stands for the Lie bracketdefined by (0.7).

Deformation theory and quantization. We now come to a second striking exam-ple of Poisson structures making an unexpected appearance, and playing next a verydefinite role in the statement of the problem and in its final solution. We give a purelyalgebraic, but very simple and natural, introduction to the problem. In this approach,one speaks of “deformation theory”. A physical interpretation of the problem willbe given afterwards; one then speaks of “quantization” or of “deformation quanti-zation”.

We start with a commutative associative algebra A = (A , ·). The reader maythink of the example of the polynomial algebra A = F[x1, . . . ,xd ]. We suppose thatthe product in A is part of a family of associative (not necessarily commutative)products, say we have a family �t of bilinear maps

�t : A ×A → A(a,b) �→ a�t b

(0.9)

such that, for each fixed t ∈F, the product �t is associative, with �0 being the originalassociative commutative product on A , i.e., (A , ·)= (A ,�0). We refer to the familyof associative products �t on A as a deformation of A . It is assumed that the familydepends nicely on t, in a way which we do not make precise here. We consider, forevery t ∈ F, the commutator of �t , which is defined for all a,b ∈ A by

[a,b]t := a�t b−b�t a .

Introduction xix

Since �t is associative, the commutator [· , ·]t satisfies for each t ∈ F the followingidentities, valid for all a,b,c ∈ A :

[a�t b,c]t = a�t [b,c]t +[a,c]t �t b ,

[[a,b]t ,c]t +[[b,c]t ,a]t +[[c,a]t ,b]t = 0 .(0.10)

Consider the skew-symmetric bilinear map {· , ·} : A ×A → A , which is definedfor a,b ∈ A by

{a,b} :=ddt |t=0

[a,b]t =ddt |t=0

(a�t b−b�t a) .

Then the first equation in (0.10) implies that {· , ·} satisfies the Leibniz prop-erty (0.8), while the second equation in (0.10) implies that {· , ·} satisfies the Jacobiidentity. Combined, it means that (A , ·,{· , ·}) is a Poisson algebra. Thus, a fam-ily of deformations of a given commutative associative algebra leads naturally to aPoisson bracket on this algebra! In the particular case of the associative algebras ofoperators of quantum mechanics, in which t is a multiple of Planck’s constant h, theoperators become in the limit h → 0 commuting classical variables and the aboveprocedure yields a Poisson bracket on the algebra of classical variables.

The main problem of deformation theory is to invert this procedure; in the contextof physics, this is called the quantization problem or, more precisely, the problemof quantization by deformation. In its algebraic version, the first question is whethergiven a Poisson algebra (A , ·,{· , ·}) there exists a family of associative products �t

on A , such that (A , ·) = (A ,�0) and such that {· , ·}, obtained from �t by the aboveprocedure, is a Poisson bracket on A . The second question is to classify all such de-formations, for a given (A , ·,{· , ·}). The physical interpretation of these questionsis that, starting from the algebra of classical variables, which inherits from phasespace a Poisson bracket, as explained above in the context of mechanical systems,one wishes to (re-)construct the corresponding algebra of quantum mechanical op-erators.

A mathematical simplification of this problem is to ask whether the above pro-cedure can be inverted formally. Stated in a precise way, this means the following.Let (A , ·,{· , ·}) be a Poisson algebra and consider the vector space A [[ν]] of for-mal power series in one variable ν , whose coefficients belong to A . Let μ� denotean associative product on A [[ν]], i.e., an F[[ν]]-bilinear map from A [[ν]] to itself,which is associative. Given a,b ∈ A , we can write

μ�(a,b) = ∑k∈N

μk(a,b)νk ,

which defines bilinear maps μi : A ×A →A for i∈N. By definition, μ� is a formaldeformation of (A , ·,{· , ·}) if μ0(a,b) = a · b and μ1(a,b)− μ1(b,a) = {a,b}, forall a,b ∈ A . The main question, stated formally, is then if every Poisson algebra(A , ·,{· , ·}) admits a formal deformation μ�.

xx Introduction

Consider for exampleR2r, with coordinates (x1, . . . ,x2r)= (q1, . . . ,qr, p1, . . . , pr).The Moyal product is the R[[ν]]-bilinear product on C∞(R2r)[[ν]], which is given forF,G ∈C∞(R2r) by

μ�(F,G) := ∑k∈N

∑1�i1, j1,...,ik, jk�2r

Ji1, j1 . . .Jik, jk∂ kF

∂xi1 . . .∂xir

∂ kG∂x j1 . . .∂x jr

νk

k!,

where J is the skew-symmetric matrix of size 2r, given by J :=

(0 1r

−1r 0

). The

Moyal product, also called the Moyal–Weyl product, defines an associative producton C∞(R2r)[[ν]], whose leading terms satisfy μ0(F,G) = FG and

μ1(F,G)−μ1(G,F) =r

∑j=1

(∂F∂q j

∂G∂ p j

− ∂G∂q j

∂F∂ p j

),

where the latter right-hand side is the Poisson bracket (0.1). This shows that theclassical Poisson bracket admits a formal deformation.

It was shown by De Wilde and Lecomte in [57] that the Poisson algebra of anarbitrary symplectic manifold (regular Poisson manifolds of maximal rank) admitsa formal deformation. A geometrical proof of their result, which also works in thecase of arbitrary regular Poisson manifolds, was given by Fedosov [73]. Finally, itwas Kontsevich [107] who proved, using ideas which come from string theory, thatthe algebra of functions on any Poisson manifold admits a formal deformation. Heproved in fact a quite stronger result, which says that for a given Poisson manifold(M,π), the equivalence classes of formal deformations of the algebra of smoothfunctions on M are classified by the equivalence classes of formal deformations ofthe Poisson structure π .

Summarizing, when one deforms a commutative associative algebra, a Poissonstructure shows up and it plays a dominant role in the entire deformation process.

The examples. The main idea which led us to the writing of this book, and to its ac-tual structure, is that Poisson structures come in big classes (families) where roughlyspeaking each class has its own tools and is related to a very definite part of mathe-matics, hence leading to specific questions. As a consequence, the main part of thebook (about half of it) is Part II, dedicated to six classes of examples. They maketheir appearance in six different chapters, namely Chapters 6–11, as follows.

• Constant Poisson structures, such as Poisson’s original bracket (0.1), are accord-ing to the Darboux theorem the model (normal form) for Poisson structures on realor complex manifolds, in the neighborhood of any point where the rank is locallyconstant. Regular Poisson manifolds and symplectic manifolds (such as cotangentbundles and Kahler manifolds) are the main examples, which exhibit even in theabsence of singularities some of the main phenomena in Poisson geometry, whichdistinguish it from Riemannian geometry and show its pertinence for classical andquantum mechanics. Constant and regular Poisson structures are studied in Chap-ter 6.

Introduction xxi

• Linear Poisson structures are in one-to-one correspondence with Lie algebras andmany features of Lie algebras are most naturally approached in terms of the canon-ical Poisson bracket on the dual of a (finite-dimensional) Lie algebra, the so-calledLie–Poisson structure. The coadjoint orbits, for example, of a Lie algebra are thesymplectic leaves of the Lie–Poisson structure, showing on the one hand that theyare even dimensional, and on the other hand that they carry a canonical symplecticstructure, the Kostant–Kirillov–Souriau symplectic structure. Linear Poisson struc-tures appear also on an arbitrary Poisson manifold (M,π) at any point x where thePoisson structure vanishes: the tangent space TxM inherits a linear Poisson struc-ture, the linearization of π at x. It leads to the linearization problem, which inquiresif π and its linearization at x are isomorphic, at least on a neighborhood of x. LinearPoisson structures and their relation to Lie algebras are the subject of Chapter 7.

• Many Poisson structures of interest are neither constant nor linear (or affine). Infact, while the basic properties of the latter Poisson structures can be traced back to(known) properties of bilinear forms and of Lie algebras, new phenomena appearwhen considering quadratic Poisson structures (i.e., homogeneous Poisson struc-tures of degree two) and higher degree Poisson structures, which often turn out tobe weight homogeneous. A prime example of this is the transverse Poisson struc-ture to an adjoint orbit in a semi-simple Lie algebra. Moreover, this Poisson structurearises in the case of the subregular orbit from a Nambu–Poisson structure, definedby the invariant functions of the Lie algebra. Higher degree Poisson structures arestudied in Chapter 8.

• In dimension two, every bivector field is a Poisson structure, yet many questionsabout Poisson structures on surfaces are non-trivial. For example, their local clas-sification has up to now only been accomplished under quite strong regularity as-sumptions on the singular locus of the Poisson structure. In dimension three, theJacobi identity can be stated as an integrability condition of a distribution (or ofa one-form), which eventually leads to the symplectic foliation. A surface in C

3

which is defined by the zero locus of a polynomial ϕ inherits a Poisson structurefrom the standard Nambu–Poisson structure on C

3, with ϕ as Casimir. In the caseof singular surfaces C

2/G, where G is a finite subgroup of SL2(C), this Poissonstructure coincides with the Poisson structure obtained from the canonical symplec-tic structure on C

2 by reduction. Thus, rather than being trivial, Poisson structures indimensions two and three form a rich playground on which a variety of phenomenacan be observed. Poisson structures in dimensions two and three are discussed inChapter 9.

• In a Lie-theoretical context, a linear Poisson structure, different from the canoni-cal Lie–Poisson structure, often pops up. This Poisson structure appears on the Liealgebra g at hand, so that the dual g∗ of g comes equipped with a Lie algebra struc-ture, or it appears on g∗, so that g is equipped with a second Lie algebra structure.The underlying operator, which relates either of these Lie structures with the orig-inal Lie bracket on g is in the first case an element r ∈ g⊗ g, called an r-matrix,while it is in the second case a (vector space) endomorphism R of g, called an R-matrix. These structures appeared first in the theory of integrable systems, but are

xxii Introduction

nowadays equally important in the theory of Lie–Poisson groups (see the next item)and of quantum groups. In the context of Lie algebras which come from associativealgebras, r-matrices and R-matrices also lead to a quadratic Poisson structure onthe Lie algebra, which plays an important role in the theory of Lie–Poisson groups;they also lead to a cubic Poisson structure, whose virtue seems at this point still verymysterious. Poisson structures coming from r-matrices or R-matrices are the subjectof Chapter 10.

• A Poisson–Lie group is a Lie group G, equipped with a Poisson structure π forwhich the group multiplication G×G → G is a Poisson map. The Poisson struc-ture π leads to a Lie algebra structure on the dual of the Lie algebra g of G, making g

into a Lie bialgebra, a structure generalizing the structure which comes from an r-matrix. Conversely, every finite-dimensional Lie bialgebra is obtained in this wayfrom a Poisson–Lie group, a result which extends Lie’s third theorem (every finite-dimensional Lie algebra is the Lie algebra of a Lie group). Poisson–Lie groups havemany applications, for example in the description of the Schubert cells of a Liegroup. Poisson–Lie groups are discussed in Chapter 11.

Theoretical foundations. The examples (Chapters 6–11) are preceded by a firstpart (5 chapters) with a systematic exposition of the general theory of Poisson struc-tures. We used two principles in the writing of these chapters. The first one is that wewanted to develop both the algebraic and geometric points of view of the theory. Infact, there has for the last decade been an increasing interest in the algebraic aspectsof Poisson structures, while of course geometrical intuition underlies all construc-tions and the vast majority of examples. In the algebraic context, we have a Poissonalgebra, whose underlying space is an F-vector space, where F is an arbitrary fieldof characteristic zero. In the geometric context, we have a Poisson manifold, wherethe manifold is either real or complex. In between the algebraic and geometric con-texts, Poisson varieties appear, whose underlying object can both be viewed as avariety or as a finitely generated algebra. Throughout this first part, the reader willexperience that although most of the results are formally very similar in the alge-braic and in the geometric setting, their concrete implementation (including proofsof the propositions) is quite different and needs to be worked out in detail in bothsettings.

The second principle which we used was to keep the prerequisites in (commu-tative and Lie) algebra and in (differential and algebraic) geometry to a minimum.Thus, for example, when we describe Poisson structures geometrically as bivectorfields, we explain the notion of a bivector field by first recalling the notion of a vec-tor field, rather than simply saying that a bivector field is a section of the exteriorsquare of the tangent bundle to the manifold. Similarly, while the basic facts aboutLie groups and Lie algebras are supposed to be known, the interplay between Liegroups and Lie algebras is recalled in detail, as it is further amplified in the presenceof Poisson structures. We added an appendix at the end of the book with some factson multilinear algebra (tensor products, wedges, algebra and coalgebra structures)and an appendix which recalls the basic facts about differential geometry. The ob-

Introduction xxiii

jects and properties which are recalled in these appendices are used throughout thebook.

Chapters 1 and 2 contain the basic definitions and constructions. A good famil-iarity with these chapters should suffice for reading a good part of any other chapterof the book. Chapter 3 deals with multi-derivations and their geometrical analog,multivector fields, and with Kahler forms, which are the algebraic analogs of dif-ferential forms. Chapter 4 deals with Poisson cohomology and Chapter 5 is devotedto reduction in the context of Poisson structures. At the end of each of these fivechapters, we give a list of exercises, so that the reader can test his understanding ofthe theory.

Applications. The third part of the book contains the two major applications, whichwe discussed above: integrable systems are discussed in Chapter 12, while de-formation quantization is detailed in Chapter 13. They are however not the onlyapplications which are given in the book. In fact, we end each example chapter(Chapters 6–11) with an application of the class of Poisson structures, studied inthat chapter.

What is absent from this book. The main topic about Poisson structures which isabsent from this book is what should be called “Poisson geometry”. By this we meanthe global geometry of Poisson structures, which involves the integration of Poissonbrackets, Poisson connections, stability of symplectic leaves and related topics. Forthis, we refer to [51, 53, 74].

Equally absent from this book are the many generalizations of Poisson structures.Quasi-Poisson structures and Poisson structures with background are not only math-ematically speaking very interesting, see [112], they have in addition non-trivial ap-plications in physics, see [119, 162, 176], and are themselves a particular case ofDirac structures [48, 89]. The theory of quantum groups, initiated in the seminalICM talk [59] by Drinfel’d, has several connections with the topics which are de-veloped in this book, but adding several chapters to the book would not have beensufficient to give a fair account of this subject; we simply refer to the excellent books[40, 103].

Equally absent from this book is the theory of Lie algebroids, introduced byPradines [171], which are both a particular case and a generalization of Poissonmanifolds. In particular, after the pioneering work of [47], the problem of the in-tegration of Poisson manifolds, which claims that symplectic Lie groupoids are toPoisson manifolds what Lie groups are to Lie algebras, has gained a long attention.For a definite solution of this problem, given by Cranic and Fernandes, see [49, 50].

The notes and the references. When writing the history of the theory of Poissonbrackets, giving each of the main results in the theory and examples the “right”credit is a challenge which is beyond the scope of this book. We claim no originalityin this book, but since none of the proofs which are given are copy-and-paste proofsfrom existing proofs, we did not think it was necessary to cite the proof in the lit-erature which is closest to the proof that we give. Also, many properties of Poisson

xxiv Introduction

structures, and many examples, have been through a long series of transformationsand generalizations, think for example of Weinstein’s splitting theorem or the Pois-son and Poisson–Dirac reduction theorems; structuring the long list of importantintermediate results which led to the final results is certainly interesting from theepistemological point of view, but this was not our main focus when writing thebook.

However, at the end of each chapter, we have put a section called “Notes”, inwhich we indicate some references, including the main references which we knowof, for further reading on what has been treated in the chapter. These notes are alsoused to situate, a posteriori, the treated subjects in the scientific literature and to tracesome of their historical developments. Finally, the notes are also used to indicatesome further connections to other fields of mathematics or physics. Since Poissonstructures are a very active field of research, it is clear that the list of connectionswith other fields is not exhaustive.