Lectures on Poisson Geometry

Marius Crainic Rui Loja Fernandes

Ioan M˘arcut,

Mathematical Institute, Utrecht University E-mail address: [email protected]

Department of Mathematics, University of Illinois at Urbana- Champaign E-mail address: [email protected]

Institute for Mathematics, AstrophysicsMarch 11, and 2021 Particle Physics, Radboud University Nijmegen E-mail address: [email protected]

Draft March 11, 2021

Draft To our families

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Draft Contents

Preface xiii

List of notations and symbols xvii

List of conventions xix

Part 1. Basic Concepts Chapter 1. Poisson BracketsMarch 11, 2021 3 §1.1. Poisson brackets 3 §1.2. Orbits 5 §1.3. Poisson and Hamiltonian diffeomorphisms 6 §1.4. Examples 8 §1.5. PoissonDraft actions and quotients 14 Problems 19

Chapter 2. Poisson Bivectors 23 §2.1. The point of view of bivectors 23 §2.2. A slight twist: π] 28 §2.3. Poisson maps and bivector fields 30 §2.4. Examples 31 Problems 40

Chapter 3. Local Structure of Poisson Manifolds 43 §3.1. The Weinstein Splitting Theorem 43 §3.2. Regular points 46

vii viii Contents

§3.3. Singular points 47 §3.4. The isotropy Lie algebra 50 §3.5. Linearization of Poisson structures 53 Problems 55 Notes and references for Part 1 57

Part 2. Poisson Geometry around Leaves Chapter 4. Symplectic Leaves and the Symplectic Foliation 61 §4.1. The symplectic foliation 61 §4.2. Regular Poisson structures 67 §4.3. More examples of symplectic foliations 71 §4.4. The coupling construction 74 Problems 79 Chapter 5. Poisson Transversals 83 §5.1. Slices and Poisson transversals 83 §5.2. The transverse Poisson structure to a leaf 89 §5.3. Poisson maps and Poisson transversalsMarch 11, 2021 93 Problems 96 Chapter 6. Symplectic Realizations 99 §6.1. Definition 99 §6.2. Examples 101 §6.3. SymplecticDraft realizations of linear Poisson structures 109 §6.4. Libermann’s Theorem and dual pairs 115 §6.5. Local existence 122 Problems 126 Chapter 7. Dirac Geometry 129 §7.1. Constant Dirac structures 130 §7.2. Dirac structures 133 §7.3. Pullbacks of Dirac structures 138 §7.4. Pushforwards of Dirac structures 141 §7.5. Gauge equivalences 144 Problems 148 Chapter 8. Submanifolds in Poisson Geometry 153 §8.1. Poisson submanifolds 153 Contents ix

§8.2. Poisson-Dirac submanifolds 161 §8.3. Coregular Poisson-Dirac submanifolds 165 §8.4. Coisotropic submanifolds 168 §8.5. Example: fixed point sets 176 §8.6. Pre-Poisson submanifolds 179 Problems 184 Notes and references for Part 2 187

Part 3. Global Aspects Chapter 9. Poisson Cohomology 191 §9.1. The cotangent Lie algebroid 191 §9.2. The Poisson differential and Poisson cohomology 193 §9.3. Low degrees 195 §9.4. Shadows of Poisson cohomology 200 §9.5. The cohomological obstruction to linearization 207 Problems 212 March 11, 2021 Chapter 10. Poisson Homotopy 217 §10.1. Cotangent paths 217 §10.2. Cotangent maps 219 §10.3. Integration and the contravariant Stokes’ Theorem 222 §10.4. Cotangent path-homotopy 225 §10.5. PoissonDraft homotopy and homology groups 231 §10.6. Variation of symplectic area 238 Problems 243 Chapter 11. Contravariant Geometry and Connections 247 §11.1. Contravariant connections on vector bundles 247 §11.2. Parallel transport along cotangent paths 250 §11.3. Flat contravariant connections 254 §11.4. Geodesics for contravariant connections 259 §11.5. Existence of symplectic realizations 263 Problems 267 Notes and references for Part 3 271

Part 4. Symplectic Groupoids x Contents

Chapter 12. Complete Symplectic Realizations 277 §12.1. The infinitesimal action 278 §12.2. Case study: linear Poisson structures 280 §12.3. Case study: the zero Poisson structure 282 §12.4. Case study: non-degenerate Poisson structures 283 §12.5. Completeness 286 §12.6. The Poisson homotopy groupoid 293 §12.7. Lagrangian fibrations 296 Problems 303 Chapter 13. A Crash Course on Lie Groupoids 307 §13.1. Lie groupoids 307 §13.2. Lie groupoids: examples and basic constructions 310 §13.3. The Lie algebroid of a Lie groupoid 320 §13.4. Lie algebroids: examples and basic constructions 324 §13.5. Duals of Lie algebroids 333 §13.6. The Lie philosophy 339 §13.7. The non-Hausdorff settingMarch 11, 2021 342 Problems 351 Chapter 14. Symplectic Groupoids 355 §14.1. Symplectic groupoids and Poisson structures 355 §14.2. Examples 362 §14.3. IntegrabilityDraft of Poisson structures I 371 §14.4. Symplectic groupoid actions 381 §14.5. Hausdorffness issues 386 §14.6. The Poisson homotopy groupoid 388 §14.7. Morita equivalence 398 §14.8. Integrability of Poisson structures II 401 Problems 407 Notes and references for Part 4 411

Part 5. Appendices Appendix A. Lie Groups 417 §A.1. Lie groups 417 §A.2. Lie group actions 420 Contents xi

§A.3. Time-dependent vector fields 425 Appendix B. Symplectic Structures 429 §B.1. Symplectic forms 429 §B.2. Symplectic and Hamiltonian actions 433 Appendix C. Foliations 439 §C.1. Regular foliations 439 §C.2. Foliated differential forms 444 §C.3. Singular foliations 446 Appendix D. Groupoids: conventions and choices 451 Bibliography 455 Index 465

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Draft Preface

The aim of this book is to provide an introduction to Poisson geometry. The book grew out of several sets of lecture notes that we have prepared along many years while teaching master and graduate level courses at our home institutions and mini-courses at various Poisson geometry schools. In particular, the writing of the book was influenced by our experiences teaching the material and by the interactions we have had with the students who attended those lectures. Although it is fair to say that the book has grown and includes a bit more materialMarch that one 11, can 2021 actually hope to cover in class during a one semester course, the aim remains the same: to provide lecture notes for a graduate level course giving an introduction to Poisson geometry, addressed to students and researchers which have some familiarity with classical differential geometry and differentiable manifolds. Some basic knowledge of algebraic topology and symplectic geometry would be a plus, but not a requirement, to fully grasp some parts of the book. Some standard topics from differentialDraft geometry that we need, but might be missing from an introductory course, are summarized in the appendices at the end of the text. Poisson geometry emerged from the mathematical formulation of clas- sical mechanics. Historically, it all started with the work of Sim´eonDenis Poisson on the mechanics of particles which led him to the discovery in 1809 of the so-called Poisson bracket as a method for obtaining new integrals of the motion. Poisson computations occupied many pages, and his results where rediscovered and simplified two decades later by Carl Gustav Jacob Jacobi, who was the first to realize the fundamental role played by the Pois- son bracket in rational mechanics and who identified its main properties: an operation (bracket) which associates to any two observables f and g a

xiii xiv Preface new observable {f, g}, and which satisfies the Leibniz and Jacobi identity. Jacobi’s work on Poisson brackets, including the discovery of his famous identity, the commutator of derivations, etc., greatly influenced Sophus Lie in his foundational study at the end of the 19th century of symmetries of partial differential equations, which led him to the discovery of Lie groups and Lie algebras (see [102]). Linear Poisson structures correspond to Lie algebra structures, so Lie was in fact the first to study them and it is remark- able how deep Lie’s work dives into Poisson geometric aspects. For instance, Lie explicitly poses the realization problem for linear Poisson structures, a problem which turns out to be the same as that of searching for a Lie group integrating a Lie algebra. However, perhaps somewhat surprisingly, the first geometric, systematic, study of Poisson structures occurred much more re- cently in the work of Andr´eLichnerowicz [108] in the 1970s, which marks the birth of Poisson geometry in its modern formulation. The spectacular development of Poisson geometry from the last few decades owes much to the foundational work of Alan Weinstein [146] in the 80s and his discovery of symplectic groupoids as the global objects be- hind Poisson structures [150]. In retrospective, this discovery follows the same path as in Lie’s work: the search for non-degenerate (symplectic) real- izations led to the discovery of interesting global structures. In some sense, this book can be seen as an updated andMarch expanded 11, exposition 2021 of Weinstein’s pioneer work. In particular, our aim here is not to provide a survey of the vast amount of work done in this subject in the last 30-40 years, but rather to provide an introduction to the subject that will allow the reader to plunge into any of these recent exciting developments, some of which are mentioned throughout the text. We have triedDraft to provide our own insight into the subject while resisting the temptation of concentrating on our contributions. Our philosophy can be summarized as follows: Poisson geometry is an amalgam of foliation theory (partition into leaves), symplectic geometry (along the leaves) and Lie theory (transverse to the leaves). In particular, it provides the framework in which these geometries get to interact with each other in a beautiful symbiosis. While this is already, we believe, the main message in Weinstein’s foundational paper [146], the full extent of this interaction came to life later with the discovery of the global counterparts to Poisson structures: symplectic Lie groupoids. These objects codify all these 3 different aspects and we have organized the book so that one is led naturally to uncover them, giving an upgraded view on Weinstein’s and Lichnerowicz’s works. The monograph by Vaisman [140] was for a long period of time the only text book on Poisson geometry, apart from an earlier account by Bhaskara and Viswanath [15]. The book by Cannas da Silva and Weinstein [30] Preface xv contains a nice elementary introduction to the subject, aimed towards non- commutative geometry and quantization. A more up-to-date account of Poisson geometry, with a strong emphasis on local normal forms, was pro- vided by Dufour and Zung in their research monograph [59]. More recently, appeared the beautiful book by Laurent-Gengoux, Pichereau and Vanhaecke [105], which is highly recommended for people with an algebraic-geometric background. As the authors point out in the introduction, “The main topic about Poisson structures which is absent from this book is what should be called Poisson geometry.” We hope that our book provides an introduc- tion to Poisson geometry, which can be assimilated during a semester-long course, or can be used as material for self-study of the topic. The main body of the book is divided into four parts, followed by the appendices that were already mentioned. Each part ends with a small set of notes containing brief historical comments and directions for further read- ing. The best overview of the book is its table of contents. Still, we would like to emphasize that we payed special attention to the way we introduce those basic concepts in the theory that are more complex and require a deeper thought process. Take for example the notion of symplectic leaf: set-theoretically, we introduce them right away in the first chapter as the orbits of Hamiltonian diffeomorphisms, promising the reader that the actual structure (smooth, symplectic) will beMarch discussed 11, later. 2021 In the second chap- ter, we take advantage of the bivector field point of view to indicate how the smooth structure may arise from a Frobenius type theorem. However, the actual local result that is needed, the Weinstein Splitting Theorem, is then dealt with in chapter three. Finally, we discuss properly their smooth and symplectic structure in the fourth chapter of the book. We have also paid special attention to examples and exercises - at the price of increasing the size of the book.Draft Several sections of the book are called “Examples” or “Case study”, and there are well over 200 exercises, split into two types: the ones spread throughout the text, called “Exercises”, which are helpful in understanding the main material, and the ones listed at the end of each chapter, called “Problems”, which are useful in consolidating the material and providing further examples. We have tried to fill in a gap in the existing literature by providing a longer list of concrete examples of symplectic real- izations and symplectic groupoids. We have made an effort to include full proofs for all the results we discuss, the exception being Lie’s 3rd Theorem for Lie algebroids. Some of the arguments used in the proofs are new, others simplify and fill some gaps in the literature (see the notes and references at the end of each Part). There are a few topics, which may be now considered standard in Pois- son geometry, which we have decided not to include, such as Poisson-Lie xvi Preface groups, deformation quantization, generalized complex structures or inte- grable systems. They go beyond our purpose here and they deserve a sep- arate volume. We hope that our book will provide a solid background for learning such topics, or for moving to more advanced ones in the cutting edge of research. Acknowledgments We would like to express our appreciation to the many students, in par- ticular to our PhD students, who took part in the various courses on Poisson geometry we have taught in Utrecht, Urbana-Champaign and Nijmegen, as well as in various summer schools on Poisson geometry. Their comments and feedback were invaluable to the writing of this book. There are many colleagues and collaborators with whom we have in- teracted throughout the years. These interactions have shaped our views of Poisson geometry and mathematics in general, and from them we have learned many ideas which have influenced the writing of this book. We are grateful to all of them! We would like to thank also our home institutions for providing a stim- ulating atmosphere and work environment and being welcoming during our mutual visits. March 11, 2021 We have been generously supported by the NWO and NSF during the writing of this book, most recently by NWO Vici grant 639.033.312 and NSF grant DMS-2003223. March 11, 2021

Draft Marius Crainic Rui Loja Fernandes Ioan M˘arcut,