Progress in MathematicsVolume 232

Series EditorsHyman BassJoseph OesterleAlan Weinstein

The Breadth of Symplecticand Poisson GeometryFestschrift in Honor of Alan Weinstein

Jerrold E. MarsdenTudor S. RatiuEditors

BirkhauserBoston • Basel • Berlin

Jerrold E. MarsdenCalifornia Institute of TechnologyDepartment of Engineeringand Applied Science

Control and Dynamical SystemsPasadena, CA 91125U.S.A.

Tudor S. RatiuEcole Polytechnique Federale de LausanneDepartement de MathematiquesCH-1015 LausanneSwitzerland

AMS Subject Classifications: 53Dxx, 17Bxx, 22Exx, 53Dxx, 81Sxx

Library of Congress Cataloging-in-Publication DataThe breadth of symplectic and Poisson geometry : festschrift in honor of Alan Weinstein /Jerrold E. Marsden, Tudor S. Ratiu, editors.p. cm. – (Progress in mathematics ; v. 232)

Includes bibliographical references and index.ISBN 0-8176-3565-3 (acid-free paper)1. Symplectic geometry. 2. Geometric quantization. 3. Poisson manifolds. I. Weinstein,

Alan, 1943- II. Marsden, Jerrold E. III. Ratiu, Tudor S. IV. Progress in mathematics(Boston, Mass.); v. 232.

QA665.B74 2004516.3’.6-dc22 2004046202

ISBN 0-8176-3565-3 Printed on acid-free paper.

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Margo, Alan, and Asha in Paris at the lovely Fontaine des Quatre Parties du Monde.

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAcademic genealogy of Alan Weinstein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiAbout Alan Weinstein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvStudents of Alan Weinstein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvAlan Weinstein’s publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi

Dirac structures, momentum maps, and quasi-Poisson manifoldsHenrique Bursztyn, Marius Crainic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Construction of Ricci-type connections by reduction and inductionMichel Cahen, Simone Gutt, Lorenz Schwachhöfer . . . . . . . . . . . . . . . . . . . . . . . 41

A mathematical model for geomagnetic reversalsJ. J. Duistermaat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Nonholonomic systems via moving frames: Cartan equivalence and ChaplyginHamiltonizationKurt Ehlers, Jair Koiller, Richard Montgomery, Pedro M. Rios . . . . . . . . . . . . . 75

Thompson’s conjecture for real semisimple Lie groupsSam Evens, Jiang-Hua Lu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

The Weinstein conjecture and theorems of nearby and almost existenceViktor L. Ginzburg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

Simple singularities and integrable hierarchiesAlexander B. Givental, Todor E. Milanov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

Momentum maps and measure-valued solutions (peakons, filaments, andsheets) for the EPDiff equationDarryl D. Holm, Jerrold E. Marsden . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

viii Contents

Higher homotopies and Maurer–Cartan algebras: Quasi-Lie–Rinehart,Gerstenhaber, and Batalin–Vilkovisky algebrasJohannes Huebschmann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

Localization theorems by symplectic cutsLisa Jeffrey, Mikhail Kogan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

Refinements of the Morse stratification of the normsquare of the moment mapFrances Kirwan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

Quasi, twisted, and all that… in Poisson geometry and Lie algebroid theoryYvette Kosmann-Schwarzbach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363

Minimal coadjoint orbits and symplectic inductionBertram Kostant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391

Quantization of pre-quasi-symplectic groupoids and their Hamiltonian spacesCamille Laurent-Gengoux, Ping Xu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423

Duality and triple structuresKirill C. H. Mackenzie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455

Star exponential functions as two-valued elementsY. Maeda, N. Miyazaki, H. Omori, A. Yoshioka . . . . . . . . . . . . . . . . . . . . . . . . . . 483

From momentum maps and dual pairs to symplectic and Poisson groupoidsCharles-Michel Marle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493

Construction of spectral invariants of Hamiltonian paths on closed symplecticmanifoldsYong-Geun Oh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525

The universal covering and covered spaces of a symplectic Lie algebra actionJuan-Pablo Ortega, Tudor S. Ratiu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571

Poisson homotopy algebra: An idiosyncratic survey of homotopy algebraictopics related to Alan’s interestsJim Stasheff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583

Dirac submanifolds of Jacobi manifoldsIzu Vaisman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603

Quantum maps and automorphismsSteve Zelditch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623

Preface

Alan Weinstein is one of the top mathematicians in the world working in the areaof symplectic and differential geometry. His research on symplectic reduction, La-grangian submanifolds, groupoids, applications to mechanics, and related areas hashad a profound influence on the field. This area of research remains active and vi-brant today and this volume is intended to be a reflection of that vigor. In additionto reflecting the vitality of the field, this is a celebratory volume to honor Alan’s60th birthday. His birthday was also celebrated in August, 2003 with a wonderfulweek-long conference held at the ESI: the Erwin Schrödinger International Institutefor Mathematical Physics in Vienna.

Alan was born in New York in 1943. He was an undergraduate at MIT and agraduate student at UC Berkeley, where he was awarded his Ph.D. in 1967 under thedirection of S. S. Chern. After spending postdoctoral years at IHES near Paris, MIT,and the University of Bonn, he joined the faculty at UC Berkeley in 1969, becominga full Professor in 1976.

Alan has received many honors, including an Alfred P. Sloan Foundation Fel-lowship, a Miller Professorship (twice), a Guggenheim Fellowship, election to theAmerican Academy of Arts and Sciences in 1992, and an honorary degree at theUniversity of Utrecht in 2003.

At the ESI conference, S. S. Chern, Alan’s advisor, sent the following words tocelebrate the occasion:

“I am glad about this celebration and I think Alan richly deserves it. Alancame to me in the early sixties as a graduate student at the University ofCalifornia at Berkeley. At that time, a prevailing problem in our geometrygroup, and the geometry community at large, was whether on a Riemannianmanifold the cut locus and the conjugate locus of a point can be disjoint.Alan immediately showed that this was possible. The result became part ofhis Ph.D. thesis, which was published in the Annals of Mathematics. Hereceived his Ph.D. degree in a short period of two years. I introduced himto IHES and the French mathematical community. He stays close with themand with the mathematical ideas of Charles Ehresmann. He is original and

x Preface

often came up with ingenious ideas. An example is his contribution to thesolution of the Blaschke conjecture. I am very proud to count him as one ofmy students and I hope he will remain interested in mathematics up to myage, which is now 91.’’

Alan’s technical contributions are wide ranging and deep. As many of his earlypapers in his publication list illustrate, he started off in his thesis and the years im-mediately following in pure differential geometry, a topic he has come back to fromtime to time throughout his career.

Already starting with his postdoc years and his early career at Berkeley, he becameinterested in symplectic geometry and mechanics. In this area he rapidly establishedhimself as one of the world’s authorities, producing important and deep results rangingfrom reduction theory to Lagrangian and Poisson manifolds to studies of periodicorbits in Hamiltonian systems. He also did important work in fluid mechanics andplasma physics and through this work, he established warm relations with the Berkeleyphysicists Allan Kaufman and Robert Littlejohn.

Alan’s important work on periodic orbits in Hamiltonian systems led him even-tually to formulate the “Weinstein conjecture,’’ namely that for a given Hamiltonianflow on a symplectic manifold, there must be at least one closed orbit on a regularcompact contact type level set of the Hamiltonian. Along with Arnold’s conjecture,the Weinstein conjecture has been one of the driving forces in symplectic topologyover the last two decades.

Alan kept up his interest in symplectic reduction theory throughout his later work.For instance, he laid some important foundation stones in the theory of semidirectproduct reduction as well as in singular reduction through his work on Satake’sV -manifolds, along with finding important links with singular structures in modulispaces.

Intertwined with his work on symplectic geometry and mechanics, he did exten-sive work on geometric PDE, eigenvalues, the Schrödinger operator and geometricquantization. Alan took the point of view of microlocal analysis and phase spacestructures in his work in this area, emphasizing the links with quantum mechanics.

Preface xi

His work on the limit distribution of eigenvalue clusters in terms of the geodesic Radontransform of the potential inspired a large number of related articles. He showed thatthe geodesic flow of a Zoll surface was symplectically equivalent to that of a roundsphere, and hence that its Laplacian could be conjugated globally to the round Lapla-cian plus a pseudodifferential potential. This work inspired many other results onconjugacies.

One of Alan’s fundamental contributions to Poisson geometry was the introduc-tion of symplectic groupoids in 1987, which marks the official beginning of his “oids’’period. In these works, he makes sweeping generalizations about a wide variety ofconstructions in symplectic geometry, including (with Courant) the important notionof Dirac structures. During this period of generalizations he constantly returned tospecific topics in symplectic and Poisson geometry, such as geometric phases andPoisson Lie groups, in addition to making other key links. For instance, symplec-tic groupoids are used to link Poisson geometry to noncommutative geometry, andgroupoids are also intimately related to many other areas, including symmetries andreduction, dual pairs, quantization and the theory of sigma models. One of the centralideas is that the usual theory of Hamiltonian actions, momentum maps, and sym-plectic reduction makes sense in the more general context of actions of symplecticgroupoids; in this setting, momentum maps are Poisson maps taking values in generalPoisson manifolds, rather than just Lie–Poisson manifolds (that is, duals of Lie alge-bras). Alan has raised the question of whether this framework can be further extendedto include new notions of momentum maps such as quasi-Poisson manifolds withgroup-valued momentum maps as well as optimal momentum maps.

Alan is well known not only for his brilliant papers and conjectures, but alsofor his general philosophy, such as the symplectic creed: Everything is a Lagrangiansubmanifold . Those of us who know him well also appreciate his very special insight.For example, in the middle of a discussion (for instance, as we both had in ourjoint works on semidirect product reduction as well as stability theory) he will saysomething like what you are really doing is. . . and then give us some usually veryspecial insight that invariably substantially improves the whole project.

Alan also has a very interesting and charming sense of humor that even makesits way into his papers from time to time. For instance, Alan had great fun in hispapers with the “East Coast–West Coast’’ discussions of whether one should use theterm momentum map or moment map. He also gave us a good laugh with the termsymplectic bones as it relates to the French translation of Poisson as Fish.

Alan is a great educator. His lectures, even on Calculus, are always a treat and arevery inspiring for their special insight, their wit and lively presentation. His enthu-siasm for mathematics is infectious. One story that comes to mind on the educationfront is this: during the days when he was exceptionally keen about groupoids, hewas preparing a lecture for undergraduates on the subject. Some of us convinced himto present it as a colloquium lecture for faculty, keeping in mind the old advice “nocolloquium talk can be too simple.’’ It was, in fact, not only a beautiful colloquiumtalk, but was perfectly pitched for the faculty, and it became a popular article in theNotices of the American Mathematical Society. Part of being a good educator is being

xii Preface

cognizant of history.Alan excels in this area. For instance, his research into the historyof Lie is what led directly to the introduction of the term “Lie–Poisson’’ bracket.

The papers in this volume were selected by invitation and all of them underwenta rigorous refereeing process. While this process took some time, it resulted in highquality papers. We thank all of the referees for their diligent and helpful work. Theauthors of this volume represent some of the best workers in the subject and theircontributions span a wide range of the topics covered by symplectic and Poissongeometry and mechanics, broadly interpreted.

The intended audience for the book includes active researchers in the generalarea of symplectic geometry and mechanics, as well as aspiring graduate studentswho wish to learn where the subject is headed and what some of the current researchtopics are.

Alan and Margo have a special relationship to Paris. They have spent many happytimes there, and we wish them all the best and many more happy visits in the yearsto come.

We wish to thank Ann Kostant for her expert editorial guidance throughout theproduction of this volume. Of course, we also thank all the authors for their contribu-tions as well as their helpful guidance and advice. The referees are also thanked fortheir valuable comments and suggestions.

Jerry Marsden and Tudor RatiuSeptember, 2004

Academic genealogy of Alan Weinstein

Otto MenckenUniversität Leipzig, 1668

Thomae Hobbesii Epicureismum historicedelineatum sistit

Johann C. WichmannshausenUniversität Leipzig, 1685

Disputationem Moralem De Divortiis SecundumJus Naturae

Christian A. HausenMartin-Luther-Universität Halle-Wittenberg, 1713

De corpore scissuris figurisque non cruetando ductu

Abraham G. KaestnerUniversität Leipzig, 1739

Theoria radicum in aequationibus

Johann F. PfaffGeorg-August-Universität Gottingen, 1786

Commentatio de ortibus et occasibus siderum apudauctores classicos commemoratis

August F. MöbiusUniversität Leipzig, 1815

De computandis occultationibus fixarum per planetas

Otto W. FiedlerUniversität Leipzig, 1859

Emil WeyrUniversity of Prague, 1870

Johannes FrischaufUniversität Wien, 1861

Karl FriesachUniversität Wien, 1846

Gustav Ritter von EscherichTechnische Universität Graz, 1873

Die Geometrie auf Flachen constanternegativer Krummung

Wilhelm WirtingerUniversität Wien, 1887

onUber eine spezielle Tripelinvoluti in der Ebene

Wilhelm BlaschkeUniversität Wien, 1908

Alan D. WeinsteinUniversity of California at Berkeley, 1967The Cut Locus and Conjugate Locus of a

Riemannian Manifold

Shiing-Shen ChernUniversität Hamburg, 1936

Eine Invariantentheorie der Dreigewebe ausr-dimensionalen Mannigfaltigkeiten im R2r

About Alan Weinstein

Alan David Weinstein

Ph.D.: University of California at Berkeley, 1967Dissertation: The Cut Locus and Conjugate Locus of a Riemannian ManifoldAdvisor: Shiing-Shen Chern

Students of Alan Weinstein

1. Jair Koiller, Studies on the Spring-Pendulum, 19752. Otto Ruiz, Existence of Brake-Orbits in Finsler Mechanical Systems, 19753. Yilmaz Akyildiz, Dynamical Symmetries of the Kepler Problem, 19764. Gerald Chachere, Numerical Experiments Concerning the Eigenvalues of the

Laplacian on a Zoll Surface, 19775. John Jacob, Geodesic Symmetries of Homogeneous Kahler Manifolds, 19776. Steven Zelditch, Reconstruction of Singularities for Solutions of Schrödinger’s

Equations, 19817. Enrique Planchart, Analogies in Symplectic Geometry of Some Results of Cartan

in Representation Theory, 19828. Barry Fortune, A Symplectic Fixed Point Theorem for Complex Projective Spaces,

19849. Stephen Omohundro (Department of Physics), Geometric Perturbation Theory

in Physics, 198510. Theodore Courant, Dirac Manifolds, 198711. Yong-Geun Oh, Nonlinear Schrödinger Equations with Potentials: Evolution,

Existence, and Stability of Semi-Classical Bound States, 198812. Viktor Ginzburg, On Closed Characteristics of 2-Forms, 199013. Milton Lopes Filho, Microlocal Regularity and Symbols for Distributions, 199014. Jiang-Hua Lu, Multiplicative and Affine Poisson Structures on Lie Groups, 199015. Ping Xu, Morita Equivalence of Poisson Manifolds, 1990

xvi About Alan Weinstein

16. Sean Bates, Symplectic End Invariants and C0 Symplectic Topology, 199417. Agust Egilsson, On Embedding a Stratified Symplectic Space in a Smooth Poisson

Manifold , 199518. Dong Yan, Yang–Mills Theory on Symplectic Manifolds, 199519. Zhao-Hui Qian, Groupoids, Midpoints and Quantizations, 199720. Vinay Kathotia, Universal Formulae for Deformation Quantization and The

Campbell–Baker–Hausdorff Formula, 199821. Dmitry Roytenberg, Courant Algebroids, Derived Brackets And Even Symplectic

Supermanifolds, 199922. Mélanie Bertelson-Volckaert (Stanford University), FoliationsAssociated to Reg-

ular Poisson Structures, 200023. Benjamin Davis, On Poisson Spaces Associated to Finitely Generated Poisson

R-Algebras, 200124. Henrique Bursztyn, Morita Equivalence in Deformation Quantization, 200125. Olga Radko, Some Invariants of Poisson Manifolds, 200226. Xiang Tang, Quantization of Noncommutative Poisson Manifolds, 200327. Marco Zambon, Submanifold Averaging in Riemannian, Symplectic, and Contact

Geometry, 200328. Chenchang Zhu, Integrating Lie Algebroids via Stacks and Applications to Jacobi

Manifolds, 2003

Alan Weinstein’s publications

[1] Weinstein, A., On the homotopy type of positively pinched manifolds, Arch. Math., 18(1967), 523–524.

[2] Weinstein, A., A fixed point theorem for positively curved manifolds, J. Math. Mech.,18 (1968), 149–153.

[3] Weinstein, A., The cut locus and conjugate locus of a riemannian manifold, Ann. Math.87 (1968), 29–41.

[4] Weinstein, A., Symplectic structures on Banach manifolds, Bull. Amer. Math. Soc., 75(1969), 1040–1041.

[5] Weinstein, A., and Simon, U., Anwendungen der de rhamschen Zerlegung auf Problemeder localen Flächentheorie, Manuscripta Math., 1 (1969), 139–146.

[6] Marsden, J., and Weinstein, A., A comparison theorem for hamiltonian vector fields,Proc. Amer. Math. Soc., 26 (1970), 629–631.

[7] Weinstein,A., The generic conjugate locus, in Global Analysis, Proceedings of Symposiaon Pure Mathematics, Vol. 15, American Mathematical Society, Providence, RI, 1970,299–301.

[8] Weinstein, A., Positively curved n-manifolds in Rn+2, J. Differential Geom., 4 (1970),1–4.

[9] Weinstein, A., Positively curved deformations of invariant Riemannian metrics, Proc.Amer. Math. Soc., 26 (1970), 151–152.

[10] Weinstein, A., Sur la non-densité des géodésiques fermées, C. R. Acad. Sci. Paris, 271(1970), 504.

[11] Roels, J., and Weinstein, A., Functions whose Poisson brackets are constants, J. Math.Phys., 12 (1971), 1482–1486.

About Alan Weinstein xvii

[12] Weinstein, A., Singularities of families of functions, in Proceedings of the Conference“Differentialgeometrie im Grossen,’’ Mathematische Forschungsinstitut, Oberwolfach,Germany, 1971, 323–330.

[13] Weinstein, A., Perturbation of periodic manifolds of Hamiltonian systems, Bull. Amer.Math. Soc., 77 (1971), 814–818.

[14] Weinstein, A., Remarks on curvature and the Euler integrand, J. Differential Geom., 6(1971), 259–262.

[15] Weinstein, A., Symplectic manifolds and their lagrangian submanifolds, Adv. Math., 6(1971), 329–346.

[16] Weinstein, A., On the invariance of Poincaré’s generating function for canonical trans-formations, Invent. Math., 16 (1972), 202–213.

[17] Weinstein, A., Distance spheres in complex projective spaces, Proc. Amer. Math. Soc.,39 (1973), 649–650.

[18] Weinstein, A., Lagrangian submanifolds and hamiltonian systems, Ann. Math., 98(1973), 377–410.

[19] Weinstein, A., Normal modes for nonlinear hamiltonian systems, Invent. Math., 20(1973), 47–57.

[20] Marsden, J., and Weinstein, A., Reduction of symplectic manifolds with symmetry, Rep.Math. Phys., 5 (1974), 121–130.

[21] Weinstein, A., Application des opérateurs intégraux de Fourier aux spectres des variétésriemanniennes, C. R. Acad. Sci. Paris, 279 (1974), 229–230.

[22] Weinstein, A., On the volume of manifolds, all of whose geodesics are closed, J. Dif-ferential Geom., 9 (1974), 513–517.

[23] Weinstein, A., Quasi-classical mechanics on spheres, Sympos. Math., 14 (1974), 25–32.[24] Weinstein, A., On Maslov’s quantization condition, in Proceedings of the International

Symposium on Fourier Integral Operators (Nice, May 1974), Lecture Notes in Mathe-matics, Vol. 469, Springer-Verlag, New York, 1975, 341–372.

[25] Guillemin, V., and Weinstein, A., Eigenvalues associated with a closed geodesic, Bull.Amer. Math. Soc., 82 (1976), 92–94.

[26] Weinstein, A., Fourier integral operators, quantization, and the spectrum of a Rieman-nian manifold, in Géométrie Symplectique et Physique Mathématique, Colloque Inter-nationale de Centre National de la Recherche Scientifique No. 237, CNRS, Paris, 1976,289–298.

[27] Weinstein, A., The principal symbol of a distribution, Bull. Amer. Math. Soc., 82 (1976),548–550.

[28] Weinstein, A., Symplectic V -manifolds, periodic orbits of Hamiltonian systems, and thevolume of certain Riemannian manifolds, Comm. Pure Appl. Math., 30 (1977), 265–271.

[29] Weinstein, A., Lectures on Symplectic Manifolds, Regional Conference Series in Math-ematics, Vol. 29, American Mathematical Society, Providence, RI, 1977.

[30] Weinstein, A., Asymptotics of eigenvalue clusters for the Laplacian plus a potential,Duke Math. J., 44 (1977), 883–892.

[31] Weinstein, A., The order and symbol of a distribution, Trans. Amer. Math. Soc., 241(1978), 1–54.

[32] Weinstein,A., Simple periodic orbits, Amer. Inst. Phys. Conf. Proc., 46 (1978), 260–263.[33] Weinstein, A., Eigenvalues of the laplacian plus a potential, in Proceedings of the Inter-

national Congress of Mathematicians, Helsinki, 1978, 803–805.[34] Weinstein, A., A universal phase space for particles in Yang–Mills fields, Lett. Math.

Phys., 2 (1978), 417–420.[35] Weinstein, A., Periodic orbits for convex hamiltonian systems, Ann. Math., 108 (1978),

507–518.

xviii About Alan Weinstein

[36] Weinstein, A., Bifurcations and Hamilton’s principle, Math. Z., 159 (1978), 235–248.[37] Weinstein, A., On the hypotheses of Rabinowitz’ periodic orbit theorems, J. Differential

Equations 33 (1979), 353–358.[38] Marsden, J. E., and Weinstein, A., Review of Geometric Asymptotics and Symplectic

Geometry and Fourier Analysis, Bull. Amer. Math. Soc., 1 (1979), 545–553.[39] Marsden, J., and Weinstein, A., Calculus, Benjamin/Cummings, San Francisco, 1980.[40] Weinstein, A., Fat bundles and symplectic manifolds, Adv. Math., 37 (1980), 239–250.[41] Weinstein, A., Nonlinear stabilization of quasimodes, in Proceedings of the AMS Sym-

posium on Geometry of the Laplacian, Hawaii, 1979, Proceedings of Symposia on PureMathematics, Vol. 36, American Mathematical Society, Providence, RI, 1980, 301–318.

[42] Marsden, J., and Weinstein, A., Calculus Unlimited , Benjamin/Cummings, San Fran-cisco, 1981.

[43] Croke, C., and Weinstein, A., Closed curves on convex hypersurfaces and periods ofnonlinear oscillations, Invent. Math., 64 (1981), 199–202.

[44] Marsden, J., Morrison, P., and Weinstein,A., Comments on: The Maxwell–Vlasov equa-tions as a continuous Hamiltonian system, Phys. Lett., 96A (1981), 235–236.

[45] Stanton, R. J., and Weinstein A., On the L4 norm of spherical harmonics, Math. Proc.Cambridge Philos. Soc., 89 (1981), 343–358.

[46] Weinstein, A., Symplectic geometry, Bull. Amer. Math. Soc. (N.S.), 5 (1981), 1–13.[47] Weinstein, A., Neighborhood classification of isotropic embeddings, J. Differential

Geom., 16 (1981), 125–128.[48] Marsden. J., and Weinstein, A., The hamiltonian structure of the Maxwell–Vlasov equa-

tions, Physica, 4D (1982), 394–406.[49] Weinstein, A., Gauge groups and Poisson brackets for interacting particles and fields,

Amer. Inst. Phys. Conf. Proc., 88 (1982), 1–11.[50] Weinstein, A., What is microlocal analysis?, Math. Intel., 4 (1982), 90–92.[51] Weinstein, A., The symplectic “category,’’ in Doebner, H.-D., Andersson, S. I., and

Petry, H. R., eds., Differential Geometric Methods in Mathematical Physics (Clausthal,Germany 1980), Lecture Notes in Mathematics, Vol. 905, Springer-Verlag, Berlin, 1982,45–50.

[52] Weinstein, A., and Zelditch, S., Singularities of solutions of some Schrödinger equationson Rn, Bull. Amer. Math. Soc., (1982).

[53] Gotay, M. J., Lashof, R., Sniatycki, J., and Weinstein, A., Closed forms on symplecticfibre bundles, Comm. Math. Helv., 58 (1983), 617–621.

[54] Marsden, J., Ratiu, T., Schmid, R., Spencer, R., and Weinstein, A., Hamiltonian sys-tems with symmetry, coadjoint orbits, and plasma physics, Atti Acad. Sci. Torino, 117-Supplemento (1983), 289–340.

[55] Marsden, J., and Weinstein, A., Coadjoint orbits, vortices and Clebsch variables forincompressible fluids, Physica, 7D (1983), 305–323.

[56] Sniatycki, J., and Weinstein, A., Reduction and quantization for singular momentummappings, Lett. Math. Phys., 7 (1983), 159–161.

[57] Weinstein, A., A symplectic rigidity theorem, Duke Math. J., 50 (1983), 1121–1125.[58] Weinstein, A., Hamiltonian structure for drift waves and geostrophic flow, Phys. Fluids,

26, (1983), 388–390.[59] Weinstein, A., Sophus Lie and symplectic geometry, Expos. Math., 1 (1983), 95–96.[60] Weinstein, A., Removing intersections of lagrangian immersions, Illinois J. Math., 27

(1983), 484–500.[61] Weinstein,A., The local structure of Poisson manifolds, J. Differential Geom., 18 (1983),

523–557.

About Alan Weinstein xix

[62] Marsden, J., Morrison, P., and Weinstein, A., The Hamiltonian structure of the BBGKYhierarchy equations, Contemp. Math., 28 (1984), 115–124.

[63] Marsden, J., Ratiu, T., and Weinstein,A., Reduction and Hamiltonian structures on dualsof semidirect product Lie algebras, Contemp. Math., 28 (1984), 55–100.

[64] Marsden. J., Ratiu, T., and Weinstein, A., Semidirect products and reduction in mechan-ics, Trans. Amer. Math. Soc., 281 (1984), 147–177.

[65] Weinstein, A., Equations of plasma physics (notes by Stephen Omohundro), inChern, S. S., ed., Seminar on Nonlinear Partial Differential Equations, MSRI Pub-lications, Springer-Verlag, New York, 1984, 359–373.

[66] Weinstein, A., Stability of Poisson-Hamiltonian equilibria, Contemp. Math., 28 (1984),3–13.

[67] Weinstein, A., C0 perturbation theorems for symplectic fixed points and lagrangianintersections, Travaux en Cours, 3 (1984), 140–144.

[68] Fortune, B., and Weinstein, A., A symplectic fixed point theorem for complex projectivespaces, Bull. Amer. Math. Soc., 12 (1985), 128–130.

[69] Holm. D. D., Marsden, J. E., Ratiu, T., and Weinstein, A., Nonlinear stability of fluidand plasma equilibria, Phys. Rep., 123-1–2 (1985), 1–116.

[70] Marsden, J., and Weinstein, A., Calculus I, II, III, 2nd ed., Springer-Verlag, New York,1985.

[71] Weinstein,A.,Asymbol calculus for some Schrödinger equations onRn, Amer. J. Math.,(1985), 1–21.

[72] Weinstein, A., A global invertibility theorem for manifolds with boundary, Proc. Roy.Soc. Edinburgh Sect. A, 99 (1985), 283–284.

[73] Weinstein, A., Periodic nonlinear waves on a half-line, Comm. Math. Phys., 99 (1985),385–388.

[74] Weinstein, A., Poisson structures and Lie algebras, Astérisque, hors série (1985), 421–434.

[75] Weinstein, A., Symplectic reduction and fixed points, in Séminaire Sud-Rhodanien deGéométrie, Rencontre de Balaruc I, Travaux en Cours, Hermann, Paris, 1985, 140–148.

[76] Weinstein, A., Three dimensional contact manifolds with vanishing torsion tensor (ap-pendix to a paper by S. S. Chern and R. Hamilton), in Hirzebruch, F., Schwermer J., andSuter S., eds., Proceedings of the Meeting held by the Max-Planck-Institut für Math-ematik, Bonn. June 15–22, 1984, Lecture Notes in Mathematics, Vol. 1111, Springer-Verlag, Berlin, 1985, 306–308.

[77] Floer, A., and Weinstein A., Nonspreading wave packets for the nonlinear Schrödingerequation with a bounded potential, J. Functional Anal., 69 (1986), 397–408.

[78] Weinstein, A., On extending the Conley-Zehnder theorem to other manifolds, Proc.Sympos. Pure Math., 45 (1986), 541–544.

[79] Weinstein, A., Critical point theory, symplectic geometry, and hamiltonian systems, inProceedings of the 1983 Beijing Symposium on Differential Geometry and DifferentialEquations, Science Press, Beijing, 1986, 261–289.

[80] Coste, A., Dazord, P., and Weinstein, A., Groupoïdes symplectiques, Publ. Dép. Math.Univ. Claude Bernard-Lyon I, 2A (1987), 1–62.

[81] Weinstein, A., ed., Some Problems in Symplectic Geometry, Séminaire Sud-Rhodaniende Géométrie VI, Travaux en Cours, Hermann, Paris, 1987.

[82] Weinstein, A., Standing and travelling waves for nonlinear wave equations, TransportTheory Stat. Phys., 16 (1987), 267–277.

[83] Weinstein, A., The Geometry of Poisson Brackets (Notes by K. Ono and K. Sugiyama),Surveys in Geometry, Tokyo, 1987.

xx About Alan Weinstein

[84] Weinstein, A., Poisson geometry of the principal series and nonlinearizable structures,J. Differential Geom., 25 (1987), 55–73.

[85] Weinstein, A., Symplectic groupoids and Poisson manifolds, Bull. Amer. Math. Soc., 16,(1987), 101–104.

[86] Courant, T. J., and Weinstein, A., Beyond Poisson structures, in Séminaire Sud-Rhodanien de Géométrie VIII, Travaux en Cours, Vol. 27, Hermann, Paris, 1988, 39–49.

[87] Mikami, K., and Weinstein,A., Moments and reduction for symplectic groupoid actions,Publ. RIMS Kyoto Univ., 24 (1988), 121–140.

[88] Weinstein, A., Coisotropic calculus and Poisson groupoids, J. Math. Soc. Japan, 40(1988), 705–727.

[89] Weinstein, A., Some remarks on dressing transformations, J. Fac. Sci. Univ. Tokyo.Sect. 1A Math., 36 (1988), 163–167.

[90] Lu, J.-H., and Weinstein, A., Groupoïdes symplectiques doubles des groupes de Lie-Poisson, C. R. Acad. Sci. Paris, 309 (1989), 951–954.

[91] Weinstein, A., Cohomology of symplectomorphism groups and critical values of hamil-tonians, Math. Z., 201 (1989), 75–82.

[92] Weinstein, A., Blowing up realizations of Heisenberg–Poisson manifolds, Bull. Sci.Math., 113 (1989), 381–406.

[93] Lu, J.-H., and Weinstein, A., Poisson Lie groups, dressing transformations, and theBruhat decomposition, J. Differential Geom., 31 (1990), 501–526.

[94] Weinstein, A., Connections of Berry and Hannay type for moving lagrangian submani-folds, Adv. Math., 82 (1990), 133–159.

[95] Weinstein, A., Affine Poisson structures, Internat. J. Math., 1 (1990), 343–360.[96] Dazord, P., Lu, J.-H., Sondaz, D., and Weinstein, A., Affinoïdes de Poisson, C. R. Acad.

Sci. Paris, 312 (1991), 523–527.[97] Hofer, H., Weinstein, A., and Zehnder, E., Andreas Floer, 1956–1991 (obituary), Notices

Amer. Math. Soc., 38 (1991), 910–911.[98] Lu, J.-H., and Weinstein, A., Classification of SU(2)-covariant Poisson structures on

S2 (appendix to a paper of A. J.-L. Sheu), Comm. Math. Phys., 135 (1991), 229–232.[99] Weinstein, A., Symplectic groupoids, geometric quantization, and irrational rotation al-

gebras, in Symplectic Geometry, Groupoids, and Integrable Systems: Séminaire Sud-Rhodanien de Géométrie à Berkeley (1989), Dazord, P., and Weinstein, A., eds.,Springer–MSRI Series, Springer-Verlag, New York, 1991, 281–290.

[100] Weinstein, A., Contact surgery and symplectic handlebodies, Hokkaido Math. J., 20(1991), 241–251.

[101] Weinstein, A., Noncommutative geometry and geometric quantization, in Donato, P.,Duval, C., Elhadad, J., and Tuynman, G. M., eds., Symplectic Geometry and Mathe-matical Physics: Actes du Colloque en l’Honneur de Jean-Marie Souriau, Progress inMathematics, Birkhäuser, Basel, 1991, 446–461.

[102] Weinstein, A., and Xu, P., Extensions of symplectic groupoids and quantization, J. ReineAngew. Math., 417 (1991), 159–189.

[103] Ginzburg, V. L., and Weinstein, A., Lie–Poisson structure on some Poisson Lie groups,J. Amer. Math. Soc., 5 (1992), 445–453.

[104] Weinstein, A., and Xu, P., Classical solutions of the quantum Yang–Baxter equation,Comm. Math. Phys., 148 (1992), 309–343.

[105] Mardsen, J. E., Tromba,A. J., and Weinstein,A., Basic Multivariable Calculus, Springer-Verlag and W. H. Freeman, New York, 1993.

[106] Weinstein, A., Traces and triangles in symmetric symplectic spaces, Contemp. Math.,179 (1994), 261–270.

About Alan Weinstein xxi

[107] Maeda, Y., Omori, H., and Weinstein, A., eds., Symplectic Geometry and Quantization:Two Symposia on Symplectic Geometry and Quantization Problems, July 1993, Japan,Contemporary Mathematics, Vol. 179, American Mathematical Society, Providence, RI,1994.

[108] Weinstein, A., Classical theta functions and quantum tori, Publ. RIMS Kyoto Univ., 30(1994), 327–333.

[109] Birnir, B., McKean, H., and Weinstein,A., The rigidity of sine-Gordon breathers, Comm.Pure Appl. Math., 47 (1994), 1043–1051.

[110] Scovel, C., andWeinsteinA., Finite dimensional Lie–Poisson approximations toVlasov–Poisson equations, Comm. Pure Appl. Math., 47 (1994), 683–709.

[111] Emmrich, C., and Weinstein, A., The differential geometry of Fedosov’s quantization, inBrylinski, J. L., Brylinski, R., Guillemin, V., and Kac, V., eds., Lie Theory and Geometry:In Honor of B. Kostant, Progress in Mathematics, Birkhäuser, Boston, 1994, 217–239.

[112] Weinstein, A., Deformation quantization, Astérisque, 227 (1995) (Séminaire Bourbaki,46ème année, 1993–94, no. 789), 389–409.

[113] Bates, S., and Weinstein, A., Lectures on the Geometry of Quantization, Berkeley Math-ematics Lecture Notes, American Mathematical Society, Providence, RI, 1997.

[114] Weinstein, A., The symplectic structure on moduli space, in Hofer, H., Taubes, C.,Weinstein, A., and Zehnder, E., eds., The Floer Memorial Volume, Birkhäuser, Basel,1995, 627–635.

[115] Weinstein, A., Lagrangian mechanics and groupoids, in Shadwick, W. F., Krish-naprasad, P. S., and Ratiu, T. S., eds., Mechanics Day, Fields Institute Communications,Vol. 7., American Mathematical Society, Providence, RI, 1995, 207–231.

[116] Emmrich, C., and Weinstein, A., Geometry of the transport equation in multicomponentWKB approximations, Comm. Math. Phys., 176 (1996), 701–711.

[117] Weinstein,A., Groupoids: Unifying internal and external symmetry, Notices Amer. Math.Soc., 43 (1996), 744–752; reprinted in Contemp. Math., 282 (2001), 1–19.

[118] Reshetikhin, N., Voronov, A. A., and Weinstein, A., Semiquantum geometry, Algebraicgeometry 5, J. Math. Sci., 82 (1996), 3255–3267.

[119] Guruprasad, K., Huebschmann, J., Jeffrey, L., and Weinstein, A., Group systems,groupoids, and moduli spaces of parabolic bundles, Duke Math. J., 89 (1997), 377–412.

[120] Weinstein, A., Tangential deformation quantization and polarized symplectic groupoids,in Gutt, S., Rawnsley, J., and Sternheimer, D., eds., Deformation Theory and SymplecticGeometry, Mathematical Physics Studies, Vol. 20, Kluwer, Dordrecht, the Netherlands,1997, 301–314.

[121] Liu, Z.-J., Weinstein, A., and Xu, P., Manin triples for Lie bialgebroids, J. DifferentialGeom., 45 (1997), 547–574.

[122] Weinstein, A., The modular automorphism group of a Poisson manifold, J. Geom. Phys.,23 (1997), 379–394.

[123] Weinstein, A., Some questions about the index of quantized contact transformations,RIMS Kôkyûroku, 1014 (1997), 1–14.

[124] Weinstein, A., and Xu, P., Hochschild cohomology and characteristic classes for star-products, in Khovanskii, A., Varchenko, A., and Vassiliev, V., eds., Geometry of Differ-ential Equations, American Mathematical Society, Providence, RI, 1997, 177–194.

[125] Liu, Z.-J., Weinstein, A., and Xu, P., Dirac structures and Poisson homogeneous spaces,Comm. Math. Phys., 192 (1998), 121–144.

[126] Weinstein, A., Poisson geometry, Differential Geom. Appl., 9 (1998), 213–238.[127] Roytenberg, D., and Weinstein, A., Courant algebroids and strongly homotopy Lie al-

gebras, Lett. Math. Phys., 46 (1998), 81–93.

xxii About Alan Weinstein

[128] Weinstein, A., From Riemann Geometry to Poisson Geometry and Back Again, Lectureat Chern Symposium, Mathematical Sciences Research Institute, Berkeley, CA, 1998;available from http://msri.org/publications/video/contents.html and on CD-ROM.

[129] Cannas da Silva, A., and Weinstein, A., Geometric Models for Noncommutative Al-gebras, Berkeley Mathematics Lecture Notes, American Mathematical Society, Provi-dence, RI, 1999.

[130] Nistor, V., Weinstein, A., and Xu., P., Pseudodifferential operators on differentialgroupoids, Pacific J. Math. 189 (1999), 117–152.

[131] Evens, S., Lu, J.-H., and Weinstein, A., Transverse measures, the modular class, and acohomology pairing for Lie algebroids, Quart. J. Math., 50 (1999), 417–436.

[132] Fuchs, D., Eliashberg, Y., Ratiu, T., and Weinstein, A., eds., Northern California Sym-plectic Geometry Seminar, American Mathematical Society, Providence, RI, 1999.

[133] Weinstein, A., Almost invariant submanifolds for compact group actions, J. EuropeanMath. Soc., 2 (2000), 53–86.

[134] Mikami, K., and Weinstein, A., Self-similarity of Poisson structures on tori, in Pois-son Geometry, Banach Center Publications, Vol. 51, Polish Scientific Publishers PWN,Warsaw, 2000, 211–217.

[135] Weinstein, A., Linearization problems for Lie algebroids and Lie groupoids, Lett. Math.Phys., 52 (2000), 93–102.

[136] Weinstein, A., Omni-Lie algebras, RIMS Kôkyûroku, 1176 (2000), 95–102.[137] Weinstein,A., Review of Riemannian Geometry During the Second Half of the Twentieth

Century by Marcel Berger, Bull. London Math. Soc., 33 (2001), 11.[138] Kinyon, M. K., and Weinstein, A., Leibniz algebras, Courant algebroids, and multipli-

cations on reductive homogeneous spaces, Amer. J. Math., 123 (2001), 525–550.[139] Weinstein, A., Poisson geometry of discrete series orbits, and momentum convexity for

noncompact group actions, Lett. Math. Phys., 56 (2001), 17–30.[140] Marsden, J., and Weinstein, A., Some comments on the history, theory, and applications

of symplectic reduction, in Landsman, N. P., Pflaum, M., and Schlichenmaier, M., eds.,Quantization of Singular Symplectic Quotients, Birkhäuser, Basel, 2001, 1–19.

[141] Hirsch, M. W., and Weinstein, A., Fixed points of analytic actions of supersoluble Liegroups on compact surfaces, Ergodic Theory Dynam. Systems, 21 (2001), 1783–1787.

[142] Ševera, P., and Weinstein, A., Poisson geometry with a 3-form background, Progr. The-oret. Phys. Suppl. Ser., 144 (2002), 145–154.

[143] Weinstein,A., Linearization of regular proper groupoids, J. Inst. Math. Jussieu, 1 (2002),493–511.

[144] Newton, P. K., Holmes, P., and Weinstein,A., eds., Geometry, Mechanics, and Dynamics:Special Volume in Honor of the 60th Birthday of J. E. Marsden, Springer-Verlag, NewYork, 2002.

[145] Bursztyn, H., and Weinstein, A., Picard groups in Poisson geometry, Moscow Math. J.,4 (2004), 39–66.

[146] Weinstein, A., The geometry of momentum, in Proceedings of the Conference on“Geometry in the 20th Century: 1930–2000,’’ (Paris, September 2001), to appear;math.SG/0208108.

[147] Bursztyn, H., Crainic, M., Weinstein, A., and Zhu, C., Integration of twisted Diracbrackets, Duke Math. J., 123 (2004), 549–607.

[148] Tang, X., and Weinstein, A., Quantization and Morita equivalence for constant Diracstructures on tori, Ann. Inst. Fourier, 54 (2004), to appear; math.QA/0305413.

[149] Weinstein, A., The Maslov gerbe, Lett. Math. Phys., to appear; math.SG/0312274.

About Alan Weinstein xxiii

[150] Bursztyn, H., and Weinstein, A., Poisson geometry and Morita equivalence, in PoissonGeometry, Deformation Quantization, and Group Representations, London Mathemati-cal Society Lecture Note Series, Cambridge University Press, Cambridge, UK, to appear;preprint math.SG/0402347.

[151] Weinstein, A., Integrating the nonintegrable, in Proceedings of the Workshop “Feuil-letages: Quantification Géométrique,’’ Maison des Science de l’Homme, Paris, 2004.

Dirac structures, momentum maps, and quasi-Poissonmanifolds

Henrique Bursztyn1 and Marius Crainic2

1 Department of MathematicsUniversity of TorontoToronto, ON M5S 3G3Canadahenrique@math.toronto.edu

2 Department of MathematicsUtrecht UniversityP. O. Box 80.010, 3508 TAUtrechtThe Netherlandscrainic@math.uu.nl

Dedicated to Alan Weinstein for his 60th birthday.

Abstract. We extend the correspondence between Poisson maps and actions of symplecticgroupoids, which generalizes the one between momentum maps and Hamiltonian actions, to therealm of Dirac geometry.As an example, we show how Hamiltonian quasi-Poisson manifolds fitinto this framework by constructing an “inversion’’ procedure relating quasi-Poisson bivectorsto twisted Dirac structures.

1 Introduction

This paper builds on three ideas pursued by Alan Weinstein in some of his many fun-damental contributions to Poisson geometry: First, Lie algebroids play a prominentrole in the study of Poisson manifolds [8, 30]; second, Poisson maps can be re-garded as generalized momentum maps for actions of symplectic groupoids [25, 31];third, Poisson structures on manifolds are particular examples of more general objectscalled Dirac structures [12, 13, 28]. The main objective of this paper is to combinethese three ideas in order to extend the notion of “momentum map’’ to the realm ofDirac geometry. As an application, we obtain an alternative approach to Hamiltonianquasi-Poisson manifolds [2] which answers many of the questions posed in [28, 31],shedding light on the relationship between various notions of generalized Poissonstructures, Hamiltonian actions and reduced spaces.

Let g be a Lie algebra, and consider its dual g∗, equipped with its Lie–Poissonstructure. The central ingredients in the formulation of classical Hamiltonian g-actions

2 H. Bursztyn and M. Crainic

are a Poisson manifold (Q, πQ) and a Poisson map J : Q → g∗, which we use todefine an action of g onQ by Hamiltonian vector fields:

g −→ X (Q), v �→ XJv := idJv (πQ), (1.1)

where Jv ∈ C∞(Q) is given by Jv(x) = 〈J (x), v〉. For the global picture, we assumethat J is a complete Poisson map [8, Sec. 6.2], in which case the infinitesimal action(1.1) can be integrated to an action of the connected, simply connected Lie group Gwith Lie algebra g, in such a way that J becomes G-equivariant with respect to thecoadjoint action of G on g∗. The map J is called a momentum map for the G-actionon Q, and we refer to the G-action as Hamiltonian. A key observation, described in[25, 31], is that this construction of a Hamiltonian action out of a Poisson map holdsin much more generality: one may replace g∗ by any Poisson manifold, as long asLie groups are replaced by symplectic groupoids [29]. In this sense, any Poisson mapcan be seen as a “Poisson-manifold valued moment map.’’

In this paper, we show that the correspondence between Poisson maps and Hamil-tonian actions by symplectic groupoids can be further extended to the context of Diracgeometry: in this setting, Poisson maps must be replaced by special types of Diracmaps called Dirac realizations (see Definition 3.11); for the associated global actions,twisted presymplectic groupoids [6] (alternatively called quasi-symplectic groupoids[33]) play the role of symplectic groupoids. Our main results show that various im-portant notions of generalized Hamiltonian actions, such as the “quasi’’ objects of[2, 3], fit nicely into the Dirac geometry framework.

We organize our results as follows.In Section 2, we discuss important connections between Lie algebroids and bivec-

tor fields. Our main result is that, just as ordinary Poisson structures give rise to Liealgebroid structures on their cotangent bundles, a quasi-Poisson manifold [2, Def. 2.1](M, π) defines a Lie algebroid structure on T ∗M ⊕ g, where g is the Lie algebra ofthe Lie group acting onM . The leaves of this Lie algebroid coincide with the leavesof the “quasi-Hamiltonian foliation’’ of [2, Sec. 9] in the Hamiltonian case, though,in our framework, we make no assumption about the existence of group-valued mo-ment maps.

In Section 3, we study Hamiltonian actions in the context of Dirac geometry at theinfinitesimal level. We observe that Dirac realizations, like Poisson maps, are alwaysassociated with Lie algebroid actions. (This is, in fact, the guiding principle in our def-inition of Dirac realizations.) After discussing how classical notions of infinitesimalHamiltonian actions fit into this framework, we prove the main result of the section:Dirac realizations of Cartan–Dirac structures on Lie groups [6, 28] are equivalentto quasi-Poisson g-manifolds carrying group-valued moment maps. This equivalenceinvolves an “inversion’’procedure relating twisted Dirac structures and quasi-Poissonbivectors, revealing that these two objects are in a certain sense “mirror’’ to one an-other. The main ingredients in this discussion are the Lie algebroids of Section 2 andthe bundle maps which appear in [6] as infinitesimal versions of multiplicative 2-forms. This result explains, in particular, the relationship between Cartan–Dirac andquasi-Poisson structures on Lie groups; on the other hand, it recovers the correspon-dence proved in [2, Thm. 10.3] between “nondegenerate’’Hamiltonian quasi-Poisson

Dirac structures, momentum maps, and quasi-Poisson manifolds 3

manifolds (i.e., those for which the Lie algebroids of Section 2 are transitive) andquasi-Hamiltonian spaces [3].

In Section 4, we study moment maps in Dirac geometry from a global point ofview. We show that complete Dirac realizations “integrate’’to presymplectic groupoidactions, which are natural extensions of those studied in [33]. As our main example,we show that the “integration’’ of Dirac realizations of Cartan–Dirac structures onLie groups results in Hamiltonian quasi-Poisson G-manifolds. Finally, we show thatthe natural reduction procedure in the setting of Dirac geometry encompasses var-ious classical reduction theorems [21, 24, 25] as well as their “quasi’’ counterparts[2, 3, 33].

We remark, following an observation of E. Meinrenken, that the results concerningquasi-Poisson manifolds in this paper only require the Lie algebras to be quadratic, incontrast with some of the constructions in [2], in which the positivity of the bilinearforms plays a key role. (In particular, our results hold for quasi-PoissonG-manifoldswhen G is a noncompact semisimple Lie group.) Most of our constructions can becarried out in the more general setting of [1], but this will be discussed in a separatepaper.

A work which gave initial motivation and is closely related to the present paper isthat of Xu [33], in which a Morita theory of quasi-symplectic groupoids is developedin order to compare “moment map theories.’’ Our results show that twisted Diracstructures complement Xu’s picture in two ways: on the one hand, by providing theinfinitesimal framework for Morita equivalence; on the other hand, by leading to moregeneral “modules’’ (i.e., Hamiltonian spaces).

It is a pleasure to dedicate this paper toAlan Weinstein, whose work and insightfulideas have been an unlimited source of inspiration to us.

Notation. We use the following conventions for bundle maps: if π is a bivector fieldonM , thenπ� : T ∗M → TM ,α �→ π(α, ·); ifω is a 2-form, thenω� : TM → T ∗M ,X �→ ω(X, ·).

The space of k-multivector fields onM is denoted by X k(M).On a Lie groupG, with Lie algebra g, (·, ·)g will denote a bi-invariant nondegen-

erate quadratic form; we writeφG for the associated Cartan 3-form, andχG ∈ �3g forthe dual trivector. The Lie algebrag is identified with right-invariant vector fields onG.

2 Lie algebroids, bivector fields, and Poisson geometry

2.1 Lie algebroids

A Lie algebroid over a manifold M is a vector bundle A → M together with a Liealgebra bracket [·, ·] on the space of sections �(A), and a bundle map ρ : A→ TM ,called the anchor, satisfying the Leibniz identity

[ξ, f ξ ′] = f [ξ, ξ ′] + Lρ(ξ)(f )ξ ′ for ξ, ξ ′ ∈ �(A) and f ∈ C∞(M). (2.1)

Whenever there is no risk of confusion, we will write Lρ(ξ) simply as Lξ .

4 H. Bursztyn and M. Crainic

If M is a point, then a Lie algebroid over M is a Lie algebra in the usual sense.An important feature of Lie algebroids A → M is that the image of the anchor,ρ(A) ⊆ TM , defines a generalized integrable distribution, determining a singularfoliation of M . The leaves of this foliation are the orbits of the Lie algebroid. Thefollowing example plays a key role in the study of Hamiltonian actions and momentmaps.

Example 2.1 (transformation Lie algebroids). Consider an infinitesimal action of aLie algebra g on a manifold M , given by a Lie algebra homomorphism ρ : g →X (M). The transformation Lie algebroid associated with this action is the trivialvector bundleM × g, with anchor (x, v) �→ ρ(x, v) := ρ(v)(x) and Lie bracket on�(M × g) = C∞(M, g) defined by

[u, v](x) := [u(x), v(x)]g + (ρ(u(x)) · v)(x)− (ρ(v(x)) · u)(x). (2.2)

We often denote a transformation Lie algebroid by g �M .Note that [·, ·] is uniquely determined by the condition that it coincides with [·, ·]g

on constant functions and the Leibniz identity. The orbits of g �M are the g-orbitsonM .

The remainder of this section is devoted to examples of Lie algebroids closelyrelated to Poisson manifolds.

2.2 Bivector fields and Poisson structures

If (M, π) is a Poisson manifold, then T ∗M carries a Lie algebroid structure withanchor

π� : T ∗M → TM, β(π�(α)) = π(α, β), (2.3)

and bracket[α, β] = Lπ�(α)(β)− Lπ�(β)(α)− dπ(α, β), (2.4)

uniquely characterized by [df, dg] = d{f, g} and the Leibniz identity (2.1). Here, asusual, {f, g} = π(df, dg) is the Poisson bracket on C∞(M). In this case, the orbitsof T ∗M are the symplectic leaves ofM , i.e., the integral manifolds of the distributiondefined by the Hamiltonian vector fields Xf = π�(df ).Example 2.2 (Lie–Poisson structures). Let (g, [·, ·]) be a Lie algebra, and considerg∗ equipped with the associated Lie–Poisson structure

{f, g}(µ) := 〈µ, [df (µ), dg(µ)]〉, µ ∈ g∗. (2.5)

Under the identification T ∗g∗ ∼= g∗×g, one can see that the Lie algebroid structure onT ∗g∗ induced by (2.5) is that of a transformation Lie algebroid g � g∗ (see Example2.1) and a direct computation reveals that the action in question is the coadjointaction.

Dirac structures, momentum maps, and quasi-Poisson manifolds 5

If π ∈ X 2(M) is an arbitrary bivector field, let us consider π�, [·, ·], {·, ·} andXf as defined by the previous formulas, and let χπ ∈ X 3(M) be the trivector fielddefined by

χπ := [π, π], (2.6)

i.e., χπ satisfies

1

2χπ(df, dg, dh) = {f, {g, h}} + {g, {h, f }} + {h, {f, g}} = {f, {g, h}} + c.p.,

where we use c.p. to denote cyclic permutations.

Lemma 2.3. For any bivector field π onM , one has

π�([α, β]) = [π�(α), π�(β)] − 1

2iα∧β(χπ), (2.7)

[α, [β, γ ]] + c.p. = 1

2(Liα∧β(χπ )(γ )+ c.p.)− d(χπ(α, β, γ )), (2.8)

for α, β, γ ∈ 1(M). As a result, the following are equivalent:

(i) π is a Poisson tensor;(ii) π� : 1(M)→ X (M) preserves the brackets;

(iii) the bracket [·, ·] on 1(M) satisfies the Jacobi identity;(iv) (T ∗M,π�, [·, ·]) is a Lie algebroid.

Proof. The key remark is that the difference between the left- and right-hand sides ofeach of (2.7) and (2.8) is C∞(M)-multilinear in α, β and γ . So it is enough to provethe identities on exact forms, which is immediate. �Example 2.4 (twisted Poisson manifolds). Consider a closed 3-form φ ∈ 3(M). Aφ-twisted Poisson structure on M [19, 27] consists of a bivector field π ∈ X 2(M)

satisfying1

2[π, π] = π�(φ).

Here, we abuse notation and write π� to denote the map induced by (2.3) on exte-rior algebras. We know from Lemma 2.3 that the bracket (2.4) induced by π is notpreserved by π� and does not satisfy the Jacobi identity. However,

π�([α, β] + iπ�(α)∧π�(β)(φ)) = [π�(α), π�(β)].Hence, if we define a “twisted’’ version of the bracket (2.4),

[α, β]φ := [α, β] + iπ�(α)∧π�(β)(φ),

then π� will preserve this new bracket, and [·, ·]φ satisfies the Jacobi identity. Asa result, (T ∗M,π�, [·, ·]φ) is a Lie algebroid. We leave it to the reader to prove a“twisted’’ version of Lemma 2.3.

6 H. Bursztyn and M. Crainic

2.3 The Lie algebroid of a quasi-Poisson manifold

LetG be a Lie group with Lie algebra g, equipped with a bi-invariant nondegeneratequadratic form (·, ·)g. Let φG be the bi-invariant Cartan 3-form on G, and let χG ∈�3g be its dual trivector. On Lie algebra elements u, v,w ∈ g, we have

φG(u, v,w) = χG(u∨, v∨, w∨) = 1

2(u, [v,w])g,

where u∨, v∨, w∨ are dual to u, v,w via (·, ·)g; when (·, ·)g is a metric and ea is anorthonormal basis of g, we can write1

χG = 1

12

∑(ea, [eb, ec])gea ∧ eb ∧ ec.

A quasi-Poisson G-manifold [2] consists of a G-manifold M together with a G-invariant bivector field π satisfying

χπ = ρM(χG), (2.9)

where ρM : g −→ X (M) is the associated infinitesimal action, and we keep the samenotation for the induced maps of exterior algebras. WhenM is just a g-manifold, wecall the corresponding object a quasi-Poisson g-manifold. The two notions are relatedby the standard procedure of integration of infinitesimal actions; in particular, theycoincide ifM is compact and G is simply connected.

In analogy with ordinary or twisted Poisson manifolds, are quasi-Poisson struc-tures also associated with Lie algebroids? As we now discuss, the answer is yes. Letus consider a more general setup: let M be a g-manifold and let π ∈ X 2(M) bean arbitrary bivector field. Motivated by [2, Sec. 9], we consider on T ∗M ⊕ g the“anchor’’ map

r : T ∗M ⊕ g −→ TM, r(α, v) = π�(α)+ ρM(v), (2.10)

combining the bivector field and the action. On sections of T ∗M ⊕ g, we considerthe bracket defined by

[(α, 0), (β, 0)] =([α, β], 1

2iρ∗M(α∧β)(χG)

), (2.11)

[(0, v), (0, v′)] = (0, [v, v′]), (2.12)

[(0, v), (α, 0)] = (LρM(v)(α), 0), (2.13)

for all 1-forms α, β ∈ 1(M) and all v, v′ ∈ g (thought of as constant sections inC∞(M, g)). As in Example 2.1, the definition of the bracket on general elements in�(T ∗M ⊕ g) = 1(M) ⊕ C∞(M, g) is obtained from the Leibniz formula (2.1).With these definitions, we obtain a quasi-Poisson analogue of Lemma 2.3.

1 More generally, with no positivity assumptions on (·, ·)g, we can write χG =1

12∑(ea, [eb, ec])gfa ∧ fb ∧ fc, where fa is a basis of g satisfying (fa, eb)g = δab.

A similar observation holds for (2.25).

Dirac structures, momentum maps, and quasi-Poisson manifolds 7

Theorem 2.5. LetM be a g-manifold equipped with a bivector field π . The followingare equivalent:

(i) (M, π) is a quasi-Poisson g-manifold;(ii) r : 1(M)⊕ C∞(M, g)→ X (M) preserves brackets;

(iii) the bracket [·, ·] on 1(M)⊕ C∞(M, g) satisfies the Jacobi identity;(iv) (T ∗M ⊕ g, r, [·, ·]) is a Lie algebroid.

Proof. Note that r preserves the bracket (2.12), since ρM is an action. From theidentity (2.7) in Lemma 2.3, it follows that r preserves the bracket of type (2.11) ifand only if χπ = ρM(χG). On the other hand, r preserves the bracket of type (2.13)if and only if π�LρM(v)(ξ) = LρM(v)π�(ξ), which is equivalent to the g-invarianceof π . This shows that (i) and (ii) are equivalent.

Let us prove that (i) implies (iii); from the proof, the converse will be clear.Assuming (i), we must show that [·, ·] on 1(M) ⊕ C∞(M, g) satisfies the Jacobiidentity. On elements of type (0, v), this reduces to the Jacobi identity for g (or,alternatively, for g�M). On elements (0, v), (0, w) and (α, 0), the Jacobi identity of[·, ·] reduces to the fact that ρM is an action. Computing the “jacobiator’’ for elementsof type (0, v), (α, 0), (β, 0), we see that the first component is

[LρM(v)(α), β] + [α,LρM(v)(β)] − LρM(v)([α, β]). (2.14)

Using the Leibniz identity, we see that the C∞(M)-linearity of (2.14) with respect toβ is equivalent to π�Lρ(v)(β) = Lρ(v)π�(β), i.e., to the g-invariance of π . Hence, ifπ is invariant, (2.14) is C∞(M)-linear on α and β, and then one can check that it iszero by looking at the particular case when α and β are exact. The second componentof the jacobiator of (0, v), (α, 0), (β, 0) can be computed similarly.

To complete the proof that (i) implies (iii), we must deal with the Jacobi identity forelements of type (α, 0), (β, 0), (γ, 0). To this end, we first need to find the expressionfor the bracket between elements of type (0, v) and (α, 0), with v ∈ C∞(M, g) notnecessarily constant: pairing dv ∈ 1(M; g) with an element µ ∈ C∞(M, g∗) givesus a 1-form onM , denoted by Av(µ), satisfying the following two properties:

Af v(µ) = fAv(µ)+ µ(v)df, and Av(fµ) = fAv(µ),for f ∈ C∞(M). We claim that

[(0, v), (α, 0)] = (LρM(v)(α)− Av(ρ∗M(α)),−Lπ�(α)(v)). (2.15)

To see this, note that (2.15) holds when v is constant, and the difference between theleft- and right-hand sides is C∞(M)-linear in v. We remark that

Aiµ∧µ′ (χG)(µ′′)+ c.p. = 2d(χG(µ,µ

′, µ′′)). (2.16)

Again, it is easy to check this identity when µ, µ′ and µ′′ are constant, so (2.16)follows fromC∞(M)-linearity. Also, denoting χM := ρM(χG), a direct computationshows that

8 H. Bursztyn and M. Crainic

ρM(iρ∗M(α∧β)(χG)) = iα∧β(χM).We now turn to the computation of the jacobiator of the elements (α, 0), (β, 0) and(γ, 0), that we denote by Jac(α, β, γ ). For the first component of Jac(α, β, γ ), weobtain

([α, [β, γ ]]+c.p.)− 1

2(Liα∧β(χM)(γ )+c.p.)+

1

2(Aiα∧β(χM)(ρ

∗M(γ ))+c.p.). (2.17)

Combining the second identity of Lemma 2.3 with (2.16), we get that (2.17) equals

d((ρM(χG)− χπ)(α, β, γ )),which vanishes by the condition χπ = ρM(χG). So we are left with proving that thesecond component of Jac(α, β, γ ) vanishes, which amounts to showing that

iρ∗M([α,β]∧γ )(χG)+ c.p. = Lπ�(γ )iρ∗M(α∧β)(χG)+ c.p.. (2.18)

In order to do that, consider the operators iρ∗M([α,β]) and Lπ�(α)iρ∗M(β) −Lπ�(β)iρ∗M(α)acting on C∞(M,�g), for α, β ∈ 1(M).

Claim 2.6. On �g, seen as constant functions in C∞(M,�g), we have

iρ∗M([α,β]) = Lπ�(α)iρ∗M(β) − Lπ�(β)iρ∗M(α). (2.19)

Proof. Both operators are derivations of degree −1 on �g, hence it suffices to show(2.19) for elements v ∈ g. As we now check, this follows from the definition of thebracket induced by π and the invariance of π : on the one hand,

iρ∗M([α,β])(v) = [α, β](ρM(v))= iρM(v)Lπ�(α)(β)− iρM(v)Lπ�(β)(α)− iρM(v)dπ(α, β). (2.20)

Using that i[X,Y ] = LXiY − iYLX for vector fields X, Y , we have

iρM(v)Lπ�(α)(β) = Lπ�(α)(β(ρM(v)))− β([π�(α), ρM(v)])= Lπ�(α)iρ∗M(β)(v)− π(LρM(v)(α), β), (2.21)

where the last equality follows from the g-invariance of π . Using the identity (2.21)(and its analogue for α and β interchanged) in (2.20), (2.19) follows. �

Using the claim, we see that

iρ∗M([α,β]∧γ ) + c.p. = iρ∗M(γ )iρ∗M([α,β]) + c.p.= (iρ∗M(γ )Lπ�(α)iρ∗M(β) − iρ∗M(γ )Lπ�(β)iρ∗M(α))+ c.p., (2.22)

when restricted to constant elements in C∞(M,�g) . On the other hand, it followsfrom (2.19) that, on �g, we can write

iρ∗M([α,β]) = [Lπ�(α), iρ∗M(β)] − [Lπ�(β), iρ∗M(α)] (2.23)

Dirac structures, momentum maps, and quasi-Poisson manifolds 9

since the Lie derivatives are zero on constant functions. But both sides of (2.23) areC∞(M)-linear, so this equality is valid for all C∞(M, g). So we can write

iρ∗M([α,β]∧γ ) + c.p. = −iρ∗M([α,β])iρ∗M(γ ) + c.p.= −([Lπ�(α), iρ∗M(β)] − [Lπ�(β), iρ∗M(α)])iρ∗M(γ ) + c.p.,

from which we deduce that

iρ∗M([α,β]∧γ ) + c.p. = 2(Lπ�(α)iρ∗M(β∧γ ) + c.p.)− (iρ∗M(γ )Lπ�(α)iρ∗M(β) − iρ∗M(β)Lπ�(α)iρ∗M(γ ) + c.p.).

On constant functions, we can use (2.22) to conclude that

(iρ∗M([α,β]∧γ ) + c.p.) = 2(Lπ�(α)iρ∗M(β∧γ ) + c.p.)− (iρ∗M([α,β]∧γ ) + c.p.),i.e., iρ∗M([α,β]∧γ ) + c.p. = Lπ�(α)iρ∗M(β∧γ ). Evaluating this identity at χG proves(2.18), showing that (i) implies (iii). Looking back at the proof, one can check thatthe same formulas show the converse, so that (i) and (iii) are equivalent.

Since (iii) and the Leibniz identity for [·, ·] are together equivalent to (iv), itfollows that (i)–(iv) are equivalent to each other. �Corollary 2.7. If (M, π) is a quasi-Poisson g-manifold, then the generalized distri-bution

π�(α)+ ρM(v) ⊆ TM, for α ∈ T ∗M, v ∈ g,

is integrable.

This result shows that the singular distribution discussed in [2, Thm. 9.2] inthe context of Hamiltonian quasi-Poisson manifolds is integrable even without thepresence of a moment map (and without the positivity of (·, ·)g). As in the case ofordinary Poisson manifolds, we call a quasi-Poisson manifold nondegenerate if itsassociated Lie algebroid is transitive (i.e., its anchor map is onto).

Example 2.8 (quasi-Poisson structures on Lie groups). LetG be a Lie group with Liealgebra g, which we assume to be equipped with an invariant nondegenerate quadraticform (·, ·)g. We consider G acting on itself by conjugation. As shown in [2, Sec. 3],the bivector field πG, defined on left invariant 1-forms by

πG(dl∗g−1(µ), dl

∗g−1(ν)) := 1

2((Adg−1 − Adg)(µ

∨), ν∨)g, (2.24)

where lg denotes left multiplication by g ∈ G, µ, ν ∈ g∗, and µ∨ is the element in gdual to µ via (·, ·)g, makes G into a quasi-Poisson G-manifold. If (·, ·)g is a metric,then we can write

πG = 1

2

∑ela ∧ era, (2.25)

where ea is an orthonormal basis of g and era (respectively, ela) are the correspondingright (respectively, left) translations.