Progress in Mathematics Volume232 SeriesEditors HymanBass JosephOesterle´ AlanWeinstein The Breadth of Symplectic and Poisson Geometry Festschrift in Honor of Alan Weinstein Jerrold E. Marsden Tudor S. Ratiu Editors Birkha¨user Boston • Basel • Berlin JerroldE.Marsden TudorS.Ratiu CaliforniaInstituteofTechnology EcolePolytechniqueFe´de´raledeLausanne DepartmentofEngineering De´partementdeMathe´matiques andAppliedScience CH-1015Lausanne ControlandDynamicalSystems Switzerland Pasadena,CA91125 U.S.A. AMSSubjectClassifications:53Dxx,17Bxx,22Exx,53Dxx,81Sxx LibraryofCongressCataloging-in-PublicationData ThebreadthofsymplecticandPoissongeometry:festschriftinhonorofAlanWeinstein/ JerroldE.Marsden,TudorS.Ratiu,editors. p.cm.–(Progressinmathematics;v.232) Includesbibliographicalreferencesandindex. ISBN0-8176-3565-3(acid-freepaper) 1.Symplecticgeometry.2.Geometricquantization.3.Poissonmanifolds.I.Weinstein, Alan,1943-II.Marsden,JerroldE.III.Ratiu,TudorS.IV.Progressinmathematics (Boston,Mass.);v.232. QA665.B742004 516.3’.6-dc22 2004046202 ISBN0-8176-3565-3 Printedonacid-freepaper. (cid:1)c2005Birkha¨userBoston Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten permissionofthepublisher(Birkha¨userBoston,c/oSpringerScience+BusinessMediaInc.,Rights andPermissions,233SpringStreet,NewYork,NY10013,USA),exceptforbriefexcerptsincon- nectionwithreviewsorscholarlyanalysis.Useinconnectionwithanyformofinformationstorage andretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodologynow knownorhereafterdevelopedisforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarksandsimilarterms,evenifthey arenotidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyare subjecttoproprietaryrights. 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Contents Preface ............................................................ ix AcademicgenealogyofAlanWeinstein ................................. xiii AboutAlanWeinstein................................................ xv StudentsofAlanWeinstein............................................ xv AlanWeinstein’spublications ......................................... xvi Diracstructures,momentummaps,andquasi-Poissonmanifolds HenriqueBursztyn,MariusCrainic ................................... 1 ConstructionofRicci-typeconnectionsbyreductionandinduction MichelCahen,SimoneGutt,LorenzSchwachhöfer....................... 41 Amathematicalmodelforgeomagneticreversals J.J.Duistermaat.................................................. 59 Nonholonomicsystemsviamovingframes:CartanequivalenceandChaplygin Hamiltonization KurtEhlers,JairKoiller,RichardMontgomery,PedroM.Rios............. 75 Thompson’sconjectureforrealsemisimpleLiegroups SamEvens,Jiang-HuaLu .......................................... 121 TheWeinsteinconjectureandtheoremsofnearbyandalmostexistence ViktorL.Ginzburg ................................................ 139 Simplesingularitiesandintegrablehierarchies AlexanderB.Givental,TodorE.Milanov .............................. 173 Momentummapsandmeasure-valuedsolutions(peakons,filaments,and sheets)fortheEPDiffequation DarrylD.Holm,JerroldE.Marsden.................................. 203 viii Contents Higher homotopies and Maurer–Cartan algebras: Quasi-Lie–Rinehart, Gerstenhaber,andBatalin–Vilkoviskyalgebras JohannesHuebschmann ............................................ 237 Localizationtheoremsbysymplecticcuts LisaJeffrey,MikhailKogan ......................................... 303 RefinementsoftheMorsestratificationofthenormsquareofthemomentmap FrancesKirwan .................................................. 327 Quasi,twisted,andallthat…inPoissongeometryandLiealgebroidtheory YvetteKosmann-Schwarzbach ....................................... 363 Minimalcoadjointorbitsandsymplecticinduction BertramKostant .................................................. 391 Quantizationofpre-quasi-symplecticgroupoidsandtheirHamiltonianspaces CamilleLaurent-Gengoux,PingXu................................... 423 Dualityandtriplestructures KirillC.H.Mackenzie ............................................. 455 Starexponentialfunctionsastwo-valuedelements Y.Maeda,N.Miyazaki,H.Omori,A.Yoshioka .......................... 483 FrommomentummapsanddualpairstosymplecticandPoissongroupoids Charles-MichelMarle ............................................. 493 ConstructionofspectralinvariantsofHamiltonianpathsonclosedsymplectic manifolds Yong-GeunOh.................................................... 525 TheuniversalcoveringandcoveredspacesofasymplecticLiealgebraaction Juan-PabloOrtega,TudorS.Ratiu ................................... 571 Poissonhomotopyalgebra:Anidiosyncraticsurveyofhomotopyalgebraic topicsrelatedtoAlan’sinterests JimStasheff...................................................... 583 DiracsubmanifoldsofJacobimanifolds IzuVaisman...................................................... 603 Quantummapsandautomorphisms SteveZelditch .................................................... 623 Preface Alan Weinstein is one of the top mathematicians in the world working in the area of symplectic and differential geometry. His research on symplectic reduction, La- grangiansubmanifolds,groupoids,applicationstomechanics,andrelatedareashas had 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 addition to reflecting the vitality of the field, this is a celebratory volume to honor Alan’s 60th birthday. His birthday was also celebrated inAugust, 2003 with a wonderful week-longconferenceheldattheESI:theErwinSchrödingerInternationalInstitute forMathematicalPhysicsinVienna. Alan was born in New York in 1943. He was an undergraduate at MIT and a graduatestudentatUCBerkeley,wherehewasawardedhisPh.D.in1967underthe directionofS.S.Chern.AfterspendingpostdoctoralyearsatIHESnearParis,MIT, andtheUniversityofBonn,hejoinedthefacultyatUCBerkeleyin1969,becoming afullProfessorin1976. Alan has received many honors, including anAlfred P. Sloan Foundation Fel- lowship, a Miller Professorship (twice), a Guggenheim Fellowship, election to the American Academy of Arts and Sciences in 1992, and an honorary degree at the UniversityofUtrechtin2003. AttheESIconference,S.S.Chern,Alan’sadvisor,sentthefollowingwordsto celebratetheoccasion: “IamgladaboutthiscelebrationandIthinkAlanrichlydeservesit.Alan came to me in the early sixties as a graduate student at the University of California at Berkeley.At that time, a prevailing problem in our geometry group,andthegeometrycommunityatlarge,waswhetheronaRiemannian manifold the cut locus and the conjugate locus of a point can be disjoint. Alanimmediatelyshowedthatthiswaspossible.Theresultbecamepartof his Ph.D. thesis, which was published in the Annals of Mathematics. He received his Ph.D. degree in a short period of two years. I introduced him toIHESandtheFrenchmathematicalcommunity.Hestaysclosewiththem andwiththemathematicalideasofCharlesEhresmann.Heisoriginaland x Preface often came up with ingenious ideas.An example is his contribution to the solutionoftheBlaschkeconjecture.Iamveryproudtocounthimasoneof mystudentsandIhopehewillremaininterestedinmathematicsuptomy age,whichisnow91.’’ Alan’stechnicalcontributionsarewideranginganddeep.Asmanyofhisearly papersinhispublicationlistillustrate,hestartedoffinhisthesisandtheyearsim- mediatelyfollowinginpuredifferentialgeometry,atopichehascomebacktofrom timetotimethroughouthiscareer. AlreadystartingwithhispostdocyearsandhisearlycareeratBerkeley,hebecame interestedinsymplecticgeometryandmechanics.Inthisareaherapidlyestablished himselfasoneoftheworld’sauthorities,producingimportantanddeepresultsranging from reduction theory to Lagrangian and Poisson manifolds to studies of periodic orbits in Hamiltonian systems. He also did important work in fluid mechanics and plasmaphysicsandthroughthiswork,heestablishedwarmrelationswiththeBerkeley physicistsAllanKaufmanandRobertLittlejohn. Alan’simportantworkonperiodicorbitsinHamiltoniansystemsledhimeven- tuallytoformulatethe“Weinsteinconjecture,’’namelythatforagivenHamiltonian flow on a symplectic manifold, there must be at least one closed orbit on a regular compactcontacttypelevelsetoftheHamiltonian.AlongwithArnold’sconjecture, theWeinsteinconjecturehasbeenoneofthedrivingforcesinsymplectictopology overthelasttwodecades. Alankeptuphisinterestinsymplecticreductiontheorythroughouthislaterwork. For instance, he laid some important foundation stones in the theory of semidirect product reduction as well as in singular reduction through his work on Satake’s V-manifolds, along with finding important links with singular structures in moduli spaces. Intertwinedwithhisworkonsymplecticgeometryandmechanics,hedidexten- siveworkongeometricPDE,eigenvalues,theSchrödingeroperatorandgeometric quantization. Alan took the point of view of microlocal analysis and phase space structuresinhisworkinthisarea,emphasizingthelinkswithquantummechanics. Preface xi HisworkonthelimitdistributionofeigenvalueclustersintermsofthegeodesicRadon transformofthepotentialinspiredalargenumberofrelatedarticles.Heshowedthat thegeodesicflowofaZollsurfacewassymplecticallyequivalenttothatofaround sphere,andhencethatitsLaplaciancouldbeconjugatedgloballytotheroundLapla- cian plus a pseudodifferential potential. This work inspired many other results on conjugacies. OneofAlan’sfundamentalcontributionstoPoissongeometrywastheintroduc- tionofsymplecticgroupoidsin1987,whichmarkstheofficialbeginningofhis“oids’’ period. In these works, he makes sweeping generalizations about a wide variety of constructionsinsymplecticgeometry,including(withCourant)theimportantnotion of Dirac structures. During this period of generalizations he constantly returned to specific topics in symplectic and Poisson geometry, such as geometric phases and Poisson Lie groups, in addition to making other key links. For instance, symplec- tic groupoids are used to link Poisson geometry to noncommutative geometry, and groupoidsarealsointimatelyrelatedtomanyotherareas,includingsymmetriesand reduction,dualpairs,quantizationandthetheoryofsigmamodels.Oneofthecentral ideas is that the usual theory of Hamiltonian actions, momentum maps, and sym- plectic reduction makes sense in the more general context of actions of symplectic groupoids;inthissetting,momentummapsarePoissonmapstakingvaluesingeneral Poissonmanifolds,ratherthanjustLie–Poissonmanifolds(thatis,dualsofLiealge- bras).Alanhasraisedthequestionofwhetherthisframeworkcanbefurtherextended to include new notions of momentum maps such as quasi-Poisson manifolds with group-valuedmomentummapsaswellasoptimalmomentummaps. Alan is well known not only for his brilliant papers and conjectures, but also forhisgeneralphilosophy,suchasthesymplecticcreed:EverythingisaLagrangian submanifold.Thoseofuswhoknowhimwellalsoappreciatehisveryspecialinsight. For example, in the middle of a discussion (for instance, as we both had in our joint works on semidirect product reduction as well as stability theory) he will say somethinglikewhatyouarereallydoingis... andthengiveussomeusuallyvery specialinsightthatinvariablysubstantiallyimprovesthewholeproject. Alan also has a very interesting and charming sense of humor that even makes its way into his papers from time to time. For instance,Alan had great fun in his paperswiththe“EastCoast–WestCoast’’discussionsofwhetheroneshouldusethe term momentum map or moment map. He also gave us a good laugh with the term symplecticbonesasitrelatestotheFrenchtranslationofPoissonasFish. Alanisagreateducator.Hislectures,evenonCalculus,arealwaysatreatandare veryinspiringfortheirspecialinsight,theirwitandlivelypresentation.Hisenthu- siasmformathematicsisinfectious.Onestorythatcomestomindontheeducation front is this: during the days when he was exceptionally keen about groupoids, he waspreparingalectureforundergraduatesonthesubject.Someofusconvincedhim topresentitasacolloquiumlectureforfaculty,keepinginmindtheoldadvice“no colloquiumtalkcanbetoosimple.’’Itwas,infact,notonlyabeautifulcolloquium talk,butwasperfectlypitchedforthefaculty,anditbecameapopulararticleinthe NoticesoftheAmericanMathematicalSociety.Partofbeingagoodeducatorisbeing