introduction to lie groups and symplectic geometry - bryant r.l.

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1、AnIntroductiontoLieGroupsandSymplecticGeometryAseriesofninelecturesonLiegroupsandsymplecticgeometrydeliveredattheRegionalGeometryInstituteinParkCity,Utah,24June–20July1991.byRobertL.BryantDukeUniversityDurham,NCbryant@math.duke.eduThisisanunofficialversionofthenotesandwaslastmodifiedon20September

2、1993.The.dvifileforthispreprintwillbeavailablebyanonymousftpfrompublications.math.duke.eduinthedirectorybryantuntilthemanuscriptisacceptedforpublication.YoushouldgettheReadMefilefirsttoseeiftheversionthereismorerecentthanthisone.Pleasesendanycomments,correctionsorbugreportstotheabovee-mailaddress

3、.IntroductionThesearethelecturenotesforashortcourseentitled“IntroductiontoLiegroupsandsymplecticgeometry”whichIgaveatthe1991RegionalGeometryInstituteatParkCity,Utahstartingon24Juneandendingon11July.Thecoursereallywasdesignedtobeanintroduction,aimedatanaudienceofstu-dentswhowerefamiliarwithbasi

4、cconstructionsindifferentialtopologyandrudimentarydifferentialgeometry,whowantedtogetafeelforLiegroupsandsymplecticgeometry.MypurposewasnottoprovideanexhaustivetreatmentofeitherLiegroups,whichwouldhavebeenimpossibleevenifIhadhadanentireyear,orofsymplecticmanifolds,whichhaslatelyundergonesomethin

5、gofarevolution.Instead,ItriedtoprovideanintroductiontowhatIregardasthebasicconceptsofthetwosubjects,withanemphasisonexampleswhichdrovethedevelopmentofthetheory.Ideliberatelytriedtoincludeafewtopicswhicharenotpartofthemainstreamsubject,suchasLie’sreductionoforderfordifferentialequationsanditsrel

6、ationwiththenotionofasolvablegroupontheonehandandintegrationofODEbyquadratureontheother.Ialsotried,inthelaterlecturestointroducethereadertosomeoftheglobalmethodswhicharenowbecomingsoimportantinsymplecticgeometry.However,afulltreatmentofthesetopicsinthespaceofninelecturesbeginningattheelementar

7、ylevelwasbeyondmyabilities.Afterthelectureswereover,Icontemplatedreworkingthesenotesintoacomprehen-siveintroductiontomodernsymplecticgeometryand,aftersomesoul-searching,finallydecidedagainstthis.Thus,Ihavecontentedmyselfwithmakingonlymin