Sophus Lie and Felix Klein: The Erlangen Program and Its Impact in Mathematics and Physics PDF
Preview Sophus Lie and Felix Klein: The Erlangen Program and Its Impact in Mathematics and Physics
Sophus Lie and Felix Klein: The Erlangen Program and Its Impact in Mathematics and Physics Lizhen Ji Athanase Papadopoulos Editors Editors: Lizhen Ji Athanase Papadopoulos Department of Mathematics Institut de Recherche Mathématique Avancée University of Michigan CNRS et Université de Strasbourg 530 Church Street 7 Rue René Descartes Ann Arbor, MI 48109-1043 67084 Strasbourg Cedex USA France 2010 Mathematics Subject Classification: 01-00, 01-02, 01A05, 01A55, 01A70, 22-00, 22-02, 22-03, 51N15, 51P05, 53A20, 53A35, 53B50, 54H15, 58E40 Key words: Sophus Lie, Felix Klein, the Erlangen program, group action, Lie group action, symmetry, projective geometry, non-Euclidean geometry, spherical geometry, hyperbolic geometry, transitional geometry, discrete geometry, transformation group, rigidity, Galois theory, symmetries of partial differential equations, mathematical physics ISBN 978-3-03719-148-4 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2015 European Mathematical Society Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum SEW A27 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: [email protected] Homepage: www.ems-ph.org Typeset using the authors’ TEX files: le-tex publishing services GmbH, Leipzig, Germany Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 9 8 7 6 5 4 3 2 1 Preface The Erlangen program provides a fundamental point of view on the place of trans- formation groups in mathematics and physics. Felix Klein wrote the program, but SophusLiealsocontributedtoitsformulation,andhiswritingsareprobablythebest exampleofhowthisprogramisusedinmathematics. Thepresentbookgivesthefirst modernhistoricalandcomprehensivetreatmentofthescope,applicationsandimpact of the Erlangen program in geometry and physics and the roles played by Lie and Kleininitsformulationanddevelopment. Thebookisalsointendedasanintroduc- tiontotheworksandvisionsofthesetwomathematicians. Itaddressesthequestion of what is geometry, how are its various facets connectedwith each other, and how aregeometryandgrouptheoryinvolvedinphysics.BesidesLieandKlein,thenames ofBernhardRiemann, HenriPoincare´, HermannWeyl, E´lie Cartan, Emmy Noether andothermajormathematiciansappearatseveralplacesinthisvolume. A conference was held at the University of Strasbourg in September 2012, as the90thmeetingoftheperiodicEncounterbetweenMathematiciansandTheoretical Physicists, whose subject was the same as the title of this book. The book does not faithfully reflect the talks given at the conference, which were generally more specialized. Indeed,ourplanwastohaveabookinterestingforawideaudienceand weaskedthepotentialauthorstoprovidesurveysandnottechnicalreports. We would like to thank Manfred Karbe for his encouragement and advice, and Hubert Goenner and Catherine Meusburger for valuable comments. We also thank Goenner,MeusburgerandArnfinnLaudalforsendingphotographsthatweuseinthis book. This work was supported in part by the French program ANR Finsler, by the GEARnetworkoftheNationalScienceFoundation(GEometricstructuresAndRep- resentationvarieties)andbyastayofthetwoeditorsattheErwinSchro¨dingerInsti- tuteforMathematicalPhysics(Vienna). LizhenJiandAthanasePapadopoulos AnnArborandStrasbourg,March2015 Contents Preface v Introduction xi 1 SophusLie,agiantinmathematics . . . . . . . . . . . . . . . . . . . . . 1 LizhenJi 2 FelixKlein: hislifeandmathematics . . . . . . . . . . . . . . . . . . . . 27 LizhenJi 3 KleinandtheErlangenProgramme . . . . . . . . . . . . . . . . . . . . 59 JeremyJ.Gray 4 Klein’s“ErlangerProgramm”:dotracesofitexistinphysicaltheories? 77 HubertGoenner 5 OnKlein’sSo-calledNon-Euclideangeometry . . . . . . . . . . . . . . . 91 NorbertA’Campo,AthanasePapadopoulos 6 WhataresymmetriesofPDEsandwhatarePDEsthemselves? . . . . . 137 AlexandreVinogradov 7 Transformationgroupsinnon-Riemanniangeometry . . . . . . . . . . 191 CharlesFrances 8 Transitionalgeometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 NorbertA’Campo,AthanasePapadopoulos 9 Ontheprojectivegeometryofconstantcurvaturespaces . . . . . . . . 237 AthanasePapadopoulos,SumioYamada 10 TheErlangenprogramanddiscretedifferentialgeometry . . . . . . . . 247 YuriB.Suris 11 Three-dimensionalgravity–anapplicationofFelixKlein’sideas inphysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 CatherineMeusburger 12 Invariancesinphysicsandgrouptheory . . . . . . . . . . . . . . . . . . 307 Jean-BernardZuber viii Contents ListofContributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 SophusLie. FelixKlein. Introduction The Erlangen program is a perspective on geometry through invariants of the auto- morphismgroupofaspace.TheoriginalreferencetothisprogramisapaperbyFelix Kleinwhichis usuallypresentedasthe exclusivehistoricaldocumentin this matter. EventhoughKlein’sviewpointwasgenerallyacceptedbythemathematicalcommu- nity,itsre-interpretationinthelightofmoderngeometries,andespeciallyofmodern theoriesofphysics,iscentraltoday. Therearenobooksonthemoderndevelopments ofthisprogram. Ourbookisonemodeststeptowardsthisgoal. The history of the Erlangen program is intricate. Klein wrote this program, but SophusLie madea verysubstantialcontribution,in promotingandpopularizingthe ideasitcontains. TheworkofLieongroupactionsandhisemphasisontheirimpor- tancewerecertainlymoredecisivethanKlein’scontribution. ThisiswhyLie’sname comes first in the title of the present volume. Another major figure in this story is Poincare´,andhisroleinhighlightingtheimportanceofgroupactionsisalsocritical. Thus, groups and group actions are at the center of our discussion. But their importance in mathematics had already been crucial before the Erlangen program wasformulated. From its earlybeginningin questionsrelatedto solutionsofalgebraicequations, group theory is merged with geometry and topology. In fact, group actions existed and were important before mathematicians gave them a name, even though the for- malizationofthenotionofagroupanditssystematicuseinthelanguageofgeometry tookplaceinthe19thcentury. Ifweconsidergrouptheoryandtransformationgroups as an abstraction of the notion of symmetry, then we can say that the presence and importanceofthisnotioninthesciencesandintheartswasrealizedinancienttimes. Today,thenotionofgroupisomnipresentinmathematicsand,infact,ifwewant to name one single concept which runs through the broad field of mathematics, it is the notion of group. Among groups, Lie groups play a central role. Besides their mathematical beauty, Lie groups have many applications both inside and out- sidemathematics. Theyareacombinationofalgebra,geometryandtopology. Besidesgroups,oursubjectincludesgeometry. Unlike the word “group” which, in mathematics has a definite significance, the word “geometry” is not frozen. It has several meanings, and all of them (even the mostrecentones)canbeencompassedbythemoderninterpretationofKlein’sidea. In the first version of Klein’s Erlangen program, the main geometries that are em- phasized are projective geometry and the three constant curvature geometries (Eu- clidean,hyperbolicandspherical),whichareconsideredthere,like affinegeometry, aspartofprojectivegeometry. Thisisduetothefactthatthetransformationgroups of all these geometries can be viewed as restrictions to subgroups of the transfor- mation groupof projectivegeometry. After these first examplesof group actions in geometry, the stress shifted to Lie transformation groups, and it gradually included manynewnotions,likeRiemannianmanifolds,andmoregenerallyspacesequipped xii Introduction with affine connections. Thereis awealth ofgeometrieswhichcanbe describedby transformation groups in the spirit of the Erlangen program. Several of these ge- ometries were studied by Klein and Lie; among them we can mention Minkowski geometry,complexgeometry,contactgeometryandsymplecticgeometry. Inmodern geometry, besides the transformations of classical geometry which take the form of motions, isometries, etc., new notions of transformations and maps between spaces arose. Today, there is a wealth of new geometries that can be described by trans- formation groupsin the spirit of the Erlangen program, including modern algebraic geometry where, according to Grothendieck’s approach, the notion of morphism is moreimportantthanthenotionofspace.1 Asaconcreteexampleofthisfact,onecan comparetheGrothendieck–Riemann–RochtheoremwiththeHirzebruch–Riemann– Roch. Theformer,whichconcernsmorphisms,ismuchstrongerthanthelatter,which concernsspaces. Besides Lie and Klein, several other mathematicians must be mentioned in this venture. Lie created Lie theory, but others’ contributions are also immense. About twodecadesbeforeKleinwrotehisErlangenprogram,Riemannhadintroducednew geometries,namely,inhisinaugurallecture,U¨berdieHypothesen,welchederGeo- metrie zu Grunde liegen (On the hypotheses which lie at the bases of geometry) (1854). These geometries, in which groups intervene at the level of infinitesimal transformations, are encompassed by the program. Poincare´, all across his work, highlightedtheimportanceofgroups. InhisarticleontheFutureofmathematics2,he wrote: “Amongthewordsthatexertedthemostbeneficialinfluence,Iwillpointout thewordsgroupandinvariant. Theymadeusforeseetheveryessenceofmathemat- ical reasoning. They showed us that in numerous cases the ancient mathematicians consideredgroupswithoutknowingit,andhow,afterthinkingthattheywerefaraway fromeachother,theysuddenlyendedupclosetogetherwithoutunderstandingwhy.” Poincare´ stressed several times the importance of the ideas of Lie in the theory of group transformations. In his analysis of his own works,3 Poincare´ declares: “Like Lie, Ibelievethatthenotion,moreorlessunconscious,ofacontinuousgroupisthe uniquelogicalbasisofourgeometry.” Killing,E´.Cartan,Weyl,Chevalleyandmany others refined the structuresof Lie theory andthey developedits globalaspectsand applications to homogeneous spaces. The generalization of the Erlangen program to these new spaces uses the notions of connections and gauge groups, which were 1SeeA.Grothendieck,ProceedingsoftheInternationalCongressofMathematicians,14–21August1958, Edinburgh, ed. J.A.Todd,CambridgeUniversityPress,p.103–118. Inthattalk, Grothendiecksketched his theoryofcohomologyofschemes. 2H. Poincare´, L’Avenir des mathe´matiques, Revue ge´ne´rale des sciences pures et applique´es 19 (1908) p.930–939.[Parmilesmotsquiontexerce´laplusheureuseinfluence,jesignaleraiceuxdegroupeetd’invariant. Ilsnousontfaitapercevoirl’essencedebiendesraisonnementsmathe´matiques;ilsnousontmontre´danscom- biendecaslesanciensmathe´maticiens conside´raientdesgroupessanslesavoir,etcomment,secroyantbien e´loigne´slesunsdesautres,ilssetrouvaienttouta`couprapproche´ssanscomprendrepourquoi.] 3Analysedesestravauxscientifiques,parHenriPoincare´.ActaMathematica,38(1921),p.3–135.[Comme Lie, jecrois que lanotion plus oumoins inconsciente degroupe continu estla seule baselogique denotre ge´ome´trie];p.127.TherearemanysimilarquotesinPoincare´’sworks.
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