Sophus Lie and Felix Klein: The Erlangen Program and Its Impact in Mathematics and Physics

346 Pages·2015·3.785 MB·English

by  Lizhen Ji, Athanase Papadopoulos;Klein, Felix, Lie, Sophus (eds.)

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
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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.