Theory of Transformation Groups I: General Properties of Continuous Transformation Groups. A Contemporary Approach and Translation

640 Pages·2015·5.13 MB·English

by  Sophus Lie

Sophus Lie
Theory
of Transformation
Groups I
General Properties
of Continuous Transformation Groups.
A Contemporary Approach
and Translation
Editor and Translator:
Joël Merker
Theory of Transformation Groups I
Sophus Lie
Theory of Transformation
Groups I
General Properties of Continuous
Transformation Groups.
A Contemporary Approach
and Translation
With the collaboration of Friedrich Engel
ë
Editor and Translator: Jo l Merker
123
Author Editor andTranslator
SophusLie(1842–1899) JoëlMerker
Laboratoire de Mathématiques, Facultédes
Collaborator Sciencesd’Orsay
Friedrich Engel(1861–1941) UniversitéParis-Sud11
Orsay
France
First edition published in German language by B.G. Teubner, Leipzig, with the title:
Theorie derTransformationsgruppen in 1888,1930
ISBN 978-3-662-46210-2 ISBN 978-3-662-46211-9 (eBook)
DOI 10.1007/978-3-662-46211-9
LibraryofCongressControlNumber:2015931246
Mathematics Subject Classification: 22E05, 22E10, 22E60, 22-03, 01A05, 01A55, 17B30, 17B40,
17B45,17B56,17B66,17B70,22F30,12H05,14P05,14P15
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THEORIE
DER
TRANSFORMATIONSGRUPPEN
—————-
ERSTERABSCHNITT
—————-
UNTERMITWIRKUNGVONProf.Dr.FRIEDRICHENGEL
BEARBEITET VON
SOPHUS LIE,
WEIL.PROFESSORDERGEOMETRIEANDDERUNIVERSITÄTLEIPZIG
UNDPROFESSORITRANSFORMASJONSGRUPPENESTEORIANDER
KÖNIGLICHENFREDERIKSUNIVERSITÄTZUOSLO
UNVERÄNDERTER NEUDRUCK
MIT UNTERSTÜTZUNG DER KÖNIGLICHEN
FREDERIKS UNIVERSITÄT ZU OSLO
1930
VERLAGUNDDRUCKVONB.G.TEUBNERINLEIPZIG
UNDBERLIN
Foreword
This modernized English translation grew out of my old simultaneous interest in
the mathematics itself and in the metaphysical thoughts governing its continued
development.IowetothebooksofRobertHermann,PeterOlver,ThomasHawkins,
andOlleStormarkmyintroductiontoLie’soriginalvastfield.
Uptotheendofthe18th Century,theuniversallanguageofSciencewasLatin,
until its centre of gravity shifted to German during the 19th Century, while nowa-
days—needless to say—English is widespread. Being intuitively convinced that
Lie’soriginalworkscontainmuchmorethanwhathasbeenmodernizeduptonow,
threeyearsagoIstartedtolearnGermanfromscratchjustinordertoreadLie,with
twomaingoalsinmind:
(cid:2) to complete and modernize the Lie-Amaldi classification of finite-
dimensionallocalLiegroupholomorphicactionsonspacesofcomplexdimensions
1,2and3forvariousapplicationsincomplexandCauchy-Riemanngeometry;
(cid:2) tobetterunderstandtherootsofÉlieCartan’sachievements.
Then it gradually appeared to me that Lie’s mathematical thought is universal
and transhistorical, hence it deserves per se to be translated. The present adapted
English translation follows an earlier monograph1 written in French and specially
devotedtoEngelandLie’streatmentoftheso-calledRiemann-Helmholtzproblem
inVolumeIIIoftheTheoriederTransformationsgruppen.
A few observations are in order concerning the chosen format. For several rea-
sons,itwasessentiallyimpossibletodirectlytranslatethefirstfewchaptersinwhich
Lie’sintentionwastosetupthebeginningsofthetheoryinthehighestpossiblegen-
erality, especially in order to eliminate the axiom of inverse, an aspect never dealt
withinmoderntreatises.Asaresult,Idecidedinthefirstfourchapterstoreorganize
thematerialandtoreprovetherelevantstatements,neverthelessretainingallofthe
embraced mathematical content. But starting with Chap. 5, Engel and Lie’s expo-
sitionissosmooth,sorigorous,sounderstandable,sosystematic,soastonishingly
wellorganized—sobeautifulforthought—thatapuretranslationisessential.
1 Merker,J.:SophusLie,FriedrichEngeletleproblèmedeRiemann-Helmholtz,HermannÉditeur
desSciencesetdesArts,Paris,xxiii+325pp,2010.
vii
viii Foreword
Lastly,theauthorisgratefultoGautamBharali,PhilipBoalch,EgmontPorten,
andMasoudSabzevari forafewfinesuggestions concerning thelanguage andfor
misprint chasing, but is of course solely responsible for the lack of idiomatic En-
glish.
Paris,ÉcoleNormaleSupérieure, JoëlMerker
16March2010
Contents
PartI ModernPresentation
1 ThreePrinciplesofThought
GoverningtheTheoryofLie .................................... 3
References..................................................... 12
2 LocalTransformationEquations
andEssentialParameters ....................................... 13
2.1 GenericRankoftheInfiniteCoefficientMapping................ 13
2.2 QuantitativeCriterion
fortheNumberofSuperfluousParameters...................... 15
2.3 TheAxiomofInverseandEngel’sCounterexample.............. 19
References..................................................... 22
3 FundamentalDifferentialEquations
forFiniteContinuousTransformationGroups..................... 23
3.1 TheConceptofaLocalTransformationGroup.................. 24
3.1.1 TransformationGroupAxioms......................... 24
3.1.2 SomeConventions ................................... 26
3.2 ChangesofCoordinatesandofParameters ..................... 27
3.3 GeometricIntroductionofInfinitesimalTransformations.......... 31
3.4 DerivationofFundamentalPartialDifferentialEquations ......... 33
3.4.1 RestrictingConsiderationstoaSingleSystemofParameters 34
3.4.2 ComparingDifferentFrames
ofInfinitesimalTransformations........................ 35
3.5 EssentializingtheGroupParameters........................... 36
3.6 TheFirstFundamentalTheorem .............................. 40
3.7 FundamentalDifferentialEquations
fortheInverseTransformations .............................. 42
3.8 TransferofIndividualInfinitesimalTransformations
bytheGroup .............................................. 44
ix
x Contents
3.8.1 ASynthetic,GeometricCounterpartoftheComputations... 46
3.8.2 TransferofGeneralInfinitesimalTransformations......... 47
3.8.3 TowardstheAdjointAction ........................... 48
3.9 SubstitutingtheAxiomofInverse
foraDifferentialEquationsAssumption ....................... 50
3.9.1 SpecifyingDomainsofExistence....................... 51
3.9.2 The Group Composition Axiom and Fundamental
DifferentialEquations ................................ 54
3.9.3 TheDifferentialEquationsAssumption
anditsConsequences................................. 57
3.9.4 TowardsTheorem26 ................................. 58
3.9.5 MetaphysicalLinkswithSubstitutionTheory............. 59
References..................................................... 60
4 One-TermGroups
andOrdinaryDifferentialEquations ............................. 61
4.1 MechanicalandMentalImages............................... 62
4.2 StraighteningofFlowsandtheExponentialFormula ............. 64
4.2.1 TheExponentialAnalyticFlowFormula................. 66
4.2.2 ActiononFunctions.................................. 67
4.3 ExponentialChangeofCoordinatesandtheLieBracket .......... 70
4.3.1 FlowsasChangesofCoordinates....................... 72
4.4 EssentialityofMultipleFlowParameters....................... 73
4.5 Generationofanr-TermGroupbyitsOne-TermSubgroups....... 80
4.6 ApplicationstotheEconomyofAxioms ....................... 82
References..................................................... 91
PartII EnglishTranslation
5 CompleteSystemsofPartialDifferentialEquations ................ 95
§ 21. ....................................................... 97
§ 22. ....................................................... 98
§ 23. .......................................................102
§ 24. .......................................................104
§ 25. .......................................................106
§ 26. .......................................................109
6 NewInterpretationoftheSolutionsofaCompleteSystem ..........111
§ 27. .......................................................111
§ 28. .......................................................114
§ 29. .......................................................120
7 DeterminationofAllSystemsofEquations
WhichAdmitGivenInfinitesimalTransformations ................123
§ 30. .......................................................123
§ 31. .......................................................130
Contents xi
§ 32. .......................................................133
§ 33. .......................................................136
§ 34. .......................................................138
§ 35. .......................................................143
§ 36. .......................................................147
8 CompleteSystemsWhichAdmit
AllTransformations
ofaOne-termGroup ...........................................151
§ 37. .......................................................152
§ 38. .......................................................157
9 CharacteristicRelationships
BetweentheInfinitesimalTransformationsofaGroup .............161
§ 39. .......................................................161
§ 40. .......................................................166
§ 41. .......................................................170
§ 42. .......................................................173
§ 43. .......................................................174
§ 44. .......................................................178
§ 45. .......................................................180
§ 46. .......................................................184
10 SystemsofPartialDifferentialEquations
theGeneralSolutionofWhich
DependsOnlyUponaFiniteNumber
ofArbitraryConstants .........................................187
§ 47. .......................................................188
§ 48. .......................................................195
11 TheDefiningEquations
fortheInfinitesimalTransformationsofaGroup ..................199
§ 49. .......................................................199
§ 50. .......................................................201
§ 51. .......................................................203
§ 52. .......................................................208
§ 53. .......................................................211
12 DeterminationofAllSubgroupsofanr-termGroup ...............217
§ 54. .......................................................217
§ 55. .......................................................219
§ 56. .......................................................221
§ 57. .......................................................222