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, Joël Merker, Friedrich Engel

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
Editor and Translator: Joël Merker
123
Author Editor and Translator
Sophus Lie (1842–1899) Joël Merker
Laboratoire de Mathématiques, Faculté des
Collaborator Sciences d’Orsay
Friedrich Engel (1861–1941) Université Paris-Sud 11
Orsay
France
First edition published in German language by B.G. Teubner, Leipzig, with the title:
Theorie der Transformationsgruppen in 1888, 1930
ISBN 978-3-662-46210-2 ISBN 978-3-662-46211-9 (eBook)
DOI 10.1007/978-3-662-46211-9
Library of Congress Control Number: 2015931246
Mathematics Subject Classification: 22E05, 22E10, 22E60, 22-03, 01A05, 01A55, 17B30, 17B40,
17B45, 17B56, 17B66, 17B70, 22F30, 12H05, 14P05, 14P15
Springer Heidelberg New York Dordrecht London
© Springer-Verlag Berlin Heidelberg 1888, 1930, 2015
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part
of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission
or information storage and retrieval, electronic adaptation, computer software, or by similar or
dissimilar methodology now known or hereafter developed.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publication does not imply, even in the absence of a specific statement, that such names are exempt
from the relevant protective laws and regulations and therefore free for general use.
The publisher, the authors and the editors are safe to assume that the advice and information in this
book are believed to be true and accurate at the date of publication. Neither the publisher nor the
authors or the editors give a warranty, express or implied, with respect to the material contained
herein or for any errors or omissions that may have been made.
Cover picture taken from “Ny illustreret Tidende”, 1886
Printed on acid-free paper
Springer-Verlag GmbH Berlin Heidelberg is part of Springer Science+Business Media
(www.springer.com)
THEORIE
DER
TRANSFORMATIONSGRUPPEN
—————-
ERSTER ABSCHNITT
—————-
UNTER MITWIRKUNG VON Prof. Dr. FRIEDRICH ENGEL
BEARBEITET VON
SOPHUS LIE,
WEIL. PROFESSOR DER GEOMETRIE AND DER UNIVERSITÄT LEIPZIG
UND PROFESSOR I TRANSFORMASJONSGRUPPENES TEORI AN DER
KÖNIGLICHEN FREDERIKS UNIVERSITÄT ZU OSLO
UNVERÄNDERTER NEUDRUCK
MIT UNTERSTÜTZUNG DER KÖNIGLICHEN
FREDERIKS UNIVERSITÄT ZU OSLO
1930
VERLAG UND DRUCK VON B.G. TEUBNER IN LEIPZIG
UND BERLIN
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. I owe to the books of Robert Hermann, Peter Olver, Thomas Hawkins,
and Olle Stormark my introduction to Lie’s original vast field.
th
Up to the end of the 18 Century, the universal language of Science was Latin,
th
until its centre of gravity shifted to German during the 19 Century, while nowa-
days — needless to say — English is widespread. Being intuitively convinced that
Lie’s original works contain much more than what has been modernized up to now,
three years ago I started to learn German from scratch just in order to read Lie, with
two main goals in mind:
□ to complete and modernize the Lie-Amaldi classification of finite-
dimensional local Lie group holomorphic actions on spaces of complex dimensions
1, 2 and 3 for various applications in complex and Cauchy-Riemann geometry;
□ to better understand the roots of Élie Cartan’s achievements.
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
1
English translation follows an earlier monograph written in French and specially
devoted to Engel and Lie’s treatment of the so-called Riemann-Helmholtz problem
in Volume III of the Theorie der Transformationsgruppen.
A few observations are in order concerning the chosen format. For several rea-
sons, it was essentially impossible to directly translate the first few chapters in which
Lie’s intention was to set up the beginnings of the theory in the highest possible gen-
erality, especially in order to eliminate the axiom of inverse, an aspect never dealt
with in modern treatises. As a result, I decided in the first four chapters to reorganize
the material and to reprove the relevant statements, nevertheless retaining all of the
embraced mathematical content. But starting with Chap. 5, Engel and Lie’s expo-
sition is so smooth, so rigorous, so understandable, so systematic, so astonishingly
well organized — so beautiful for thought — that a pure translation is essential.
1 Merker, J.: Sophus Lie, Friedrich Engel et le problème de Riemann-Helmholtz, Hermann Éditeur
des Sciences et des Arts, Paris, xxiii+325 pp, 2010.
vii
viii Foreword
Lastly, the author is grateful to Gautam Bharali, Philip Boalch, Egmont Porten,
and Masoud Sabzevari for a few fine suggestions concerning the language and for
misprint chasing, but is of course solely responsible for the lack of idiomatic En-
glish.
Paris, École Normale Supérieure, Joël Merker
16 March 2010
Contents
Part I Modern Presentation
1 Three Principles of Thought
Governing the Theory of Lie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 Local Transformation Equations
and Essential Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1 Generic Rank of the Infinite Coefficient Mapping . . . . . . . . . . . . . . . . 13
2.2 Quantitative Criterion
for the Number of Superfluous Parameters . . . . . . . . . . . . . . . . . . . . . . 15
2.3 The Axiom of Inverse and Engel’s Counterexample . . . . . . . . . . . . . . 19
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3 Fundamental Differential Equations
for Finite Continuous Transformation Groups . . . . . . . . . . . . . . . . . . . . . 23
3.1 The Concept of a Local Transformation Group . . . . . . . . . . . . . . . . . . 24
3.1.1 Transformation Group Axioms . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.1.2 Some Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2 Changes of Coordinates and of Parameters . . . . . . . . . . . . . . . . . . . . . 27
3.3 Geometric Introduction of Infinitesimal Transformations . . . . . . . . . . 31
3.4 Derivation of Fundamental Partial Differential Equations . . . . . . . . . 33
3.4.1 Restricting Considerations to a Single System of Parameters 34
3.4.2 Comparing Different Frames
of Infinitesimal Transformations . . . . . . . . . . . . . . . . . . . . . . . . 35
3.5 Essentializing the Group Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.6 The First Fundamental Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.7 Fundamental Differential Equations
for the Inverse Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.8 Transfer of Individual Infinitesimal Transformations
by the Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
ix
x Contents
3.8.1 A Synthetic, Geometric Counterpart of the Computations . . . 46
3.8.2 Transfer of General Infinitesimal Transformations . . . . . . . . . 47
3.8.3 Towards the Adjoint Action . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.9 Substituting the Axiom of Inverse
for a Differential Equations Assumption . . . . . . . . . . . . . . . . . . . . . . . 50
3.9.1 Specifying Domains of Existence . . . . . . . . . . . . . . . . . . . . . . . 51
3.9.2 The Group Composition Axiom and Fundamental
Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.9.3 The Differential Equations Assumption
and its Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.9.4 Towards Theorem 26 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.9.5 Metaphysical Links with Substitution Theory . . . . . . . . . . . . . 59
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4 One-Term Groups
and Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.1 Mechanical and Mental Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.2 Straightening of Flows and the Exponential Formula . . . . . . . . . . . . . 64
4.2.1 The Exponential Analytic Flow Formula . . . . . . . . . . . . . . . . . 66
4.2.2 Action on Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.3 Exponential Change of Coordinates and the Lie Bracket . . . . . . . . . . 70
4.3.1 Flows as Changes of Coordinates . . . . . . . . . . . . . . . . . . . . . . . 72
4.4 Essentiality of Multiple Flow Parameters . . . . . . . . . . . . . . . . . . . . . . . 73
4.5 Generation of an r-Term Group by its One-Term Subgroups . . . . . . . 80
4.6 Applications to the Economy of Axioms . . . . . . . . . . . . . . . . . . . . . . . 82
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
Part II English Translation
5 Complete Systems of Partial Differential Equations . . . . . . . . . . . . . . . . 95
§ 21. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
§ 22. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
§ 23. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
§ 24. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
§ 25. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
§ 26. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6 New Interpretation of the Solutions of a Complete System . . . . . . . . . . 111
§ 27. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
§ 28. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
§ 29. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
7 Determination of All Systems of Equations
Which Admit Given Infinitesimal Transformations . . . . . . . . . . . . . . . . 123
§ 30. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
§ 31. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
Contents xi
§ 32. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
§ 33. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
§ 34. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
§ 35. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
§ 36. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
8 Complete Systems Which Admit
All Transformations
of a One-term Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
§ 37. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
§ 38. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
9 Characteristic Relationships
Between the Infinitesimal Transformations of a Group . . . . . . . . . . . . . 161
§ 39. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
§ 40. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
§ 41. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
§ 42. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
§ 43. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
§ 44. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
§ 45. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
§ 46. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
10 Systems of Partial Differential Equations
the General Solution of Which
Depends Only Upon a Finite Number
of Arbitrary Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
§ 47. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
§ 48. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
11 The Defining Equations
for the Infinitesimal Transformations of a Group . . . . . . . . . . . . . . . . . . 199
§ 49. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
§ 50. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
§ 51. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
§ 52. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
§ 53. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
12 Determination of All Subgroups of an r-term Group . . . . . . . . . . . . . . . 217
§ 54. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
§ 55. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
§ 56. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
§ 57. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222