Lie's Theory of Transformation Groups I, Contemporary Approach and Translation

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Joël Merker, editor/translator

Theory of Transformation Groups of Lie and Engel

Modern Presentation and English Translation (Vol. I, 1888)

August 20, 2014

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Address of the editor/translator:

Prof. Joël Merker Faculté des Sciences Université Paris-Sud France

E-mail:

Joel.Merker@math.u-psud.fr Internet:

www.math.u-psud.fr/∼merker/

MSC codes and keywords in order of importance:

22E05, 22E10, 22E60, 22-03, 01A05, 01A55, 17B30, 17B40, 17B45, 17B56, 17B66, 17B70, 22F30, 12H05, 14P05, 14P15

continuous transformation groups, infinitesimal transformations, general projective group, complete systems ofPDE’s, classification of Lie algebras, local holomorphic vector fields

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v

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

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Widmung.

—————-

Die neuen Theorien, die in diesem Werke dargestellt sind, habe ich in den Jahren von 1869 bis 1884 en- twickelt, wo mir die Liberalität meines

Geburtslandes Norwegen

gestattete, ungestört meine volle Arbeitskraft der Wissenschaft zu widmen, die durch A

BELS

Werke in Norwegens wissenschaftlicher Literatur den ersten Platz erhalten hat.

Bei der Durchführung meiner Ideen im Einzelnen und bei ihrer systematischen Darstellung genoss ich seit 1884 in der Grössten Ausdehnung die uner- müdliche Unterstützung des P

ROFESSORS

Friedrich Engel

meines ausgezeichneten Kollegen and der Universität Leipzig .

Das in dieser Weise entstandene Werk widme ich Frankreichs

École Normale Supérieure

deren unsterblicher Schüler G

ALOIS

zuerst die Be- deutung des Begriffs discontinuirliche Gruppe erkannte.

Den hervorragenden Lehrern dieses Instituts, beson- ders den Herren G. D

ARBOUX

, E. P

ICARD

und J.

T

ANNERY

verdanke ich es, dass die tüchtigsten jungen Mathematiker Frankreichs wetteifernd mit einer Reihe junger deutscher Mathematiker meine Untersuchungen über continuirliche Gruppen , über Geometrie und über Differentialgleichungen studiren und mit glänzendem Erfolge verwerthen.

Sophus Lie.

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Foreword

This modernized English translation grew out of my old simultaneous interest in the mathematics themselves and in the metaphysical thoughts governing their continued development. I owe to the books of Robert Hermann, of Peter Olver, of Thomas Hawkins, and of Olle Stormark to have been introduced in Lie’s original vast field.

Up to the end of the 18^thCentury, the universal language of Science was Latin, until its centre of gravity shifted to German during the 19^thCentury while nowa- days — and needless to say — English is widespread. Being intuitively convinced that Lie’s original works contain much more than what is modernized up to now, I started three years ago to learn Germanfrom scratchjust in order to read Lie, and with two main goals in mind:

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;

better understand the roots of Élie Cartan’s achievements.

Then it gradually appeared to me thatthe mathematical thought of Lie is univer- sal and transhistorical, hence deservesper seto be translated. The present adapted English translation follows a first monograph¹written in French and specially de- voted to the treatment by Engel and Lie of the so-calledRiemann-Helmholtz prob- lemin the Volume III of theTheorie der Transformationsgruppen.

A few observations are in order concerning the chosen format. For several rea- sons, it was essentially impossible to translate directly 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 come up with the elimination of 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 concerned statements, without nevertheless removing anything from the mathematical contents embraced. But starting with Chap. 5, the exposition of Engel and Lie is so smooth, so rigorous, so

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

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understandable, so systematic, so astonishingly well organized — in one word:so beautiful for thought— that a pure translation is essential.

Lastly, the author is grateful to Gautam Bharali, to Philip Boalch, to Egmont Porten, to Masoud Sabzevari for a few fine suggestions concerning the language and for misprint chasing, but is of course the sole responsible for the lack of idiomatic English.

Paris, École Normale Supérieure, Joël Merker

16 March 2010

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Part I

Modern Presentation

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Chapter 1 Three Principles of Thought Governing the Theory of Lie

Letx= (x₁, . . . ,x_n)be coordinates on ann-dimensional real or complex euclidean

space Cⁿ or Rⁿ, considered as a source domain. The archetypal objects of Lie’s Theory of Continuous Transformation Groups arepoint transformation equations:

x⁰_i=f_i(x₁, . . .x_n;a₁, . . . ,a_r) (i=1···n),

parameterized by a finite number r of real or complex parameters (a₁, . . . ,a_r), namely each mapx⁰= f(x;a) =: f_a(x)is assumed to constitute a diffeomorphism from some domain¹ in the source space into some domain in a target space of the same dimension equipped with coordinates (x⁰₁, . . . ,x⁰_n). Thus, the functional determinant:

det Jac(f) =

∂f₁

∂x₁ ··· ^∂_∂_x^f¹_n ... . .. ...

∂fn

∂x₁ ··· ^∂_∂_x^fⁿ_n =

∑

σ∈Permn

sign(σ⁾ ∂^f1

∂^xσ(1)

∂^f2

∂^xσ(2)··· ∂^fn

∂^xσ(n)

vanishes at no point of the source domain.

§ 15. ([1], pp. 25–26)

[The concepts of transformationx⁰= f(x)and of transformation equations x⁰=f(x;a)are of purely analytic nature.] However, these concepts can receive a graphic interpretation [ANSCHAULICHAUFFASUNG], when the concept of an n-times extended space [RAUM] is introduced.

If we interpret x₁, . . . ,x_n as the coordinates for points [PUNKTCOORDI-

NATEN] of such a space, then a transformation x⁰_i = f_i(x₁, . . . ,x_n)appears as a point transformation [PUNKTTRANSFORMATION]; consequently, this transformation can be interpreted as anoperationwhich consists in that, every point x_iis transferred at the same time in the new positionx⁰_i. One expresses this as

1 Bydomain, it will always been meant aconnected, nonempty open set.

3

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follows: the transformation in question is an operation through which the points of the spacex₁, . . . ,x_nare permuted with each other.

Before introducing the continuous group axioms in Chap. 3 below, the very first question to be settled is: how many different transformationsx⁰_i= f_i(x;a)do cor- respond to the∞^r different systems of values(a₁, . . . ,ar)? Some parameters might indeed be superfluous, hence they should be removed from the beginning, as will be achieved in Chap. 2. For this purpose, it is crucial to formulate explicitly and once for allthree principles of thought concerning the admission of hypotheses that do hold throughout the theory of continuous groups developed by Lie.

General Assumption of Analyticity

Curves, surfaces, manifolds, groups, subgroups, coefficients of infinitesimal transformations, etc., all mathematical objects of the theory will be assumed to beana- lytic, i.e. their representing functions will be assumed to be locally expandable in convergent, univalent power series defined in a certain domain of an appropriateR^m.

Principle of Free Generic Relocalization

Consider a local mathematical object which is represented by functions that are analytic in some domainU₁, and suppose that a certain “generic” nice behaviour holds onU₁\D₁ outside a certain proper closed analytic subsetD₁⊂U₁; for instance:

the invertibility of a square matrix composed of analytic functions holds outside the zero-locus of its determinant. Then relocalize the considerations in some subdomain U₂⊂U₁\D₁.

D₁

D₂ _U₃

U₂

U₃

U₁ U2

Fig. 1.1 Relocalizing finitely many times in neighbourhoods of generic points

Afterwards, inU₂, further reasonings may demand to avoid another proper closed analytic subsetD₂, hence to relocalize the considerations into some subdomainU₃⊂ U₂\D₂, and so on. Most proofs of theTheorie der Transformationsgruppen, and especially the classification theorems, do allow a great number of times such relo- calizations, often without any mention, such anact of thoughtbeing considered as

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1 Three Principles of Thought Governing the Theory of Lie 5 implicitly clear, and free relocalization being justified by the necessity ofstudying at first generic objects.

Giving no Name to Domains or Neighbourhoods

Without providing systematic notation, Lie and Engel commonly wrotetheneigh- bourhood [derUMGEBUNG] (of a point), similarly as one speaks oftheneighbour- hood of a house, or ofthesurroundings of a town, whereas contemporary topology conceptualizesa(say, sufficiently small) given neighbourhood amongst an infinity.

Contrary to what the formalistic, twentieth-century mythology tells sometimes, Lie and Engel did emphasize the local nature of the concept of transformation group in terms of narrowing down neighbourhoods; we shall illustrate this especially when presenting Lie’s attempt to economize the axiom of inverse. Certainly, it is true that most of Lie’s results are stated without specifying domains of existence, but in fact also, it is moreover quite plausible that Lie soon realized that giving no name to neighbourhoods, and avoiding superfluous denotation is efficient and expeditious in order to perform far-reaching classification theorems.

Therefore, adopting the economical style of thought in Engel-Lie’s treatise, our

“modernization-translation” of the theory will, without nevertheless providing fre- quent reminders, presuppose that:

• mathematical objects are analytic;

• relocalization is freely allowed;

• open sets are often small, usually unnamed, and alwaysconnected.

Introduction

([1], pp. 1–8)

—————–

If the variablesx⁰₁, . . . ,x⁰_nare determined as functions ofx1, . . . ,xnbynequa- tions solvable with respect tox₁, . . . ,x_n:

x⁰_i=f_i(x₁, . . . ,x_n) (i=1···n),

then one says that these equations represent a transformation [TRANSFORMA-

TION] between the variablesxandx⁰. In the sequel, we will have to deal with such transformations; unless the contrary is expressly mentioned, we will re- strict ourselves to the case where the f_i areanalytic[ANALYTISCH]functions of their arguments. But because a not negligible portion of our results is independent of this assumption, we will occasionally indicate how various develop- ments take shape by taking into considerations functions of this sort.

When the functions f_i(x₁, . . . ,x_n)are analytic and are defined inside a common region [BEREICH], then according to the known studies of CAUCHY,

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WEIERSTRASS, BRIOTand BOUQUET, one can always delimit, in the mani- fold of all real and complex systems of valuesx₁, . . . ,x_n, a region(x)such that all functions f_iare univalent in the complete extension [AUSDEHNUNG] of this region, and so that as well, in the neighbourhood [UMGEBUNG] of every system of valuesx⁰₁, . . . ,x⁰_nbelonging to the region(x), the functions behave regularly [REGULÄR VERHALTEN], that is to say, they can be expanded in ordinary power series with respect tox₁−x⁰₁, . . . ,x_n−x⁰_nwith only entire positive powers.

For the solvability of the equationsx⁰_i=f_i(x), a unique condition is necessary and sufficient, namely the condition that the functional determinant:

∑

^±^∂_∂_x^f¹₁^···^∂_∂_x^fⁿ_n

should not vanish identically. If this condition is satisfied, then the region(x) defined above can specially be defined so that the functional determinant does not take the value zero for any system of values in the(x). Under this assumption, if one lets thextake gradually all systems of values in the region(x), then the equationsx⁰_i=fi(x)determine, in the domain [GEBIETE] of thex⁰, a region of such a nature thatx1, . . . ,xn, in the neighbourhood of every system of values x⁰₁⁰, . . . ,x⁰_n⁰in this new region behave regularly as functions of x⁰₁, . . . ,x⁰_n, and hence can be expanded as ordinary power series ofx₁⁰−x⁰₁⁰, . . . ,x⁰_n−x⁰_n⁰. It is well known that from this, it does not follow that thex_iare univalent functions

ofx⁰₁, . . . ,x⁰_nin the complete extension of the new region; but when necessary,

it is possible to narrow down the region(x)defined above so that two different systems of valuesx₁, . . . ,x_nof the region(x)always produce two, also different systems of valuesx⁰₁=f1(x), . . . ,x⁰_n=fn(x).

Thus, the equationsx⁰_i= fi(x) establish a univalent invertible relationship [BEZIEHUNG] between regions in the domain of thexand regions in the domain of thex⁰; to every system of values in one region, they associate one and only one system of values in the other region, and conversely.

If the equationsx⁰_i= f_i(x)are solved with respect to thex, then in turn, the resulting equations:

x_k=F_k(x⁰₁, . . . ,x⁰_n) (k=1···n)

again represent a transformation. The relationship between this transformation and the initial one is evidently a reciprocal relationship; accordingly, one says:

the two transformations areinverseone to another. From this definition, it visi- bly follows:

If one executes at first the transformation:

x⁰_i=f_i(x₁, . . . ,x_n) (i=1···n)

and afterwards the transformation inverse to it:

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1 Three Principles of Thought Governing the Theory of Lie 7

x⁰⁰_i =F_i(x⁰₁, . . . ,x⁰_n) (i=1···n), then one obtains the identity transformation:

x⁰⁰_i =x_i (i=1···n).

Here lies the real definition of the concept [BEGRIFF] of two transformations inverse to each other.

In general, if one executes two arbitrary transformations:

x⁰_i=f_i(x₁, . . . ,x_n), x⁰⁰_i =g_i(x⁰₁, . . . ,x⁰_n) (i=1···n)

one after the other, then one obtains a new transformation, namely the following one:

x⁰⁰_i =g_i f₁(x), . . . ,f_n(x)

(i=1···n).

In general, this new transformation naturally changes when one changes the order [REIHENFOLGE] of the two transformations; however, it can also happen that the order of the two transformations is indifferent. This case occurs when one has identically:

g_i f₁(x), . . . ,f_n(x)

≡f_i g₁(x), . . . ,g_n(x)

(i=1···n);

we then say, as in the process of the Theory of Substitutions [VORGANG DER

SUBSTITUTIONENTHEORIE]:the two transformations:

x⁰_i=f_i(x₁, . . . ,x_n) (i=1···n)

and:

x⁰_i=g_i(x₁, . . . ,x_n) (i=1···n)

are interchangeable[VERTAUSCHBAR]one with the other. —

A finite or infinite family[SCHAAR]of transformations between the x and the x⁰is called agroup of transformationsor atransformation groupwhen any two transformations of the family executed one after the other give a transformation which again belongs to the family.^†

A transformation group is called discontinuouswhen it consists of a discrete number of transformations, and this number can be finite or infinite. Two transformations of such a group are finitely different from each other. The discontinuous groups belong to the domain of theTheory of Substitutions, so in the sequel, they will remain out of consideration.

† SophusLIE, Gesellschaft der Wissenschaften zu Christiania 1871, p. 243.KLEIN, Vergle- ichende Betrachtungen über neuere geometrische Forschungen, Erlangen 1872.LIE, Göttinger Nachrichten 1873, 3. Decemb.

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The discontinuous groups stand in opposition to thecontinuoustransforma- tion groups, which always contain infinitely many transformations. A transformation group is calledcontinuouswhen it is possible, for every transformation belonging to the group, to indicate certain other transformations which are only infinitely little different from the transformation in question, and when by con- trast, it is not possible to reduce the complete totality [INBEGRIFF] of transformations contained in the group to a single discrete family.

Now, amongst the continuous transformation groups, we consider again two separate categories [KATEGORIEN] which, in the nomenclature [BENEN-

NUNG], are distinguished asfinite continuousgroups and asinfinite continuous groups. To begin with, we can only give provisional definitions of the two categories, and these definitions will be apprehended precisely later.

Afinite continuous transformation groupwill be represented byonesystem ofnequations:

x⁰_i=fi(x₁, . . . ,xn,a1, . . . ,ar) (i=1···n),

where the f_idenote analytic functions of the variablesx₁, . . . ,x_nand of the arbitrary parametersa₁, . . . ,a_r. Since we have to deal with a group, two transformations:

x_i⁰=fi(x₁, . . . ,xn,a1, . . . ,ar) x⁰⁰_i =f_i(x⁰₁, . . . ,x_n⁰,a₁, . . . ,a_r),

when executed one after the other, must produce a transformation which belongs to the group, hence which has the form:

x⁰⁰_i =f_i f₁(x,a), . . . ,f_n(x,a),b₁, . . . ,b_r

= f_i(x₁, . . . ,x_n,c₁, . . . ,c_r).

Here, thec_kare naturally independent of thexand so, are functions of only the aand theb.

Example.A known group of this sort is the following:

x⁰= x+a₁ a₂x+a₃,

which contains the three parametersa₁,a₂,a₃. If one executes the two transformations:

x⁰= x+a₁

a₂x+a₃, x⁰⁰= x⁰+b₁ b₂x⁰+b₃ one after the other, then one receives:

x⁰⁰= x+c1

c₂x+c₃,

wherec₁,c₂,c₃are defined as functions of theaand thebby the relations:

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c₁=a₁+b₁a₃

1+b₁a₂, c₂=b₂+a₂b₃

1+b₁a₂, c₃=b₂a₁+b₃a₃ 1+b₁a₂ . The following group with then²parametersa_ikis not less known:

x⁰_i=

∑

n k=1

a_ikx_k (i=1···n). If one sets here:

x⁰⁰_ν=

∑

n i=1

b_ν_ix⁰_i (i=1···n), then it comes:

x⁰⁰_ν=

1···n

∑

i,k

b_ν_ia_ikx_k=

∑

n k=1

c_ν_kx_k, where thec_ν_kare determined by the equations:

c_ν_k=

∑

n i=1

b_ν_ia_ik (ν,k=1···n).−

In order to arrive at a usable definition of a finite continuous group, we want at first to somehow reshape the definition of finite continuous groups. On the occasion, we use a proposition from the theory of differential equations about which, besides, we will come back later in a more comprehensive way (cf.

Chap. 10).

Let the equations:

x⁰_i= f_i(x₁, . . . ,x_n,a₁, . . . ,a_r) (i=1···n)

represent an arbitrary continuous group. According to the proposition in question, it is then possible to define the functions f_ithrough a system of differential equations, insofar as they depend upon thex. To this end, one only has to differ- entiate the equationsx⁰_i=f_i(x,a)with respect tox₁, . . . ,x_nsufficiently often and then to set up all equations that may be obtained by elimination ofa1, . . . ,ar. If one is gone sufficiently far by differentiation, then by elimination of the a, one obtains a system of differential equations forx⁰₁, . . . ,x⁰_n, whose most general system of solutions is represented by the initial equationsx⁰_i=f_i(x,a)with therarbitrary parameters. Now, since by assumption the equationsx⁰_i=f_i(x,a) define a group, it follows that the concerned system of differential equations possesses the following remarkable property: ifx⁰_i= f_i(x₁, . . . ,x_n,b₁, . . . ,b_r)is a system of solutions of it, and ifx⁰_i=f_i(x,a)is a second system of solutions, then:

x_i⁰=f_i f₁(x,a), . . . ,f_n(x,a),b₁, . . . ,b_r

(i=1···n)

is also a system of solutions.

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From this, we see that the equations of an arbitrary finite continuous transformation group can be defined by a system of differential equations which possesses certain specific properties. Firstly, from two systems of solutions of the concerned differential equations one can always derive, in the way indicated above, a third system of solutions: it is precisely in this that we have to deal with a group. Secondly, the most general system of solutions of the concerned differential equations depends only upon a finite number of arbitrary constants:

this circumstance expresses that our group is finite.

Now, we assume that there is a family of transformationsx⁰_i=f_i(x₁, . . . ,x_n) which is defined by a system of differential equations of the form:

W_k

x⁰₁, . . . ,x⁰_n,∂^x⁰1

∂^x1

, . . . ,∂²^x⁰1

∂^x²₁^{, . . .}

=0 (k=1,2···).

Moreover, we assume that this system of differential equations possesses the first one of the two mentioned properties, but not the second one; therefore, with x⁰_i=f_i(x₁, . . . ,x_n)andx⁰_i=g_i(x₁, . . . ,x_n), then always,x⁰_i=g_i f₁(x), . . . ,f_n(x) is also a system of solutions of these differential equations, and the most general system of solutions of them does not only depend upon a finite number of arbitrary constants, but also upon higher sorts of elements, as for example, upon arbitrary functions. Then the totality of all transformations which satisfy the concerned differential equations evidently forms again a group, and in general, a continuous group, though no more a finite one, but one which we callinfinite continuous.

Straightaway, we give a few simple examples of infinite continuous transformation groups.

When the differential equations which define the concerned infinite group reduce to the identity equation 0=0, then the transformations of the group read:

x⁰_i=Πi(x₁, . . . ,x_n) (i=1···n)

where theΠidenote arbitrary analytic functions of their arguments.

Also the equations:

∂^x⁰i

∂^xk

=0 (i6=k;i,k=1···n)

define an infinite group, namely the following one:

x_i⁰=Πi(x_i) (i=1···n), where again theΠiare absolutely arbitrary.

Besides, it is to be observed that the concept of an infinite continuous group can yet be understood more generally than what takes place here. Actually, one

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could call infinite continuous any continuous group which is not finite. However, this definition does not coincide with the one given above.

For instance, the equations:

x⁰=F(x), y⁰=F(y)

in whichF, for the two cases, denotes the same function of its arguments, represent a group. This group is continuous, since all its transformations are represented by a single system of equations; in addition, it is obviously not finite.

Consequently, it would be an infinite continuous group if we would interpret this concept in the more general sense indicated above. But on the other hand, it is not possible to define the family of the transformations:

x⁰=F(x), y⁰=F(y)

by differential equations that are free of arbitrary elements. Consequently, the definition stated first for an infinite continuous group does not fit to this case.

Nevertheless, we find suitable to consider only the infinite continuous groups which can be defined by differential equations and hence, we always set as fun- damental our first, tight definition.

We do not want to omit emphasizing that the concept of “transformation group” is still not at all exhausted by the difference between discontinuous and continuous groups. Rather, there are transformation groups which are subordi- nate to none of these two classes but which have something in common with each one of the two classes. In the sequel, we must at least occasionally also treat this sort of groups. Provisionally, two examples will suffice.

The totality of all coordinate transformations of a plane by which one trans- fers an ordinary right-angled system of coordinates to another one forms a group which is neither continuous, nor discontinuous. Indeed, the group in question contains two separate categories of transformations between which a continuous transition is not possible: firstly, the transformations by which the old and the new systems of coordinates are congruent, and secondly, the transformations by which these two systems are not congruent.

The first ones have the form:

x⁰−a=xcosα−ysinα^, ^y⁰−b=xsinα⁺^y^cosα^, while the analytic expression of the second ones reads:

x⁰−a=xcosα⁺^y^sinα^, ^y⁰−b=xsinα−ycosα^.

Each one of these systems of equations represents a continuous family of transformations, hence the group is not discontinuous; but it is also not continuous, because both systems of equations taken together provide all transformations

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of the group; thus, the transformations of the group decompose in two discrete families. If one imagines the planex,yin ordinary space and if one adds zas a third right-angled coordinate, then one can imagine the totality of coordinate transformations of the planez=0 is a totality of certain movements [BEWE-

GUNGEN] of the space, namely the movements during which the plane z=0 keeps its position. Correspondingly, these movements separate in two classes, namely in the class which only shifts the plane in itself and in the class which turns the plane.

As a second example of such a group, one can yet mention the totality of all projective and dualistic transformations of the plane.

—————–

According to these general remarks about the concept of transformation group, we actually turn ourselves to the consideration of the finite continuous transformation groups which constitute the object of the studies following. These studies are divided in three volumes [ABSCHNITTE]. Thefirstvol- ume treats finite continuous groups in general. Thesecondvolume treats the finite continuous groups whose transformations are so-calledcontact transformations[BERÜHRUNGSTRANSFORMATIONEN]. Lastly, in thethirdvolume, certain general problems of group theory will be carried out in great details for a small number of variables.

References

1. Engel, F., Lie, S.: Theorie der Transformationsgruppen. Erster Abschnitt. Unter Mitwirkung von Prof. Dr. Friedrich Engel, bearbeitet von Sophus Lie, Verlag und Druck von B.G. Teubner, Leipzig und Berlin, xii+638 pp. (1888). Reprinted by Chelsea Publishing Co., New York, N.Y.

(1970)

—————–

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Chapter 2 Local Transformation Equations and Essential Parameters

Abstract LetK=RorC, throughout. As said in Chap. 1, transformation equations

x⁰_i= f_i(x;a₁, . . . ,a_r),i=1, . . . ,n, which are local, analytic diffeomorphisms ofKⁿ

parametrized by a finite numberrof real or complex numbersa₁, . . . ,a_r, constitute the archetypal objects of Lie’s theory. The preliminary question is to decide whether the f_ireally depend uponallparameters, and also, to get rid of superfluous parameters, if there are any.

Locally in a neighborhood of a fixedx₀, one expandsf_i(x;a) =∑_α∈Nⁿ U_αⁱ(a) (x− x₀)^α in power series and one looks at theinfinite coefficient mapping U_∞:a7−→

U_αⁱ(a)_16i6n

α∈Nⁿ from K^r to K^∞, expected to tell faithfully the dependence with respect toa in question. If ρ∞ denotes the maximal, generic and locally constant rank of this map, with of course 06ρ∞6r, then the answer says that locally in a neighborhood of a generic a0, there exist both a local change of parameters a7→ u₁(a), . . . ,u_ρ_∞(a)

=:udecreasing the number of parameters fromrdown to ρ∞, and new transformation equations:

x⁰_i=g_i x;u₁, . . . ,u_ρ_∞

(i=1···n)

dependingonlyuponρ∞parameters which give again the old ones:

g_i x;u(a)

≡f_i(x;a) (i=1···n).

At the end of this brief chapter, before introducing precisely the local Lie group axioms, we present an example due to Engel which shows that the axiom of inverse cannot be deduced from the axiom of composition, contrary to one of Lie’sIdées fixes.

2.1 Generic Rank of the Infinite Coefficient Mapping

Thus, we consider local transformation equations:

13

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x⁰_i=f_i(x₁, . . . ,x_n;a₁, . . . ,a_r) (i=1···n).

We want to illustrate how the principle of free generic relocalization exposed just above on p. 4 helps to get rid of superfluous parametersa_k. We assume that the f_i are defined and analytic forxbelonging to a certain (unnamed, connected) domain ofKⁿand forabelonging as well to some domain ofK^r.

Expanding the f_iofx⁰_i=f_i(x;a)in power series with respect tox−x₀in some neighborhood of a pointx₀:

f_i(x;a) =

∑

α∈NⁿU_αⁱ(a) (x−x₀)^α,

we get an infinite number of analytic functionsU_αⁱ =U_αⁱ(a)of the parameters that are defined in some uniform domain of K^r. Intuitively, this infinite collection of coefficient functionsU_αⁱ(a)should show how f(x;a)does depend ona.

To make this claim precise, we thus consider the map:

U_∞: K^r3 a7−→ U_αⁱ(a)_16i6n

α∈Nⁿ ∈K^∞.

For the convenience of applying standard differential calculus in finite dimension, we simultaneously consider also all itsκ-th truncations:

U_κ : K^r3 a7−→ U_αⁱ(a)_16i6n

|α|6κ ∈Kⁿ⁽ⁿ⁺^n!^κ^κ^!^)!,

where⁽ⁿ⁺_n!_κ^κ_!^)!is the number of multiindicesα∈Nⁿwhose length|α|:=α1+···+αn

satisfies the upper bound|α|6κ^{. We call} ^Uκ,U_∞ the(in)finite coefficient mapping(s)ofx⁰_i=f_i(x;a).

TheJacobian matrix ofU_κis ther× n⁽ⁿ⁺_n!_κ^κ_!^)!

matrix:

∂U_αⁱ

∂aj (a)

_|α|6κ_,16i6n

16j6r ,

itsr rows being indexed by the partial derivatives. Thegeneric rankofU_κ is the largest integerρκ6rsuch that there is aρκ×ρκ minor of JacU_κ which does not vanish identically, but all (ρκ+1)×(ρκ+1) minors do vanish identically. The uniqueness principle for analytic functions then insures that the common zero-set of allρκ×ρκ minors is aproper closed analytic subsetD_κ (of the unnamed domain where the U_αⁱ are defined), so it is stratified by a finite number of submanifolds of codimension>1 ([8, 2, 3, 5]), and in particular, it is of empty interior, hence intuitively “thin”.

So the set of parametersaat which there is a least oneρκ×ρκ minor ofJac U_κ which does not vanish is open anddense. Consequently, “for a generic pointa”, the mapU_κ is of rank>ρκ at every pointa⁰ sufficiently close toa (since the corre- spondingρκ×ρκ minor does not vanish in a neighborhood ofa), and because all (ρκ+1)×(ρκ+1)minors of JacU_κ were assumed to vanish identically, the map

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2.2 Quantitative Criterion for the Number of Superfluous Parameters 15 U_κ happens to be in fact ofconstant rankU_κ in a (small) neighborhood of every such a generica.

Insuring constancy of a rank is one important instance of why free relocaliza- tion is useful: a majority of theorems of the differential calculus and of the classical theory of (partial) differential equations do hold under specific local constancy as- sumptions.

Asκ increases, the number of columns of JacU_κ increases, hence ρκ1 6ρκ2

for κ16κ2. Sinceρκ 6r is anyway bounded, the generic rank of U_κ becomes constant for allκ>κ0bigger than some large enoughκ0. Thus, letρ∞6rdenote this maximal possible generic rank.

Definition 2.1.The parameters(a₁, . . . ,a_r)of given point transformation equations x⁰_i=f_i(x;a)are calledessentialif, after expandingf_i(x;a) =∑α∈NⁿU_αⁱ(a) (x−x₀)^α in power series at somex₀, the generic rankρ∞ of the coefficient mappinga7−→

U_αⁱ(a)_16i6n

α∈Nⁿ is maximal, equal to the numberrof parameters:ρ∞=r.

Without entering technical details, we make a remark. It is a consequence of the principle of analytic continuation and of some reasonings with power series that the same maximal rankρ∞is enjoyed by the coefficient mappinga7→ U⁰ⁱ_α(a)_16i6n

α∈Nⁿ

for the expansion of f_i(x;a) =∑α∈NⁿU⁰ⁱ_α(a) (x−x⁰₀)^α at another, arbitrary point x⁰₀. Also, one can prove thatρ∞is independent of the choice of coordinatesx_iand of parametersa_k. These two facts will not be needed, and the interested reader is referred to [9] for proofs of quite similar statements holding true in the context of Cauchy-Riemann geometry.

2.2 Quantitative Criterion

for the Number of Superfluous Parameters

It is rather not practical to compute the generic rank of the infinite Jacobian matrix JacU_∞. To check essentiality of parameters in concrete situations, a helpful criterion due to Lie is(iii)below.

Theorem 2.1.The following three conditions are equivalent:

(i) In the transformation equations

x⁰_i= f_i(x₁, . . . ,x_n;a₁, . . . ,a_r) =

∑

α∈NⁿU_αⁱ(a) (x−x₀)^α (i=1···n), the parameters a₁, . . . ,a_rarenotessential.

(ii) (By definition) The generic rankρ∞of the infinite Jacobian matrix:

JacU_∞(a) = ∂U_αⁱ

∂^aj

(a)

_α∈Nⁿ_,16i6n

16j6r

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is strictly less than r.

(iii) Locally in a neighborhood ofevery(x₀,a₀), there exists a not identically zero analytic vector field on the parameter space:

T =

∑

n k=1

τk(a) ∂

∂^ak

which annihilates all the fi(x;a):

0≡T f_i=

∑

n k=1

τk∂^fi

∂^ak

=

∑

α∈Nⁿ

∑

r k=1

τk(a)∂U_αⁱ

∂^ak

(a) (x−x₀)^α (i=1···n). More generally, ifρ∞denotes the generic rank of the infinite coefficient mapping:

U_∞: a 7−→ U_αⁱ(a)_16i6n

α∈Nⁿ,

then locally in a neighborhood ofevery(x₀,a₀), there exist exactly r−ρ∞, and no more, analytic vector fields:

T_µ=

∑

n k=1

τµk(a) ∂

∂^ak

(µ=1···r−ρ∞), with the property that the dimension ofSpan T1

a, . . . ,T_r−ρ_∞

a

is equal to r−ρ∞

at every parameter a at which the rank ofU_∞is maximal equal toρ∞, such that the derivationsTµall annihilate the f_i(x;a):

0≡T_µ f_i=

∑

r k=1

τµk(a)∂^fi

∂^ak

(x;a) (i=1···n;µ=1···r−ρ∞).

Proof. Just by the chosen definition, we have(i)⇐⇒(ii). Next, suppose that condition(iii)holds, in which the coefficientsτk(a)of the concerned nonzero derivation T are locally defined. Reminding that the Jacobian matrix JacU_∞hasrrows and an infinite number of columns, we then see that thenannihilation equations 0≡T f_i, when rewritten in matrix form as:

0≡ τ1(a), . . . ,τr(a)∂U_αⁱ

∂^aj

(a)

_α∈Nⁿ_,16i6n

16j6r

just say that the transpose of JacU_∞(a) has nonzero kernel at each a where the vectorT

a= τ1(a), . . . ,τr(a)

is nonzero. Consequently, JacU_∞has rank strictly less thanrlocally in a neighborhood of everya₀, hence in the wholea-domain. So (iii)⇒(ii).

Conversely, assume that the generic rankρ∞of JacU_∞ is<r. Then there exist ρ∞<r“basic” coefficient functionsU_αⁱ⁽¹⁾₍₁₎, . . . ,U_αⁱ⁽₍^ρ_ρ^∞_∞⁾₎(there can be several choices) such that the generic rank of the extracted map a7→ U_α^i(l)_(l)

16l6ρ∞ equalsρ∞ al-

Lie's Theory of Transformation Groups I, Contemporary Approach and Translation

Joël Merker, editor/translator

Theory of Transformation Groups of Lie and Engel

Modern Presentation and English Translation (Vol. I, 1888)

August 20, 2014

THEORIE

TRANSFORMATIONSGRUPPEN

ERSTER ABSCHNITT

SOPHUS LIE,

1930

VERLAG UND DRUCK VON B.G. TEUBNER IN LEIPZIG

UND BERLIN

Die neuen Theorien, die in diesem Werke dargestellt sind, habe ich in den Jahren von 1869 bis 1884 en- twickelt, wo mir die Liberalität meines

Geburtslandes Norwegen

gestattete, ungestört meine volle Arbeitskraft der Wissenschaft zu widmen, die durch A

Werke in Norwegens wissenschaftlicher Literatur den ersten Platz erhalten hat.

Bei der Durchführung meiner Ideen im Einzelnen und bei ihrer systematischen Darstellung genoss ich seit 1884 in der Grössten Ausdehnung die uner- müdliche Unterstützung des P

Friedrich Engel

meines ausgezeichneten Kollegen and der Universität Leipzig .

Das in dieser Weise entstandene Werk widme ich Frankreichs

École Normale Supérieure

deren unsterblicher Schüler G

zuerst die Be- deutung des Begriffs discontinuirliche Gruppe erkannte.

Den hervorragenden Lehrern dieses Instituts, beson- ders den Herren G. D

, E. P

und J.

T

verdanke ich es, dass die tüchtigsten jungen Mathematiker Frankreichs wetteifernd mit einer Reihe junger deutscher Mathematiker meine Untersuchungen über continuirliche Gruppen , über Geometrie und über Differentialgleichungen studiren und mit glänzendem Erfolge verwerthen.

Foreword

Contents

Part I

Modern Presentation

Chapter 1

Three Principles of Thought Governing the Theory of Lie

∑

Introduction

∑

∑

∑

∑

∑

∑

References

Chapter 2

Local Transformation Equations and Essential Parameters

2.1 Generic Rank of the Infinite Coefficient Mapping

∑

2.2 Quantitative Criterion

for the Number of Superfluous Parameters

∑

∑

∑

∑

∑

∑

∑