Axiom of Inverse and Engel’s Counter-Example

do holdeverywhere, as desired. In conclusion, we have shown the implication(ii)

⇒(iii), and simultaneously, we have established the last part of the theorem. ut Corollary 2.1.Locally in a neighborhood of every generic point a₀ at which the infinite coefficient mapping a7→U_∞(a) has maximal, locally constant rank equal to its generic rank ρ∞, there exist both a local change of parameters a7→ u₁(a), . . . ,u_ρ∞(a)

=:udecreasing the number of parameters from r down 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:

gi x;u(a)

≡fi(x;a) (i=1···n).

Proof. Choose ρ∞ coefficients U_α^i(l)_(l)(a) =: u_l(a), 16 l 6ρ∞, with ∆^{(a) :=}

det ^∂_∂^u_a^l(a)

m (a)_16l6ρ∞

16m6ρ∞ 6≡0 as in the proof of the theorem. Locally in some small neighborhood of any a⁰ with ∆^(a0)6=0, the infinite coefficient map U_∞ has constant rank ρ∞, hence the constant rank theorem provides, for every (i,α^{), a} certain functionV_αⁱofρ∞variables such that:

U_αⁱ(a)≡V_αⁱ u₁(a), . . . ,u_ρ∞(a) .

Thus, we can work out the power series expansion:

f_i(x;a) =∑α∈NⁿU_αⁱ(a) (x−x₀)^α

=∑_α∈NⁿV_αⁱ(u₁(a), . . . ,u_ρ∞(a)) (x−x₀)^α

=:g_i(x,u₁(a), . . . ,u_ρ∞(a)).

which yields the natural candidate for g_i(x;u). Lastly, one may verify that any Cauchy estimate for the growth decrease of U_αⁱ(a)as |α| →∞insures a similar Cauchy estimate for the growth decrease ofb7→V_αⁱ(u), whence eachg_iis analytic,

and in fact, termwise substitution was legitimate. ut

Definition 2.2.Transformation equationsx_i⁰=f_i(x₁, . . . ,x_n;a₁, . . . ,a_r),i=1, . . . ,n, are calledr-termwhen all its parameters(a₁, . . . ,a_r)are essential.

2.3 Axiom of Inverse and Engel’s Counter-Example

Every analytic diffeomorphism of ann-times extended space permutes all the points in a certain differentiable, invertible way. Although they act on a set of infinite car-dinality, diffeomorphisms can thus be thought to be sorts of analogs of the

substi-tutionson a finite set. In fact, in the years 1873–80, Lie’s Idée fixe was to build, in the geometric realm ofn-dimensional continua, a counterpart of the Galois theory of substitutions of roots of algebraic equations ([6]).

As above, letx⁰=f(x;a₁, . . . ,a_r) =:f_a(x)be a family of (local) analytic diffeo-morphisms parameterized by a finite numberrof parameters. For Lie, the basic, single group axiom should just require that such a family beclosed under composi-tion, namely that one always has f_a f_b(x)

≡f_c(x)for somecdepending onaand onb. More precisions on this definition will given in the next chapter, but at present, we ask whether one can really economize the other two group axioms: existence of an identity element and existence of inverses.

Lemma 2.1.If H is any subset of some abstract group G withCardH<∞which is closed under group multiplication:

h₁h₂∈H whenever h₁,h₂∈H,

then H contains the identity element e of G and every h∈H has an inverse in H, so that H itself is a true subgroup of G.

Proof. Indeed, pickingh∈Harbitrary, the infinite sequenceh,h²,h³, . . . ,h^k, . . .of elements of the finite setHmust become eventually periodic:h^a=h^a+nfor some a>1 and for somen>1, whencee=hⁿ, soe∈Handhⁿ⁻¹is the inverse ofh. ut For more than thirteen years, Lie was convinced that a purely similar property should also hold with G=Diff_n being the (infinite continuous pseudo)group of analytic diffeomorphisms and withH⊂Diff_nbeing any continuous family closed under composition. We quote a characteristic excerpt of [7], pp. 444–445.

As is known, one shows in the theory of substitutions that the permutations of a group can be ordered into pairwise inverse couples of permutations. Now, since the distinction between a permutation group and a transformation group only lies in the fact that the former contains a finite and the latter an infinite number of operations, it is natural to presume that the transformations of a transforma-tion group can also be ordered into pairs of inverse transformatransforma-tions. In previous works, I came to the conclusion that this should actually be the case. But be-cause in the course of my investigations in question, certainimplicithypotheses have been made about the nature of the appearing functions, then I think that it is necessary toexpressly add the requirement that the transformations of the group can be ordered into pairs of inverse transformations. In any case, I conjecture that this is a necessary consequence of my original definition of the concept [BEGRIFF] of transformation group. However, it has been impossible for me to prove this in general.

As a proposal of counterexample that Engel devised in the first year he worked with Lie (1884), consider the family of transformation equations:

2.3 Axiom of Inverse and Engel’s Counter-Example 21 x⁰=ζ^x,

wherex,x⁰∈Cand the parameterζ∈Cis restricted to|ζ|<1. Of course, this fam-ily is closed under any composition, say:x⁰=ζ1xandx⁰⁰=ζ2x⁰=ζ1ζ2x, with in-deed|ζ2ζ1|<1 when|ζ1|,|ζ2|<1, but neither the identity element nor any inverse transformation does belong to the family. However, the requirement|ζ|<1 is here too artificial: the family extends in fact trivially as the complete group x⁰=ζ^x_ζ∈C of dilations of the line. Engel’s idea was to appeal to a Riemann map ω ^having {|ζ|=1}as a frontier of nonextendability. The map used by Engel is the follow-ing¹. Letod_kdenote the number of odd divisors (including 1) of any integerk>1.

The theory of holomorphic functions in one complex variables yields the following.

Lemma 2.2.The infinite series:

ω^(a):=

∑

ν>1

a^ν

1−a^2ν =

∑

ν>1

a^ν+a^3ν+a^5ν+a^7ν+···

=

∑

k>1

od_ka^k converges absolutely in every open disc∆ρ={z∈C: |z|<ρ}of radiusρ^<1and defines a univalent holomorphic function∆→Cfrom the unit disc∆^:={|z|<1}to Cwhichdoes notextend holomorphically across any point of the unit circle∂∆ ^:=

{|z|=1}.

In fact, any other similar Riemann biholomorphic mapζ7−→ω⁽ζ^{) =:}λ^{from the} unit disc∆onto some simply connected domainΛ^:=ω⁽∆⁾having fractal boundary not being a Jordan curve, as e.g. the Von Koch Snowflake Island, would do the job². Denote then byλ7−→χ⁽λ^{) =:}ζ the inverse of such a map and consider the family of transformation equations:

x⁰=χ⁽λ^)x_λ∈Λ^.

By construction,|χ⁽λ⁾|<1 for everyλ∈Λ. Any composition ofx⁰=χ⁽λ1)xand of x⁰⁰=χ⁽λ2)x⁰ is of the formx⁰⁰=χ⁽λ⁾x, with the uniquely defined parameter λ ^:=ω χ⁽λ1)χ⁽λ2)

, hence the group composition axiom is satisfied. However there is again no identity element, and again, none transformation has an inverse.

And furthermore crucially (and lastly), there does not exist any prolongation of the family to a larger domainΛe⊃Λ together with a holomorphic prolongationχe^ofχ toΛe ^{so that}χ^e Λ^econtains a neighborhood of{1}(in order to catch the identity) or a fortiori a neighborhood of ∆ (in order to catch inverses of transformations x⁰=χ⁽λ^)xwithλ∈Λ ^{close to}∂Λ^).

1 In the treatise [4], this example is presented at the end of Chap. 9,seebelow p. 179.

2 A concise presentation of Carathéodory’s theory may be found in Chap. 17 of [10].

Observation

In Vol. I of theTheorie der Transformationsgruppen, this example appears only in Chap. 9, on pp. 163–165, and it is written in small characters. In fact, Lie still be-lieved that a deep analogy with substitution groups should come out as a theorem.

Hencethe structure of the first nine chapters insist on setting aside, whenever pos-sible,the two axioms of existence of identity element and of existence of inverses.

To do justice to this great treatise, we shall translate in Chap. 9 how Master Lie managed to produce the Theorem 26 on p. 178, which he considered to provide the sought analogy with finite group theory, after taking Engel’s counterexample into account.

References

1. Ackerman, M., Hermann, R.: Sophus Lie’s 1880 Transformation Group paper. Math. Sci. Press, Brookline, Mass. (1975)

2. Bochnak, J., Coste, M., Roy, M.-F.: Géométrie algébrique réelle. Ergenisse der Mathematik und ihrer Grenzgebiete (3), 12. Springer-Verlag, Berlin, x+373 pp. (1987)

3. Chirka, E.M.: Complex analytic sets. Mathematics and its applications (Soviet Series), 46.

Kluwer Academic Publishers Group, Dordrecht, xx+372 pp. (1989)

4. 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)

5. Gunning, R.: Introduction to Holomorphic Functions of Several Variables, 3 vols. Wadsworth

& Brooks/Cole. I: Function theory, xx+203 pp. II: Local theory, +218 pp. III: Homological theory, +194 pp. (1990)

6. Hawkins, T.: Emergence of the theory of Lie groups. An essay in the history of mathematics 1869–1926. Sources and studies in the history of mathematics and physical sciences, Springer-Verlag, Berlin, xiii+564 pp. (2001)

7. Lie, S.: Theorie der Transformationsgruppen. Math. Ann.16, 441–528 (1880). Translated in English and commented in: [1].

8. Malgrange, B.: Ideals of Differentiable Functions. Tata Institute of Fundamental Research Studies in Mathematics, No. 3, Bombay. Oxford University Press, London, vii+106 pp. (1967) 9. Merker, J.: On the local geometry of generic submanifolds ofCⁿand the analytic reflection

principle. Journal of Mathematical Sciences (N. Y.)125, 751–824 (2005)

10. Milnor, J.: Dynamics in one complex variable. Annals of Mathematics Studies, 160. Princeton University Press, Princeton, Third Edition, viii+304 pp. (2006)

—————–

Chapter 3

Fundamental Differential Equations

for Finite Continuous Transformation Groups

Abstract Afinite continuous local transformation group in the sense of Lie is a family of local analytic diffeomorphismsx⁰_i=f_i(x;a),i=1. . . ,n, parametrized by a finite numberrof parametersa₁, . . . ,a_rthat is closed under composition and under taking inverses:

f_i f(x;a);b)

=f_i x;m(a,b)

and x_i=f_i x⁰;i(a) ,

for somegroup multiplication mapmand for somegroup inverse mapi, both local and analytic. Also, it is assumed that there existse= (e₁, . . . ,e_r)yielding the iden-titytransformation fi(x;e)≡xi.

Crucially, these requirements imply the existence offundamental partial differ-ential equations:

∂^fi

∂^ak

(x;a) =−

∑

j=1

ψk j(a)∂^fi

∂^aj

(x;e) (i=1···n,k=1···r)

which, technically speaking, are cornerstones of the basic theory. What matters here is that the group axioms guarantee that ther×rmatrix (ψk j)depends only ona and it is locally invertible near the identity. Geometrically speaking, these equations mean that therinfinitesimal transformations:

X_k^a

x=∂^f1

∂^ak

(x;a) ∂

∂^x1

+···+∂^fn

∂^ak

(x;a) ∂

∂^xn

(k=1···r)

corresponding to an infinitesimal increment of thek-th parameter computed ata:

f(x;a₁, . . . ,a_k+ε^{, . . . ,a}r)−f(x;a₁, . . . ,a_k, . . . ,a_r)≈ε^Xk^a

x

arelinear combinations, with certain coefficients−ψk j(a)depending only on the parameters, of the same infinitesimal transformations computed at the identity:

23

X_k^e

x=∂^f1

∂^ak

(x;e) ∂

∂^x1

+···+∂^fn

∂^ak

(x;e) ∂

∂^xn

(k=1···r).

Remarkably, the process of removing superfluous parameters introduced in the pre-vious chapter applies to local Lie groups without the necessity of relocalizing around a generica₀, so that everything can be achieved around the identityeitself, without losing it.