Essentiality of Multiple Flow Parameters

with a first order perturbation which is the Lie bracket

X⁰,Y⁰ .

4.4 Essentiality of Multiple Flow Parameters

At present, we again consider r arbitrary vector fields with analytic coefficients defined on a certain, unnamed domain ofKⁿwhich contains the origin:

X_k=

∑

n i=1

ξki(x₁, . . . ,x_n) ∂

∂^xi

(k=1···r).

Though the collection X1, . . . ,X_k does not necessarily stem from anr-term group, each individual X_k nonetheless generates the one-term continuous transformation groupx⁰=exp(tX_k)(x)with corresponding infinitesimal transformationsx⁰_i=x_i+ ε ξki(x); this is the reason why we shall hence sometimes alternatively refer to such general vector fieldsX_kas beinginfinitesimal transformations.

Definition 4.1.The rinfinitesimal transformations X₁, . . . ,X_r will be called inde-pendent(of each other) if they are linearly independent, namely if thenequations:

0≡e₁ξ1i(x) +···+e_rξri(x) (i=1···n)

in whiche₁, . . . ,e_rareconstants, do implye₁=···=e_r=0.

For instance, Theorem 3 on p. 40 states that anr-term continuous local trans-formation groupx⁰_i=f_i(x;a₁, . . . ,a_r)whose parametersa_kare allessentialalways gives rise to therinfinitesimal transformationsX_k:=−_∂^∂_a^f_k(x;e),k=1, . . . ,r, which areindependent of each other.

Introducingrarbitrary auxiliary constantsλ1, . . . ,λr, one may consider the one-term group generated by the general linear combination:

C:=λ1X₁+···+λrX_r, ofX₁, . . . ,X_r, namely the flow:

x⁰_i=exp(tC)(x_i) =x_i+t

1C(x_i) + t²

1·2C C(x_i) +···

=x_i+t

∑

r k=1

λkξki+t²

1...r

∑

k,j

λkλj

1·2 X_k(ξji) +···

=:h_i(x;t,λ1, . . . ,λr) (i=1···n). If theX_k≡ −_∂^∂_a^f

k(x;e)do stem from anr-term continuous groupx⁰_i=f_i(x;a₁, . . . ,a_r), a natural question is then to compare the above integrated finite equations h_i(x;t,λ1, . . . ,λr)to the original transformation equations f_i(x;a₁, . . . ,a_r). Before studying this question together with Lie and Engel, we focus our attention on a subquestion whose proof shows a beautiful, synthetical, geometrical idea: that of prolongating the action jointly to finite sets of points.

At first, without assuming that theX_kstem from anr-term continuous group, it is to be asked (subquestion) whether the parametersλ1, . . . ,λrin the above integrated transformation equationsx⁰_i=h_i(x;λ1, . . . ,λr)are all essential. In the formula just above, we notice that the r+1 parameters only appear in the formtλ1, . . . ,tλr, hence because theλkare arbitrary, there is no restriction to sett=1. We shall then simply writeh_i(x;λ1, . . . ,λr)instead ofh_i(x; 1,λ1, . . . ,λr).

4.4 Essentiality of Multiple Flow Parameters 75

Theorem 8.If the r independent infinitesimal transformations:

X_k(f) =

Proof. Here exceptionally, we observed a harmless technical incorrection in Engel-Lie’s proof ([2], pp. 62–65) about the link between the generic rank ofX₁

x, . . . ,X_r

x

and a lower bound for the number of essential parameters².

However, Lie’s main idea is clever and pertinent: it consists in the introduction of exactlyr(the number ofλk’s) copies of the same space x₁, . . . ,x_nwhose coor-dinates are labelled asx⁽₁^µ), . . . ,x⁽_n^µ) forµ^{=1, . . . ,r}and to consider the family of transformation equations induced by thesame transformation equations:

x⁽_i^µ)0=exp(C) x⁽_i^µ)

=h_i x^(µ);λ1, . . . ,λr

(i=1···n;µ=1···r)

on each copy of space, again witht =1. Geometrically, one thus views how the initial transformation equationsx⁰_i=hi(x;λ1, . . . ,λr)actsimultaneouslyonr-tuples of points. Written in greater length, these transformations read:

(5’) x⁽_i^µ)0=x⁽_i^µ)+ an idea also reveals to be fruitful in other contexts.

According to the theorem stated on p. 15, in order to check that the parameters

λ1, . . . ,λrare essential, one only has to expandx⁰in power series with respect to the

2 On page 63, it is said that if the numberrof the independent infinitesimal transformationsX_kis 6n, then ther×nmatrix ξ^ki(x)16i6n

16k6rof their coefficients is of generic rank equal tor, although this claim is contradicted withn=r=2 by the two vector fieldsx_∂x^∂ +y_∂y^∂ andxx_∂x^∂ +xy_∂y^∂. Nonetheless, the ideas and the arguments of the written proof (which does not really needs such a fact) are perfectly correct.

powers ofxat the origin:

x⁰_i=

∑

α∈Nⁿ

U_αⁱ(λ^{)xα (i=1...n),}

and to show that the generic rank of the infinite coefficient mapping λ 7−→

U_αⁱ(λ⁾¹⁶ⁱ⁶ⁿ_α∈Nn is maximal possible equal tor. Correspondingly and immediately, we get the expansion of ther-times copied transformation equations:

(5”) x⁽_i^µ)0=

∑

α∈NⁿU_α^i,(µ)(λ^{) (x(µ))α (i=1...n;µ=1...r),} with, for eachµ^{=1, . . . ,r, thesame}coefficient functions:

U_α^i,(µ)(λ⁾≡U_αⁱ(λ^{) (i=1...n;α∈Nn;µ=1...r).}

So the generic rank of the corresponding infinite coefficient matrix, which is just an r-times copy of the same mappingλ7−→ U_αⁱ(λ⁾¹⁶ⁱ⁶ⁿ_α∈Nn, does neither increase nor decrease.

Thus, the parametersλ1, . . . ,λrfor the transformation equations x⁰=h(x;λ^)are essential if and only if they are essential for the diagonal transformation equations x^(µ)0=h x^(µ);λ^,µ^{=1, . . . ,}r, induced on the r-fold copy of the space x₁, . . . ,x_n.

Therefore, we are left with the purpose of showing that the generic rank of ther times copy of the infinite coefficient matrixλ7−→ U_α^i,(µ)(λ^)16i6n,_α∈Nn ^16µ6ris equal tor. We shall in fact establish more, namely that the rank at λ ⁼0 of this map already equalsr, or equivalently, that the infinite constant matrix:

∂U_α^i,(µ)

∂λk

(0)

_16i6n,α∈Nⁿ,16µ6r 16k6r

,

whoserlines are labelled with respect to partial derivatives, has rank equal tor.

To prepare this infinite matrix, if we differentiate the expansions (5’) which iden-tify to (5”) with respect to λk atλ ⁼0, and if we expand the coefficients of our infinitesimal transformations:

ξki(x^(µ)) =

∑

α∈Nⁿξkiα(x^(µ))^α (i=1...n;k=1...r;µ=1...r)

with respect to the powers ofx₁, . . . ,x_n, we obtain a more suitable expression of it:

∂U_α^i,(µ)

∂λk

(0)

_16i6n,α∈Nⁿ_,16µ6r

16k6r ≡

ξkiα_16i6n,α∈Nⁿ

16k6r ··· ξkiα_16i6n,α∈Nⁿ

16k6r

=:

J^∞Ξ⁽⁰⁾···J^∞Ξ^(0).

4.4 Essentiality of Multiple Flow Parameters 77 As argued up to now, it thus suffices to show that this matrix has rank r. Also, we observe that this matrix identifies with theinfinite jet matrixJ^∞Ξ^{(0)of Taylor} coefficients of ther-fold copy of the samer×nmatrix of coefficients of the vector fieldsX_k:

Ξ^(x):=



ξ11(x)··· ξ1n(x)

··· ··· ···

ξr1(x)··· ξrn(x)



.

This justifies the symbolJ^∞introduced just above. At present, we can formulate an auxiliary lemma which will enable us to conclude.

Lemma 4.2.Let n>1, q>1, m>1be integers, letx∈Kⁿand let:

A(x) =



a11(x)··· a1m(x)

··· ··· ···

a_q1(x)··· a_qm(x)





be an arbitrary q×m matrix of analytic functions:

a_{i j}(x) =

∑

α∈Nⁿ

a_{i jα}x^α (i=1...q;j=1...m)

that are all defined in a fixed neighborhood of the origin inKⁿ, and introduce the q×∞constant matrix of Taylor coefficients:

J^∞A(0):= a_{i jα16j6m,α∈N}ⁿ

16i6q

whose q lines are labelled by the index i. Then the following inequality between (generic) ranks holds true:

rkJ^∞A(0)>genrkA(x).

Proof. Here, our infinite matrixJ^∞A(0)will be considered as acting byleft multi-plication onhorizontal vectors u= (u₁, . . . ,u_q), so thatuJ^∞A(0)is an∞×1 matrix, namely an infinite horizontal vector. Similarly,A(x)will act on horizontal vectors of analytic functions(u₁(x), . . . ,ur(x)).

Supposing that u = (u₁, . . . ,u_q)∈K^q is any nonzero vector in the kernel of J^∞A(0), namely: 0=uJ^∞A(0), or else in greater length:

0=u₁a_1jα+···+u_qa_{q jα} (j=1...m;α∈Nⁿ),

we then immediately deduce, after multiplying each such equation byx^α and by summing up over allα∈Nⁿ:

0≡u₁a_1j(x) +···+u_qa_{q j}(x) (j=1...m),

so that the same constant vectoru= (u₁, . . . ,u_q)also satisfies 0≡u A(x). It follows that the dimension of the kernel ofJ^∞A(0)is smaller than or equal to the dimension

of the kernel ofA(x)(at a genericx): this is just equivalent to the above inequality

between (generic) ranks. ut

Now, for eachq=1,2, . . . ,r, we want to apply the lemma with the matrixA(x) being the q-fold copy of matrices Ξ^(x(1))···Ξ^(x(q)), or equivalently in greater length:

Ξq ex_q :=



ξ₁₁⁽¹⁾··· ξ_1n⁽¹⁾ ξ₁₁⁽²⁾··· ξ_1n⁽²⁾ ··· ξ₁₁^(q) ··· ξ_1n^(q)

··· ··· ··· ··· ··· ··· ··· ··· ··· ···

ξr1⁽¹⁾··· ξrn⁽¹⁾ ξr1⁽²⁾··· ξrn⁽²⁾ ··· ξr1^(q) ··· ξrn^(q)



,

where we have abbreviated:

ex_q:= x⁽¹⁾, . . . ,x^(q) .

Lemma 4.3.It is a consequence of the fact that X1,X2, . . . ,Xrare linearly indepen-dent of each other that for every q=1,2, . . . ,r, one has:

genrk

Ξ ^x(1) Ξ ^x(2) ··· Ξ ^x(q)>q.

Proof. Indeed, forq=1, it is at first clear that genrk Θ^(x(1))>1, just because not all theξki(x)vanish identically.

We next establish by induction that, as long as they remain<r, generic ranks do increase of at least one unity at each step:

genrk

Ξq+1 ex_q+1

>1+genrk

Ξq ex_q , a fact which will immediately yield the lemma.

Indeed, if on the contrary, the generic ranks would stabilize, and still be <r, then locally in a neighborhood of a generic, fixed ex⁰_q+1, both matrices Ξq+1

and Ξq would have the same, locally constant rank. Consequently, the solutions ϑ1(ex_q)···ϑr(ex_q)

to the (kernel-like) system of linear equations written in matrix form:

0≡ ϑ1(ex_q)···ϑr(ex_q) Ξq ex_q

,

which are analytic nearex⁰_qthanks to an application of Cramer’s rule and thanks to constancy of rank, would be automatically also solutions of the extended system:

0≡ ϑ1(ex_q)···ϑr(ex_q)

Ξq(ex_q) Ξ^(x(q+1)),

whence there would existnonzerosolutions(ϑ1, . . . ,ϑr)to the linear dependence equations:

0= ϑ1···ϑr

Ξ ^x(q+1)

4.4 Essentiality of Multiple Flow Parameters 79 which areconstantwith respect to the variablex^(q+1), since they only depend upon ex_q. This exactly contradicts the assumption thatX₁^(q+1), . . . ,Xr^(q+1)are independent

of each other. ut

Lastly, we may chain up a series of (in)equalities that are now obvious conse-quences of the lemma and of the assertion:

rank

J^∞Ξ⁽⁰⁾···J^∞Ξ^(0)=rankJ∞Ξr(0)>genrkΞr ex_r

=r, and since all ranks are anyway6r, we get the promised rank estimation:

r=rank

J^∞Ξ⁽⁰⁾···J^∞Ξ^(0),

which finally completes the proof of the theorem. ut

In order to keep a memory track of the trick of extending the group action to an r-fold product of the base space, we also translate a summarizing proposition which is formulated on p. 66 of [2].

Proposition 5.If the r infinitesimal transformations:

X_k(f) =

∑

n i=1

ξki(x₁, . . . ,x_n)∂^f

∂^xi

(k=1...r)

are independent of each other, if furthermore:

x⁽₁^µ), . . . ,x⁽_n^µ) (µ=1...r)

are r different systems of n variables, and if lastly one sets for abbreviation:

X_k^(µ)(f) =

∑

n i=1

ξki x⁽₁^µ), . . . ,x⁽n^µ)

∂^f

∂^x(i^µ)

(k,µ=1...r),

then the r infinitesimal transformations:

W_k(f) =

∑

r µ=1

X_k^(µ)(f) (k=1...r)

in the nr variables x⁽_i^µ)satisfynorelation of the form:

∑

n k=1

χk x₁⁽¹⁾, . . . ,x⁽¹⁾_n , . . . ,x^(r)₁ , . . . ,x^(r)_n

W_k(f)≡0.

Also, we remark for later use as in [2], p. 65, that during the proof of the the-orem 8 above, it did not really matter that the equations (5) represented the finite

equations of a family of one-term groups. In fact, we only considered the terms of first order with respect toλ1, . . . ,λr in the finite equations (5), and the crucial Lemma 4.3 emphasized during the proof was true under the only assumption that the infinitesimal transformationsX₁, . . . ,X_r were mutually independent. Consequently, Theorem 8 can be somewhat generalized as follows.

Proposition 4.If a family of transformations contains the r arbitrary parame-ters e₁, . . . ,e_rand if its equations, when they are expanded with respect to pow-ers of e₁, . . . ,e_r, possess the form:

x⁰_i=x_i+

∑

r k=1

e_kξki(x₁, . . . ,x_n) +··· ^(i=1···n),

where the neglected terms in e1, . . . ,er are of second and of higher order, and lastly, if the functionsξki(x)have the property that the r infinitesimal transfor-mations made up with them:

X_k(f) =

∑

n i=1

ξki(x₁, . . . ,x_n)∂^f

∂^xi

(k=1···r)

are independent of each other, then those transformation equations represent∞^r different transformations, or what is the same: the r parameters e₁, . . . ,e_r are essential.

Dans le document Lie's Theory of Transformation Groups I, Contemporary Approach and Translation (Page 87-94)