Specifying Domains of Existence

Thus, we considerK-analytic transformation equationsx⁰_i= f_i(x;a)defined on a more general domain than a product∆_ρⁿ×^r_σof two small polydiscs centered at the origin. Here is how Lie and Engel specify their domains of existence on p. 14 of [1], and these domains might be global.

§ 2.

In the transformation equations:

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

let now all the parametersa₁, . . . ,a_rbe essential.

Since the f_i are analytic functions of their arguments, in the domain [GE

-BIETE] of all systems of valuesx₁, . . . ,x_nand in the domain of all systems of valuesa₁, . . . ,a_r, we can choose a region [BEREICH](x)and, respectively, a region(a)such that the following holds:

Firstly. The f_i(x,a)are single-valued [EINDEUTIG] functions of then+r variablesx₁, . . . ,x_n,a₁, . . . ,a_rin the complete extension [AUSDEHNUNG] of the two regions(x)and(a).

Secondly. Thef_i(x,a)behave regularly in the neighbourhood of every sys-tem of valuesx⁰₁, . . . ,x⁰_n,a⁰₁, . . . ,a⁰_r, hence are expandable in ordinary power se-ries with respect tox1−x⁰₁, . . . ,xn−x⁰_n, as soon asx⁰₁, . . . ,x⁰_nlies arbitrarily in the domain(x), anda⁰₁, . . . ,a⁰_r lies arbitrarily in the domain(a).

Thirdly. The functional determinant:

∑

vanishes for no combination of systems of values of x_i and ofa_k in the two domains(x)and(a), respectively.

Fourthly. If one gives to the parametersa_kin the equationsx⁰_i=f_i(x,a)any of the valuesa⁰_kin domain(a), then the equations:

x⁰_i=f_i(x₁, . . . ,x_n;a⁰₁, . . . ,a⁰_r) (i=1···n)

always produce two different systems of valuesx⁰₁, . . . ,x⁰_nfor two different sys-tems of valuesx₁, . . . ,x_nof the domain(x).

We assume that the two regions(x)and(a)are chosen in such a way that these four conditions are satisfied. If we give to the variablesx_iin the equations x⁰_i=f_i(x,a)all possible values in(x)and to the parametersa_kall possible values in(a), then in their domain, thex⁰_irun throughout a certain region, which we can denote symbolically by the equationx⁰= f (x)(a)

. This new domain has the following properties:

Firstly. Ifa⁰₁, . . . ,a⁰_ris an arbitrary system of values of(a)andx⁰₁⁰, . . . ,x⁰_n⁰an arbitrary system of values of the subregionx⁰=f (x)(a⁰)

, then in the neigh-bourhood of the system of valuesx⁰_i⁰,a⁰_k, thex₁, . . . ,x_ncan be expanded in ordi-nary power series with respect tox⁰₁−x⁰₁⁰, . . . ,x⁰_n−x⁰_n⁰,a₁−a⁰₁, . . . ,a_n−a⁰_n.

Secondly. If one gives to thea_kfixed valuesa⁰_k in domain(a), then in the equations:

x⁰_i=f_i(x₁, . . . ,x_n;a⁰₁, . . . ,a⁰_r) (i=1···n),

the quantitiesx₁, . . . ,x_nwill be single-valued functions ofx⁰₁, . . . ,x⁰_n, which be-have regularly in the complete extension of the regionx⁰= f (x)(a⁰)

.

Referring also to the excerpt p. 26, we may rephrase these basic assumptions as follows. The f_i(x;a)are defined for(x,a)belonging to the product:

X ×A ⊂Kⁿ×K^r

of two domainsX ⊂KⁿandA ⊂K^r. These functions areK-analytic in both vari-ables, hence expandable in Taylor series at every point(x⁰,a⁰). Furthermore, for every fixeda⁰, the mapx7→ f(x;a⁰)is assumed to constitute aK-analytic diffeo-morphism ofX × {a⁰}onto its image. Of course, the inverse map is also locally expandable in power series, by virtue of theK-analytic inverse function theorem.

To insure that the composition of two transformations exists, one requires that there exist nonempty subdomains:

X¹⊂X and A¹⊂A, with the property that for every fixeda¹∈A¹:

f X¹× {a¹}

⊂X,

so that for every such ana¹∈A¹and for every fixedb∈A, the composed map:

x7−→f f(x;a¹);b

is well defined for allx∈X¹and moreover, is aK-analytic diffeomorphism onto its image. In fact, it is evenK-analytic with respect to all the variables(x,a¹,b)in X¹×A¹×A. Lie’s fundamental and uniquegroup composition axiommay then be expressed as follows.

(A1) There exists aK^r-valuedK-analytic mapϕ⁼ϕ^{(a,b)defined in}A¹×A¹with ϕ A¹×A¹

⊂A such that:

f f(x;a);b

≡ f x;ϕ^{(a,b) for all x}∈X¹,a∈A¹, b∈A¹.

3.9 Substituting the Axiom of Inverse for a Differential Equations Assumption 53 Here are two further specific unmentioned assumptions that Lie presupposes, still with the goal of admitting neither the identity element, nor the existence of inverses.

(A2) There is aK^r-valuedK-analytic map(a,c)7−→b=b(a,c)defined forarunning in a certain (nonempty) subdomain A²⊂A¹ and for c running in a certain (nonempty) subdomainC²⊂A¹withb A²×C²

⊂A¹ which solvesb in terms of(a,c)in the equationsc_k=ϕk(a,b), namely which satisfies identically:

c≡ϕ ^{a,b(a,c) for all a}∈A²,c∈C².

Inversely,ϕ^(a,b)solvescin terms of(a,b)in the equationsb_k=b_k(a,c), namely more precisely: there exists a certain (nonempty) subdomainA³⊂A²(⊂A¹) and a certain (nonempty) subdomainB³⊂A¹withϕ A³×B³

⊂C²such that one has identically:

b≡b a,ϕ^{(a,b) for all a}∈A³, b∈B³.

Example.In Engel’s counterexample of the groupx⁰=χ⁽λ^)xwith a Riemann uniformizing mapω ^:∆ →Λ as on p. 21 having inverseχ^:Λ →∆, these three requirements are satisfied, and in addition, we claim that one may even takeX = KandA¹=A =Λ, with no shrinking, for composition happens to hold in fact without restriction in this case. Indeed, starting from the general composition:

x⁰⁰=χ⁽λ2)x⁰=χ⁽λ2)χ⁽λ1)x,

that is to say, fromx⁰⁰=χ⁽λ2)χ⁽λ1)x, in order to represent it under the specific form x⁰⁰=χ⁽λ3)x, it is necessary and sufficient to solveχ⁽λ3) =χ⁽λ1)χ⁽λ2), hence we may take forϕ^:

λ3=ω χ⁽λ1)χ⁽λ2)

=:ϕ⁽λ1,λ2),

without shrinking the domains, for the two inequalities|χ⁽λ1)|<1 and|χ⁽λ2)|<1 readily imply that|χ⁽λ1)χ⁽λ2)|<1 too so thatω χ⁽λ1)χ⁽λ2)

is defined. On the other hand, for solvingλ2in terms of(λ1,λ3)in the above equation, we are naturally led to define:

b(λ1,λ3):=ω χ⁽λ3) χ⁽λ1)

,

and then b=b(λ1,λ3) is defined under the specific restriction that |χ⁽λ3)|<

|χ⁽λ1)|.

However, the axiom (A2)happens to be still incomplete for later use, and one should add the following axiom in order to be able to also solveainc=ϕ^(a,b).

(A3)There is aK^m-valuedK-analytic map(b,c)7−→a=a(b,c)defined inB⁴×C⁴ withB⁴⊂A¹andC⁴⊂A¹, and witha B⁴×C⁴

⊂A¹, such that one has identically:

c≡ϕ ^{a(b,c),b for all b}∈B⁴, c∈C⁴.

Inversely,ϕ^(a,b)solvescin the equationsa_k=a_k(b,c), namely more precisely:

there existB⁵⊂B⁴andA⁵⊂A¹withϕ A⁵×B⁵

⊂C⁴such that one has

identically:

a≡a b,ϕ^{(a,b) for all a}∈A⁵, b∈B⁵, and furthermore in addition, witha B⁵×C⁴

⊂A²and withb A⁵×C⁴

⊂ B⁴such that one also has identically:

b≡b a(b,c),c

for all b∈B⁵, c∈C⁴ a≡a b(a,c),a

for all a∈A⁵,c∈C⁴.

The introduction of the numerous (nonempty) domainsA²,C²,A³,B³,B⁴, C⁴,A⁵,B⁵which appears slightly unnatural and seems to depend upon the order in which the solving mapsaandbare considered can be avoided by requiring from the beginning only that there exist two subdomainsA³⊂A²⊂A¹such that one has uniformly:

c≡ϕ ^{a,b(a,c) for all a}∈A³, c∈A³ c≡ϕ ^{a(b,c),b for all b}∈A³,c∈A³ b≡b a(b,c),c

for all b∈A³,c∈A³ b≡b a,ϕ^{(a,b) for all a}∈A³,b∈A³ a≡a b(a,c),c

for all a∈A³,c∈A³ a≡a b,ϕ^{(a,b) for all a}∈A³, b∈A³.

We will adopt these axioms in the next subsection. Importantly, we would like to point out that, although b(a,c) seems to represent the group product a⁻¹·c=m i(a),c

, the assumption (A2) neither reintroduces inverses, nor the identity element, it just means that one may solvebby means of the implicit function theorem in the parameter composition equationsc_k=ϕk(a₁, . . . ,a_r,b₁, . . . ,b_r).

3.9.2 Group Composition Axiom And Fundamental Differential