Local Transformation Equations and Essential Parameters

Let $$\mathbb {K}= \mathbb {R}$$ K = R or $$\mathbb {C}$$ C , throughout. As said in Chap. 1 , transformation equations $$x_i' = f_i ( x; \, a_1, \dots , a_r)$$ x i ′ = f i ( x ; a 1 , ⋯ , a r ) , $$i =1, \dots , n$$ i = 1 , ⋯ , n , which are local, analytic diffeomorphisms of $$\mathbb {K}^n$$ K n parametrized by a finite number $$r$$ r of real or complex numbers $$a_1, \dots , a_r$$ a 1 , ⋯ , a r , constitute the archetypal objects of Lie’s theory. The preliminary question is to decide whether the $$f_i$$ f i really depend upon all parameters, and also, to get rid of superfluous parameters, if there are any. Locally in a neighborhood of a fixed $$x_0$$ x 0 , one expands $$f_i ( x; \, a) = \sum _{ \alpha \in \mathbb {N}^n}\, \fancyscript{ U}_\alpha ^i (a) \, ( x - x_0)^\alpha $$ f i ( x ; a ) = ∑ α ∈ N n U α i ( a ) ( x - x 0 ) α in power series and one looks at the infinite coefficient mapping $$\mathsf{U}_\infty : \, a \longmapsto \big ( \fancyscript{ U}_\alpha ^i ( a) \big )_{ \alpha \in \mathbb {N}^n}^{ 1\leqslant i \leqslant n}$$ U ∞ : a ⟼ ( U α i ( a ) ) α ∈ N n 1 ⩽ i ⩽ n from $$\mathbb {K}^r$$ K r to $$\mathbb {K}^\infty $$ K ∞ , which is expected to faithfully describe the dependence with respect to $$a$$ a in question. If $$\rho _\infty $$ ρ ∞ denotes the maximal, generic and locally constant rank of this map, with of course $$0 \leqslant \rho _\infty \leqslant r$$ 0 ⩽ ρ ∞ ⩽ r , then the answer says that locally in a neighborhood of a generic $$a_0$$ a 0 , there exist both a local change of parameters $$a \mapsto \big ( \mathsf{u}_1 ( a), \dots , \mathsf{u}_{\rho _\infty } ( a) \big ) =: \mathsf{u}$$ a ↦ ( u 1 ( a ) , ⋯ , u ρ ∞ ( a ) ) = : u decreasing the number of parameters from $$r$$ r down to $$\rho _\infty $$ ρ ∞ , and new transformation equations: $$ x_i' = g_i \big (x;\,\mathsf{u}_1,\dots ,\mathsf{u}_{\rho _\infty }\big ) \ \ \ \ \ \ \ \ \ \ {{(i\,=\,1\,\cdots \,n)}} $$ x i ′ = g i ( x ; u 1 , ⋯ , u ρ ∞ ) ( i = 1 ⋯ n ) depending only upon $$\rho _\infty $$ ρ ∞ parameters which give again the old ones: $$ g_i\big (x;\,\mathsf{u}(a)\big ) \equiv f_i(x;\, a) \ \ \ \ \ \ \ \ \ \ {{(i\,=\,1\,\cdots \,n)}}. $$ g i ( x ; u ( a ) ) ≡ f i ( x ; a ) ( i = 1 ⋯ n ) . At the end of this brief chapter, before giving a precise introduction to 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’s Idées fixes.