Fundamental Differential Equations for Finite Continuous Transformation Groups

A finite continuous local transformation group in the sense of Lie is a family of local analytic diffeomorphisms $$x_i ' = f_i ( x; \, a)$$ x i ′ = f i ( x ; a ) , $$i = 1 \dots , n$$ i = 1 ⋯ , n , parametrized by a finite number $$r$$ r of parameters $$a_1, \dots , a_r$$ a 1 , ⋯ , a r that is closed under composition and under taking inverses: $$ f_i\big (f(x;\,a);\,b)\big ) = f_i\big (x;\,\mathbf m (a,b)\big ) \ \ \ \ \ \ \ \text {and} \ \ \ \ \ \ \ x_i = f_i\big (x';\,\mathbf i (a)\big ), $$ f i ( f ( x ; a ) ; b ) ) = f i ( x ; m ( a , b ) ) and x i = f i ( x ′ ; i ( a ) ) , for some group multiplication map $$\mathbf m $$ m and for some group inverse map $$\mathbf i $$ i , both local and analytic. Also, it is assumed that there exists an $$e = (e_1, \dots , e_r)$$ e = ( e 1 , ⋯ , e r ) yielding the identity transformation $$f_i ( x; \, e) \equiv x_i$$ f i ( x ; e ) ≡ x i . Crucially, these requirements imply the existence of fundamental partial differential equations: $$ \boxed { \frac{\partial f_i}{\partial a_k}(x;\,a) = - \sum _{j=1}^r\,\psi _{kj}(a)\,\frac{\partial f_i}{\partial a_j}(x;\,e)} \ \ \ \ \ \ \ \ \ \ {{(i\,=\,1\,\cdots \,n,\,\,\,k\,=\,1\,\cdots \,r)}} $$ ∂ f i ∂ a k ( x ; a ) = - ∑ j = 1 r ψ k j ( a ) ∂ f i ∂ a j ( 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 the $$r\times r$$ r × r matrix $$( \psi _{ kj})$$ ( ψ k j ) depends only on $$a$$ a and it is locally invertible near the identity. Geometrically speaking, these equations mean that the $$r$$ r infinitesimal transformations: $$ X_k^a\big \vert _x = \frac{\partial f_1}{\partial a_k}(x;\,a)\,\frac{\partial }{\partial x_1} +\cdots + \frac{\partial f_n}{\partial a_k}(x;\,a)\,\frac{\partial }{\partial x_n} \ \ \ \ \ \ \ \ \ \ \ \ \ {{(k\,=\,1\,\cdots \,r)}} $$ X k a | x = ∂ f 1 ∂ a k ( x ; a ) ∂ ∂ x 1 + ⋯ + ∂ f n ∂ a k ( x ; a ) ∂ ∂ x n ( k = 1 ⋯ r ) corresponding to an infinitesimal increment of the $$k$$ k -th parameter computed at $$a$$ a : $$ f(x;a_1,\dots ,a_k+\varepsilon ,\dots ,a_r) - f(x;\,a_1,\dots ,a_k,\dots ,a_r) \approx \varepsilon X_k^a\big \vert _x $$ f ( x ; a 1 , ⋯ , a k + ε , ⋯ , a r ) - f ( x ; a 1 , ⋯ , a k , ⋯ , a r ) ≈ ε X k a | x are linear combinations, with certain coefficients $$- \psi _{ kj} (a)$$ - ψ k j ( a ) depending only on the parameters, of the same infinitesimal transformations computed at the identity: $$ X_k^e\big \vert _x = \frac{\partial f_1}{\partial a_k}(x;\,e)\,\frac{\partial }{\partial x_1} +\cdots + \frac{\partial f_n}{\partial a_k}(x;\,e)\,\frac{\partial }{\partial x_n} \ \ \ \ \ \ \ \ \ \ \ \ \ {{(k\,=\,1\,\cdots \,r)}}. $$ X k e | x = ∂ f 1 ∂ a k ( x ; e ) ∂ ∂ x 1 + ⋯ + ∂ f n ∂ a k ( x ; e ) ∂ ∂ x n ( k = 1 ⋯ r ) . Remarkably, the process of removing superfluous parameters introduced in the previous chapter applies to local Lie groups without the necessity of relocalizing around a generic $$a_0$$ a 0 , so that everything can be achieved around the identity $$e$$ e itself, without losing it.