Fundamental Differential Equations for Finite Continuous Transformation Groups

Abstract

A finite continuous local transformation group  in the sense of Lie is a family of local analytic diffeomorphisms $x_i ' = f_i ( x; \, a)$, $i = 1 \dots , n$, parametrized by a finite number $r$ of parameters $a_1, \dots , a_r$ that is closed under composition and under taking inverses:

$$ f_i\big (f(x;\,a);\,b)\big ) = f_i\big (x;\,\mathbf m (a,b)\big ) \ \ \ \ \ \ \ \text {and} \ \ \ \ \ \ \ x_i = f_i\big (x';\,\mathbf i (a)\big ), $$

for some group multiplication map $\mathbf m$ and for some group inverse map $\mathbf i$, both local and analytic. Also, it is assumed that there exists an $e = (e_1, \dots , e_r)$ yielding the identity transformation $f_i ( x; \, e) \equiv x_i$. Crucially, these requirements imply the existence of fundamental partial differential equations:

$$ \boxed { \frac{\partial f_i}{\partial a_k}(x;\,a) = - \sum _{j=1}^r\,\psi _{kj}(a)\,\frac{\partial f_i}{\partial a_j}(x;\,e)} \ \ \ \ \ \ \ \ \ \ {{(i\,=\,1\,\cdots \,n,\,\,\,k\,=\,1\,\cdots \,r)}} $$

which, technically speaking, are cornerstones of the basic theory. What matters here is that the group axioms guarantee that the $r\times r$ matrix $( \psi _{ kj})$ depends only on $a$ and it is locally invertible near the identity. Geometrically speaking, these equations mean that the $r$ infinitesimal transformations:

$$ X_k^a\big \vert _x = \frac{\partial f_1}{\partial a_k}(x;\,a)\,\frac{\partial }{\partial x_1} +\cdots + \frac{\partial f_n}{\partial a_k}(x;\,a)\,\frac{\partial }{\partial x_n} \ \ \ \ \ \ \ \ \ \ \ \ \ {{(k\,=\,1\,\cdots \,r)}} $$

corresponding to an infinitesimal increment of the $k$-th parameter computed at $a$:

$$ f(x;a_1,\dots ,a_k+\varepsilon ,\dots ,a_r) - f(x;\,a_1,\dots ,a_k,\dots ,a_r) \approx \varepsilon X_k^a\big \vert _x $$

are linear combinations, with certain coefficients $- \psi _{ kj} (a)$ depending only on the parameters, of the same infinitesimal transformations computed at the identity:

$$ X_k^e\big \vert _x = \frac{\partial f_1}{\partial a_k}(x;\,e)\,\frac{\partial }{\partial x_1} +\cdots + \frac{\partial f_n}{\partial a_k}(x;\,e)\,\frac{\partial }{\partial x_n} \ \ \ \ \ \ \ \ \ \ \ \ \ {{(k\,=\,1\,\cdots \,r)}}. $$

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 $a_0$, so that everything can be achieved around the identity $e$ itself, without losing it.