Lie Transformation Groups -- An Introduction to Symmetry Group Analysis of Differential Equations

TL;DR

These notes present a comprehensive framework for symmetry analysis of differential equations via Lie transformation groups, emphasizing local actions and the role of non-constant rank distributions. They develop the machinery of prolongation to jet spaces, invariant theory, and dimensional analysis, culminating in criteria for when a local group is a symmetry of a differential system. The approach unifies geometric and analytic methods, linking Frobenius-type integrability, orbit structure, and quotient constructions to practical symmetry detection and reduction techniques. By grounding the theory in local transformation groups, prolongations, and invariants, the work provides a rigorous bridge from differential geometry to algorithmic symmetry analysis applicable to a broad class of differential equations.

Abstract

These are lecture notes of a course on symmetry group analysis of differential equations, based mainly on P. J. Olver's book 'Applications of Lie Groups to Differential Equations'. The course starts out with an introduction to the theory of local transformation groups, based on the Stefan-Sussman theory on the integrability of distributions of non-constant rank. The exposition is self-contained, pre-supposing only basic knowledge in differential geometry and Lie groups.