Lie's theory of transformation groups

Analytic local transformation groups, parameters, flows, complete systems, the three fundamental theorems, structure and homogeneous actions, jets and contact transformations, classification, and the Riemann–Helmholtz problem. Twelve lessons contain proofs, examples, exercises and solutions. The general Lie–Tresse finite-generation theorem and the full assigned classification catalogues are not proved here. Historical readings comprise the explicitly identified selections.

1. Transformation groups and their parameters
2. The fundamental differential equations
3. One-parameter groups
4. Complete systems and invariants
5. The second fundamental theorem and the group of parameters
6. The third fundamental theorem
7. The adjoint group, composition and isomorphism
8. Transitivity and primitivity
9. Prolongation, differential invariants and the projective group
10. Contact transformations and first-order equations
11. Lie algebras of vector fields on the line and in the plane
12. The Riemann-Helmholtz space problem

Editable lesson sources and figures (ZIP)

Original text CC0 1.0 unless the lesson states other terms.