Lie's theorems on transformation groups

Remarks

By part 1, the fundamental vector fields form a Lie algebra of vector fields of dimension at most $\dim G$. Part 1 follows by differentiating the identity $\exp(sX)\exp(tY)\exp(-sX) = \exp(t\operatorname{Ad}(\exp sX)Y)$ along the action; the minus sign comes from using a left action together with the bracket of left-invariant vector fields. Lie's original theorems were local: they concerned local groups of transformations and the vector fields that generate them. The global form of part 2 is due to Palais. Completeness of the vector fields is needed, since an incomplete field such as $x^2 \frac{d}{dx}$ on $\R$ generates no global action of $\R$. Part 3 is Lie's third theorem, which in the modern form is proved by Ado's theorem or by Cartan's method. Example: the rotation action of $SO(3)$ on $\R^3$ has as fundamental vector fields the three infinitesimal rotations $y\partial_z - z\partial_y$, $z\partial_x - x\partial_z$, $x\partial_y - y\partial_x$ and their linear combinations, a Lie algebra isomorphic to $\mathfrak{so}(3)$.

Going deeper

For a student

These results are the origin of Lie theory: Lie used them to study symmetries of differential equations. Part 1 turns a group action into a Lie algebra homomorphism into vector fields, up to a sign, which is the basic tool for computing symmetries. Part 2 integrates infinitesimal symmetries to actual group actions; without completeness one only gets a local action. Part 3 shows that no Lie algebra is missed, so the classification of finite-dimensional Lie algebras gives all the local types of transformation groups.