So, I know the following facts:
- A Lie group is a group and a Manifold, whose group structure is continuous with respect to the Manifold structure
The Lie algebra is a vector space with a Lie bracket structure on it.
Every Lie group has a corresponding Lie algebra.
The Lie algebra represents the "infinitesimal behaviour" of the Lie group.
Till here, stuff makes sense. However, this is where I lose it:
- The Vector space of all Vector fields over a Manifold form a Lie algebra with the Lie bracket of vector fields structure.
Why do I care about fact 5? Is it because it is "cute" that the Lie bracket exists? What do I gain by showing that vector fields have a lie bracket structure?