I can’t remember how this came up, I believe it involved an exchange with Shea Browne but it certainly has been motivated by readings. You see, sometimes you’ll see a Lie Algebra defined all by itself, as a vector space with a certain product structure on it. This has struck me as odd since on the other hand we know that the Lie algebra is a local linearization of a Lie group at it’s identity element. Lie algebras can seem like a funny place to start because they also don’t uniquely determine a Lie group. What gives?
I’ll advance a thought which is also a question. Feel free to chime. If you don’t I’ll figure this out eventually and post it all the same. Physics starts locally, which is expressed in its differential equations. Things like vector fields on a manifolds have a bracket structure and so make it a Lie algebra, but those vector fields are differential operators and these allow us to express differential equations. The process of solving that differential equation is to find a group of transformations that tell us what happens if you’re at a particular point on the manifold, where do you go from there? The differential equation gives you a little information, where to go (infinitesimally) and how long to do so (infinitesimally) but the group (the solution) gives you a non-infinitesimal set of directions and times. This information is still localized, I’ll need to review the existence theorems for solutions of differential equations and so maybe the fact that Lie algebras don’t uniquely determine Lie groups have something to do with this?
I know this isn’t the whole picture because you can also look at the Lie group of symmetries of a Lagrangian or differential equation so I’m missing a lot of the puzzle. Thoughts?