All we’re going to do now is try to take what we did in the last couple of posts and generalize. So instead of working on Lie groups where we have nice left invariance, and we had all our flows were complete, we’re going to try to get things for arbitrary vector fields on a manifold where the flow is not necessarily guaranteed to be complete.
First off, I’m going to want to think of right actions instead of left now. This is because in the last post I showed that flowing is the same thing as right multiplication by . From now on, I’m assuming we have a Lie group acting smoothly on a manifold on the right . We want a global flow action of on our manifold , so let . Define the action by (note: the two dots are two different actions, the one on the right comes from the Lie group).
We have an infinitesimal generator for this flow, say . i.e. . Thus we have a map by .
Let’s break down what this map really is. For any , examine by . This is a smooth, since we can identify and then it is just inclusion followed by the smooth action. Thus this is the orbit map of the action. You get everything in the orbit of by the action of . Thus .
Let’s go one step further and show that and are related (note now that , , and the action are completely arbitrary, so this is really a very general statement).
By the group law , we actually have . Now we just check:
Which shows related.
Now we easily can get that is a Lie algebra hom. Using linearity of for a fixed p, and the previous statement, we get that . Thus .
Now we are ready to state the “Fundamental Theorem on Lie Algebra Actions”. A quick term, we say a -action is complete if is complete for all .
The FT on LAA says that given any complete -action , there is a unique smooth right -action on M whose infinitesimal generator is .
I won’t prove this, but it was nice to state and a good ending place for the day.