I have a slight problem. The subject is rather specialized; any attempt on my part to sugar the pill would take half a book, so I simply throw myself on the mercy of those who have half an idea of what I am talking about.

So we let

be a Lie group and

a manifold. Then the group (left) action is given by

One sets

so that

. This all fine with me.

Now I am told told that the vector field

on

"generates" this action, which I am not quite getting. Help!

Worse, my text now says something like "let us realize the algebra

by means of the fields

induced by the mapping

". Note the muddle (in my mind) of the active "generates" and passive "induced by"

And they go to some pains to point out that they use the notation

to denote a field, i.e. an element in the set

to distinguish it from the element

.

Now I am well aware that the elements

close into a real Lie algebra, so it seems we might have TWO distinct algebras - one as the vector space of fields on

and the other as the vector space at the identity of

.

Is this right? Is it possible? I doubt it (since

has the product topology). And in any case, even if there ARE 2 algebras, surely it's wrong to conflate them in this way?

Please help (I have more questions, but later for that)