Lie Algebra differentiable manifold

[Hymne]

Okey, I have problem with the foundation of lie algebra. This is my understanding:

We have a lie group which is a differentiable manifold. This lie group can for example be SO(2), etc.

Then we have the Lie algebra which is a vectorspace with the lie bracket defined on it: [. , .].
This lie algebra will, when we put it in the exponential mapping, give the Lie group.

For example: su(2) = R(0, 1; -1, 0).

I hope this is correct.

The we come to representation space.. Well in the example above our elements of the Lie group, G, will be represented by a matrix: A = exp(t x) where x belongs to the lie algebra, and t is our parameter. Does this mean that the vector space of n,n matrices is our representation space?

 

[arkajad]

Apart of minor inexactitudes: when you have a group consisting of some set of nxn matrices, then, yes, the set of all nxn matrices can be considered as a particular representation space - usually a reducible one.