Proving Linearity of Simple Lie Groups with Trivial Center

[shybishie]

Suppose we have a simple Lie Group $G$, i.e, a Lie Group with a trivial center(the identity). Show that this group must be linear, i.e, we can map it to a Lie subgroup of $GL(N)$.So far, I have that from abstract algebra we can show a group with trivial center is isomorphic to the inner automorphisms on the group, let us say $Inn(G)$. Also, $Inn(G)$ is clearly a group of diffeomorphisms on $G$. I have a vague intuition that if we can show $Inn(G)$ is linear , that would show our desired result, although I am not sure how to make this more concrete.

Does anyone have a few hints that might get me on the right track, or tools that I am missing? I am pondering if I can tie this to the Lie algebra of $G$ in some manner.

Also to clarify, this is not part of a graded problem set. I saw it on an assignment some months ago, but didn't figure it out at the time and I want to think my way through it now. The forum rules say that graduate level coursework questions can be posted here, but if it's a problem I apologize for the misclassification.

 

[pat_connell]

Answer:One way to show that a simple Lie group G must be linear is by using the fact that every Lie group can be viewed as a differentiable manifold, and thus its tangent space at each point can be identified with the Lie algebra of the group. Since G is a simple Lie group, it has a trivial center, which implies that its Lie algebra is semisimple, i.e., it is the direct sum of simple Lie algebras. By the Ado-Iwasawa Theorem, any semisimple Lie algebra can be mapped to a linear Lie subalgebra of GL(N). This mapping can then be extended to a map from G to a linear Lie subgroup of GL(N), since the exponential map is a diffeomorphism between the Lie algebra and the Lie group.