Lie Groups and Matrices

Question

I vaguely remember (maybe I am making this up) this. Is there some sort of result about Lie groups (of a certain class) which classifies them as matrix Lie groups? In other words, given a Lie group G, there exists an isomorphism from G into a matrix Lie group? If G admits a faithful representation, then most certainly...

Answer 1

Well, this is exactly the question of which Lie groups admit a faithful representation. Compactness is a sufficient condition.

Answer 2

As Robert Israel says, this is exactly the question of which Lie groups $G$ admit a faithful finite-dimensional representation. Some comments:

• An easy sufficient condition is that $G$ is connected and has trivial center, in which case its adjoint representation is faithful.
• Another sufficient condition is that $G$ is compact; this is nontrivial and is a corollary of the Peter-Weyl theorem.
• Ado's theorem asserts that every finite-dimensional Lie algebra $\mathfrak{g}$ has a faithful finite-dimensional representation. It follows that every simply connected Lie group $G$ admits a finite-dimensional representation whose kernel is a discrete subgroup of its center.
• An example of a Lie group which does not admit a faithful finite-dimensional representation is $G = \widetilde{SL}_2(\mathbb{R})$, the universal cover of $SL_2(\mathbb{R})$. This is because $G$ has the same finite-dimensional representation theory as its Lie algebra $\mathfrak{sl}_2(\mathbb{R})$, which in turn has the same finite-dimensional representation theory as $SL_2(\mathbb{R})$. So every finite-dimensional representation of $G$ factors through $SL_2(\mathbb{R})$.