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


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*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 $SL_2(\mathbb{R})$.

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