I have a smattering of knowledge and disconnected facts about this question, so I would like to clarify the following discussion, and I also seek references and citations supporting this knowledge. Please see my specific questions at the end, after "discussion".

# Discussion and Background

In practice, groups that do not have any faithful linear representation seem to be seldom (in the sense that I believe it was not till the late 1930s that anyone found any).

By Ado's theorem, every abstract finite dimensional Lie algebra over $\mathbb{R}, \mathbb{C}$ is the Lie algebra of some matrix Lie group. All Lie groups with a given Lie algebra are covers of one another, so even groups that are not subsets of $GL\left(V\right)$ ($V = \mathbb{R}, \mathbb{C}$) are covers of matrix groups.

I know that the metaplectic groups (double covers of the symplectic groups $Sp_{2 n}$) are not matrix groups. And I daresay it is known (although I don't know) exactly which covers of semisimple groups have faithful linear representations, thanks to the Cartan classification of all semisimple groups. But is there a know general reason (i.e. theorem showing) why particular groups lack linear representations? I believe a group must be noncompact to lack linear representations, because the connected components of all compact ones are the exponentials of the Lie algebra (actually if someone could point me to a reference to a proof of this fact, if indeed I have gotten my facts straight, I would appreciate that too). But conversely, do noncompact groups always have covers which lack faithful linear representations? Therefore, here are my specific questions:

# Specific Questions

Firm answers with citations to any of the following would be highly helpful:

1) Is there a general theorem telling one exactly when a finite dimensional Lie group lacks a faithful linear representation;

2) Alternatively, which of the (Cartan-calssified) semisimple Lie groups have covers lacking faithful linear representations;

3) Who first exhibited a Lie group without a faithful representation and when;

4) Is compactness a key factor here? Am I correct that a complex group is always the exponetial of its Lie algebra (please give a citation for this). Does a noncompact group always have a cover lacking a faithful linear representation?

Many thanks in advance.

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