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
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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
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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.



When is a finite dimensional real or complex Lie Group not a matrix group
Most of the answers can be found in Hochschild's book on the structure of Lie group.


a) Every complex semisimple group has a faithful rep (Thm 3.2 in Chap. XVII)


b) A connected Lie group with Levi decomposition $G=RS$ ($R$ the solvable radical, $S$ a semisimple Levi factor) is linear iff both $R$ and $S$ are linear (Thm 4.2 in Chap. XVIII)


c) A solvable Lie group $G$ is linear iff its commutator subgroup $G'$ is closed, and $G'$ has no non-trivial compact subgroup (Thm 3.2 in Chap. XVIII)


Now, let $G$ be a semi-simple Lie group. Assume that $G$ is simply connected. Then $G$ admits a greatest linear quotient. Indeed, let $G\_{\mathbb{C}}$ be the simply connected complex group corresponding to the complexified Lie algebra of $G$. Let $L$ be the kernel of the canonical homomorphism $G\rightarrow G\_{\mathbb{C}}$; so $L$ is a finite index subgroup of the center of $G$. Then $G/L$ is the greatest linear quotient of $G$, in the sense that, if $H$ is locally isomorphic to $G$ and $p:G\rightarrow H$ is a universal covering, the group $H$ s linear iff $p$ factors through $G/L$.


