Non-algebraic Lie groups

When trying to learn about Lie groups I find that most natural examples of Lie groups are actually examples of algebraic groups.

What are some interesting examples of Lie groups which are not algebraic groups?

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1 Answer

This post was incorrect. However, as Theo has pointed out, the double cover of $SL_2(\mathbb{R})$ is an example of a Lie group which is non linear and non-abelian, and hence, if it is algebraic, it is at least neither affine nor projective.

The wikipedia page on Linear algebraic groups has a list of a few criterea which prevent a lie group from being an algebraic group.

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@Theo Buehler Every compact Lie group is linear? Is there a simple proof? I feel bad for spreading misinformation. I wish I remembered who had told me that Spin(n) wasn't linear. –  Aaron May 25 '11 at 21:42
Well, the precise statement is: a compact group with no small subgroups has a faithful linear representation. This follows from Peter-Weyl and is von Neumann's solution for Hilbert's fifth problem for compact groups. It's not very hard. However, this doesn't tell you anything about algebraicity. –  t.b. May 25 '11 at 21:46
The metaplectic group you mentioned before and the universal cover of $Sp_{n}$ would work, too, I guess. But I'm far from my expertise, here. –  t.b. May 25 '11 at 21:47