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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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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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    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.2011-05-25