# 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?

-
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 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