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