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We know that for a matrix Lie group $G$, we define it to be a closed subgroup of $GL(n,\mathbb{C})$. But Lie groups are defined as manifolds in $\mathbb{R}^n$ for some $n$, in general. The question is that, do we know any Lie group which is not a matrix Lie group? Thank you very much.

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One family of standard examples is obtained by covers of $\operatorname{SL}(2,\mathbb{R})$, see e.g. metaplectic groups. –  t.b. Mar 20 '12 at 19:20
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It is usually a good idea to wait a significant amount of time before crossposting to MO a question asked here. Moreover, please be explicit about the fact that the question has already been asked here. Also: Google is your friend! –  Mariano Suárez-Alvarez Mar 21 '12 at 4:40
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For reference, the MO question is at mathoverflow.net/questions/91789/non-linear-lie-group –  Mariano Suárez-Alvarez Mar 21 '12 at 4:41
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See also: math.stackexchange.com/q/129644/5363 –  t.b. Apr 22 '12 at 15:15
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