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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1One 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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2It 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-Álvarez Mar 21 '12 at 04:40
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1For reference, the MO question is at http://mathoverflow.net/questions/91789/non-linear-lie-group – Mariano Suárez-Álvarez Mar 21 '12 at 04:41
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1See also: http://math.stackexchange.com/q/129644/5363 – t.b. Apr 22 '12 at 15:15
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Also the universal cover of $\mathrm{SL}_3\mathbb{R}$ is an example. – Cronus Aug 18 '16 at 19:29
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It's not quite right that Lie groups are required to be submanifolds of $\mathbb R^n$ for any $n$. Rather, they are locally diffeomorphic to opens in $\mathbb R^n$... – paul garrett Aug 07 '18 at 23:17
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2Cited from there: Copied from http://planetmath.org/encyclopedia/ExamplesOfNonMatrixLieGroup.html (one click less ;-) – draks ... Apr 22 '12 at 16:04