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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](http://en.wikipedia.org/wiki/Metaplectic_group).2012-03-20
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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!2012-03-21
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    For reference, the MO question is at http://mathoverflow.net/questions/91789/non-linear-lie-group2012-03-21
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    See also: http://math.stackexchange.com/q/129644/53632012-04-22
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    Also the universal cover of $\mathrm{SL}_3\mathbb{R}$ is an example.2016-08-18
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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 $\mathbb R^n$...2018-08-07

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https://mathoverflow.net/questions/91789/non-linear-lie-group/91805#91805

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    Cited from there: *Copied from http://planetmath.org/encyclopedia/ExamplesOfNonMatrixLieGroup.html* (one click less ;-)2012-04-22