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I'm new to Lie Groups, but all the examples I found are matrix groups. Can someone show a non-matrix Lie group?

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Example: en.wikipedia.org/wiki/Metaplectic_group. Proof that $\mathrm{Mp}(2, \Bbb R)$ is not a matrix group: concretenonsense.wordpress.com/2009/07/10/…. – Henry T. Horton Oct 2 '12 at 18:03
See also this related discussion on MO: mathoverflow.net/questions/64195/… – Henry T. Horton Oct 2 '12 at 18:05
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See also: math.stackexchange.com/q/129644 – commenter Oct 3 '12 at 14:30

2 Answers

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There is the metaplactic group, which is the unique connected double cover of the symplectic group.

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Lie groups are smooth manifolds. They may or may not have matrix representations. For example, ordinary Euclidean space with vector addition is a Lie group.

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But Euclidean space has a matrix representation, so it is a matrix group. – M Turgeon Oct 2 '12 at 18:11
I misinterpreted the question. It certainly does. The universal cover of $\mathbf{SL}_2(\mathbf{R})$ is a good example. – Luis Costa Oct 2 '12 at 18:18

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