I'm new to Lie Groups, but all the examples I found are matrix groups. Can someone show a non-matrix Lie group?
Is there a non-matrix Lie group?
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$\begingroup$
lie-groups
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1Example: http://en.wikipedia.org/wiki/Metaplectic_group. Proof that $\mathrm{Mp}(2, \Bbb R)$ is not a matrix group: http://concretenonsense.wordpress.com/2009/07/10/a-lie-group-which-isnt-a-matrix-group/. – 2012-10-02
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0See also this related discussion on MO: http://mathoverflow.net/questions/64195/when-is-a-finite-dimensional-real-or-complex-lie-group-not-a-matrix-group – 2012-10-02
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1See also: http://math.stackexchange.com/q/129644 – 2012-10-03
3 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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There are no non-matrix Lie groups whose dimension is $1$ or $2$. On the other hand, consider the quotient of the Heisenberg group$$\left\{\begin{pmatrix}1&a&c\\0&1&b\\0&0&1\end{pmatrix}\,\middle|\,a,b,c\in\mathbb R\right\}$$by the normal subgroup$$\left\{\begin{pmatrix}1&0&m\\0&1&0\\0&0&1\end{pmatrix}\,\middle|\,m\in\mathbb Z\right\}.$$It is a non-matrix three-dimensional Lie group.
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Lie groups are smooth manifolds. They may or may not have matrix representations. For example, the universal cover of $\mathbf{SL}_2(\mathbf{R})$ is a Lie group that is not a matrix Lie group.