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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?

Léo
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    Example: 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/. – Henry T. Horton Oct 02 '12 at 18:03
  • See 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 – Henry T. Horton Oct 02 '12 at 18:05
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    See also: http://math.stackexchange.com/q/129644 – commenter Oct 03 '12 at 14:30

4 Answers4

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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.

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

M Turgeon
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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.

Luis Costa
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In Hall 2015 Definition 1.20 there is following example of a nonmatrix lie group: Let $$G= \mathbb{R} \times \mathbb{R} \times S^1 =\{(x,y,u)|x\in \mathbb{R}, y \in \mathbb{R}, u \in S^1\subset\mathbb{C}\}$$ Be a group with group Operation $$(x_1,y_1,u_1)\cdot(x_2,y_2,u_2)= (x_1+x_2, y_1+y_2,e^{i x_1y_2}u_1u_2) $$ Then G has the following proprieties

  • G is associative as you can check.
  • It has an Identity $e = (0,0,1)$.
  • It has an inverse $(-x,-y,e^{ixy}u^{-1})$.
  • It is a manifold beucause as defined above G is a cylinder.
  • The group operations are smooth.

So G is a lie Group.

Monterosa2
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