examples of non-matrix Lie groups

While most well-known Lie groups are matrix groups, there do in fact exist Lie groups which are not matrix groups. That is, they have no faithful finite dimensional representations.

For example, let $H$ be the real Heisenberg group

 $H=\left\{\begin{pmatrix}1&a&b\\ 0&1&c\\ 0&0&1\end{pmatrix}\mid a,b,c\in\mathbb{R}\right\},$

and $\Gamma$ the discrete subgroup

 $\Gamma=\left\{\begin{pmatrix}1&0&n\\ 0&1&0\\ 0&0&1\end{pmatrix}\mid n\in\mathbb{Z}\right\}.$

The subgroup $\Gamma$ is central, and thus normal. The Lie group $H/\Gamma$ has no faithful finite dimensional representations over $\mathbb{R}$ or $\mathbb{C}$.

Another example is the universal cover of $\mathrm{SL}_{2}\mathbb{R}$. $\mathrm{SL}_{2}\mathbb{R}$ is homotopy equivalent to a circle, and thus $\pi(\mathrm{SL}_{2}\mathbb{R})\cong\mathbb{Z}$, and thus has an infinite-sheeted cover. Any real or complex representation of this group factors through the projection map to $\mathrm{SL}_{2}\mathbb{R}$.

Title examples of non-matrix Lie groups ExamplesOfNonmatrixLieGroups 2013-03-22 13:20:48 2013-03-22 13:20:48 bwebste (988) bwebste (988) 6 bwebste (988) Example msc 17B10 AdosTheorem