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

and $\mathrm{\Gamma }$ the discrete subgroup

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

Another example is the universal cover of ${\mathrm{SL}}_2ℝ$. ${\mathrm{SL}}_2ℝ$ is homotopy equivalent to a circle, and thus $\pi ({\mathrm{SL}}_2ℝ)\cong \mathbb{Z}$, and thus has an infinite-sheeted cover. Any real or complex representation of this group factors through the projection map to ${\mathrm{SL}}_2ℝ$.