# 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

$$H=\{\left(\begin{array}{ccc}\hfill 1\hfill & \hfill a\hfill & \hfill b\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill c\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right)\mid a,b,c\in \mathbb{R}\},$$ |

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

$$\mathrm{\Gamma}=\{\left(\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill n\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right)\mid n\in \mathbb{Z}\}.$$ |

The subgroup^{} $\mathrm{\Gamma}$ is central^{}, and thus normal. The Lie group $H/\mathrm{\Gamma}$ has no faithful finite dimensional representations over $\mathbb{R}$ or $\u2102$.

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 |
---|---|

Canonical name | ExamplesOfNonmatrixLieGroups |

Date of creation | 2013-03-22 13:20:48 |

Last modified on | 2013-03-22 13:20:48 |

Owner | bwebste (988) |

Last modified by | bwebste (988) |

Numerical id | 6 |

Author | bwebste (988) |

Entry type | Example |

Classification | msc 17B10 |

Related topic | AdosTheorem |