# Borel subgroup

Let $G={\mathrm{GL}}_{n}\u2102$, the group of all automorphisms^{} of the $n$-dimensional vector space^{} over the field of complex numbers $\u2102$, and $H\le G$ a subgroup^{} of $G$. The *standard Borel subgroup* of $H$ is the subgroup of $H$ consisting of all upper triangular matrices^{} (in $H$). A *Borel subgroup* of $H$ is a conjugate^{} (in $H$) of the standard Borel subgroup of $H$.

The notion of a Borel subgroup can be generalized. Let $G$ be a complex semi-simple Lie group^{}. Then any maximal solvable^{} subgroup $B\le G$ is called a Borel subgroup. All Borel subgroups of a given group are
conjugate. Any Borel group is connected and equal to its own normalizer^{}, and contains a
unique Cartan subgroup. The intersection of $B$ with a maximal compact subgroup
$K$ of $G$ is the maximal torus of $K$.

Title | Borel subgroup |
---|---|

Canonical name | BorelSubgroup |

Date of creation | 2013-03-22 13:27:58 |

Last modified on | 2013-03-22 13:27:58 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 6 |

Author | CWoo (3771) |

Entry type | Definition |

Classification | msc 17B20 |