# Borel subalgebra

Let $\U0001d524$ be a semi-simple^{} Lie group^{}, $\U0001d525$ a Cartan subalgebra^{}, $R$ the associated root system^{}, and ${R}^{+}\subset R$ a set of positive roots. We have a root decomposition into the Cartan subalgebra and the root spaces ${\U0001d524}_{\alpha}$

$$\U0001d524=\U0001d525\oplus \left(\underset{\alpha \in R}{\oplus}{\U0001d524}_{\alpha}\right).$$ |

Now let $\U0001d51f$ be the direct sum of the Cartan subalgebra and the positive root spaces.

$$\U0001d51f=\U0001d525\oplus \left(\underset{\beta \in {R}^{+}}{\oplus}{\U0001d524}_{\beta}\right).$$ |

This is called a Borel subalgebra^{}.

Title | Borel subalgebra |
---|---|

Canonical name | BorelSubalgebra |

Date of creation | 2013-03-22 13:12:16 |

Last modified on | 2013-03-22 13:12:16 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 6 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 17B20 |