# parabolic subgroup

Let $G$ be a complex semi-simple^{} Lie group. Then any subgroup $P$ of $G$ containg
a Borel subgroup $B$ is called parabolic. Parabolics are classified in the
following manner. Let $\U0001d524$ be the Lie algebra^{} of $G$, $\U0001d525$ the unique Cartan
subalgebra^{} contained in $\U0001d51f$, the algebra^{} of $B$, $R$ the set of roots corresponding
to this choice of Cartan, and ${R}^{+}$ the set of positive roots whose root spaces are
contained in $\U0001d51f$ and let $\U0001d52d$ be the Lie
algebra of $P$. Then there exists a unique subset ${\mathrm{\Pi}}_{P}$ of $\mathrm{\Pi}$, the base of simple
roots associated to this choice of positive roots, such that
${\{\U0001d51f,{\U0001d524}_{-\alpha}\}}_{\alpha \in {\mathrm{\Pi}}_{P}}$ generates $\U0001d52d$. In other words,
parabolics containing a single Borel subgroup are classified by subsets of the
Dynkin diagram^{}, with the empty set corresponding to the Borel, and the whole graph
corresponding to the group $G$.

Title | parabolic subgroup |
---|---|

Canonical name | ParabolicSubgroup |

Date of creation | 2013-03-22 13:28:02 |

Last modified on | 2013-03-22 13:28:02 |

Owner | bwebste (988) |

Last modified by | bwebste (988) |

Numerical id | 4 |

Author | bwebste (988) |

Entry type | Definition |

Classification | msc 17B20 |