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 $𝔤$ be the Lie algebra of $G$, $𝔥$ the unique Cartan subalgebra contained in $𝔟$, 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 $𝔟$ and let $𝔭$ 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 ${\{𝔟,{𝔤}_{-\alpha }\}}_{\alpha \in {\mathrm{\Pi }}_P}$ generates $𝔭$. 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