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 $\mathfrak{g}$ be the Lie algebra of $G$ , $\mathfrak{h}$ the unique Cartan subalgebra contained in $\mathfrak{b}$ , 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 $\mathfrak{b}$ and let $\mathfrak{p}$ be the Lie algebra of $P$ . Then there exists a unique subset $\Pi_{P}$ of $\Pi$ , the base of simple roots associated to this choice of positive roots, such that $\{\mathfrak{b},\mathfrak{g}_{-\alpha}\}_{\alpha\in\Pi_{P}}$ generates $\mathfrak{p}$ . 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