parabolic subgroup

Let G be a complex semi-simplePlanetmathPlanetmath 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 algebraMathworldPlanetmath of G, 𝔥 the unique Cartan subalgebraMathworldPlanetmath contained in 𝔟, the algebraPlanetmathPlanetmath 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 ΠP of Π, the base of simple roots associated to this choice of positive roots, such that {𝔟,𝔤-α}αΠP generates 𝔭. In other words, parabolics containing a single Borel subgroup are classified by subsets of the Dynkin diagramMathworldPlanetmath, 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