Let be a complex semi-simple Lie group. Then any subgroup of containg a Borel subgroup is called parabolic. Parabolics are classified in the following manner. Let be the Lie algebra of , the unique Cartan subalgebra contained in , the algebra of , the set of roots corresponding to this choice of Cartan, and the set of positive roots whose root spaces are contained in and let be the Lie algebra of . Then there exists a unique subset of , the base of simple roots associated to this choice of positive roots, such that generates . 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 .
|Date of creation||2013-03-22 13:28:02|
|Last modified on||2013-03-22 13:28:02|
|Last modified by||bwebste (988)|