parabolic subgroup
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 .
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 |