Borel subgroup
Let , the group of all automorphisms of the -dimensional vector space
over the field of complex numbers , and a subgroup
of . The standard Borel subgroup of is the subgroup of consisting of all upper triangular matrices
(in ). A Borel subgroup of is a conjugate
(in ) of the standard Borel subgroup of .
The notion of a Borel subgroup can be generalized. Let be a complex semi-simple Lie group. Then any maximal solvable
subgroup is called a Borel subgroup. All Borel subgroups of a given group are
conjugate. Any Borel group is connected and equal to its own normalizer
, and contains a
unique Cartan subgroup. The intersection of with a maximal compact subgroup
of is the maximal torus of .
Title | Borel subgroup |
---|---|
Canonical name | BorelSubgroup |
Date of creation | 2013-03-22 13:27:58 |
Last modified on | 2013-03-22 13:27:58 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 6 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 17B20 |