Borel subgroup

Let G=GLn, the group of all automorphismsPlanetmathPlanetmathPlanetmath of the n-dimensional vector spaceMathworldPlanetmath over the field of complex numbers , and HG a subgroupMathworldPlanetmathPlanetmath of G. The standard Borel subgroup of H is the subgroup of H consisting of all upper triangular matricesMathworldPlanetmath (in H). A Borel subgroup of H is a conjugatePlanetmathPlanetmath (in H) of the standard Borel subgroup of H.

The notion of a Borel subgroup can be generalized. Let G be a complex semi-simple Lie groupMathworldPlanetmath. Then any maximal solvablePlanetmathPlanetmath subgroup BG is called a Borel subgroup. All Borel subgroups of a given group are conjugate. Any Borel group is connected and equal to its own normalizerMathworldPlanetmathPlanetmath, and contains a unique Cartan subgroup. The intersection of B with a maximal compact subgroup K of G is the maximal torus of K.

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