Borel subgroup

Let $G={\mathrm{GL}}_nℂ$, the group of all automorphisms of the $n$-dimensional vector space over the field of complex numbers $ℂ$, and $H\le G$ a subgroup of $G$. The standard Borel subgroup of $H$ is the subgroup of $H$ consisting of all upper triangular matrices (in $H$). A Borel subgroup of $H$ is a conjugate (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 group. Then any maximal solvable subgroup $B\le G$ 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 $B$ 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