Borel subalgebra

Let $\mathfrak{g}$ be a semi-simple Lie group, $\mathfrak{h}$ a Cartan subalgebra, $R$ the associated root system, and $R^{+}\subset R$ a set of positive roots. We have a root decomposition into the Cartan subalgebra and the root spaces $\mathfrak{g}_{\alpha}$

\mathfrak{g}=\mathfrak{h}\oplus\left(\bigoplus_{\alpha\in R}\mathfrak{g}_{% \alpha}\right).

Now let $\mathfrak{b}$ be the direct sum of the Cartan subalgebra and the positive root spaces.

\mathfrak{b}=\mathfrak{h}\oplus\left(\bigoplus_{\beta\in R^{+}}\mathfrak{g}_{% \beta}\right).

This is called a Borel subalgebra.

Title	Borel subalgebra
Canonical name	BorelSubalgebra
Date of creation	2013-03-22 13:12:16
Last modified on	2013-03-22 13:12:16
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	6
Author	mathcam (2727)
Entry type	Definition
Classification	msc 17B20