# 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 .

Title Borel subalgebra BorelSubalgebra 2013-03-22 13:12:16 2013-03-22 13:12:16 mathcam (2727) mathcam (2727) 6 mathcam (2727) Definition msc 17B20