So, suppose given a root system ${\Phi}$ in a euclidean space ${E}$, which arises from a semisimple Lie algebra and a Cartan subalgebra as before. The first goal of this post is to discuss the “splitting”

$\displaystyle \Phi = \Phi^+ \cup \Phi^-$

(disjoint union) in a particular way, into positive and negative roots, and the basis decomposition into simple roots. Here ${\Phi^- = - \Phi^+}$.

To do this, choose ${v \in E}$ such that ${(v, \alpha) \neq 0}$ for ${\alpha \in \Phi}$. Then define ${\Phi^+}$ to be those roots ${\alpha}$ with ${(v,\alpha)>0}$ and ${\Phi^-}$ those with ${(v,\alpha) < 0}$. This was easy. We talked about positive and negative roots before using a real-valued linear functional, which here is given by an inner product anyway.

Bases

OK. Next, I claim it is possible to choose a linearly independent set ${\Delta \subset \Phi^+}$ such that every root is a combination

$\displaystyle \alpha = \sum k_i \delta_i, \quad \delta_i \in \Delta, \ k_i \in \mathbb{Z}$

with all the ${k_i \geq 0}$ or all the ${k_i \leq 0}$.

Then ${\Delta}$ will be called a base. It is not unique, but I will show how to construct this below. (more…)