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.


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…)