Lie groups and Lie algebras: Simple roots

We define simple roots in a root system, describe how they define a Weyl chamber, and prove some properties of the simple roots. In particular, we prove that they form a basis of R^n.

[@jacovantonder7562]

Why can't the distance from the hyperplane of irrational slope decrease like 1, 3/4, 5/8, 9/16, ... approaching 1/2 not 0? I.e. why does it have to approach the hyperplane (0 distance from it)?

[@jacovantonder7562]

I guess this is prevented by the angles between mirrors having a lower bound of 30 degrees not 0.

[@jonathanevans27]

@Jaco Van Tonder - Think of it this way. Take the orthogonal projection p from R^n to R whose kernel is the hyperplane of irrational slope (so the value of p on some vector gives the orthogonal distance from that vector to the hyperplane). There are a finite number of values that p can take on positive roots because there are finitely many roots. If you decompose as a sum of positive roots: a=b+c then p(a)=p(b)+p(c) by linearity and all these numbers are strictly positive (that's what it means to be a positive root). If the process of splitting into positive roots went on forever, you'd produce a strictly decreasing sequence of values for p on positive roots, which is a contradiction regardless of whether the limit of this sequence were 0 or 1/2 or anything else, because there are only finitely many possible values for p on the positive roots.