April 14

If K is compact with discrete center, pi_1(K) is finite. Hence there are finitely
many Lie groups with that Lie algebra. Proof: use Hurewicz to connect to H^1, and Lie algebra cohomology to show that H^1 vanishes over R.

The angle between any two roots must be 90, 120, 135, 150, or 180.

There are at most two root lengths in a connected root system, and if two they differ by a factor of sqrt(2) or sqrt(3).

Next time: simple systems, Dynkin diagrams, their classification.