Lie Groups and Lie Algebras: Winter-2023 Practice Exercises 6 : Due Friday 2/17
Dynkin diagrams
S0 4613S
The goal of this problem is to construct all possible Dynkin diagrams for simple Lie algebras. For the purposes of this problem, a Dynkin diagram has the following properties:
There are \(n\) points, representing simple roots.
Two points may be connected by 0, 1, 2, or 3 lines.
The diagram is connected, that is, there is a path from every point to every other point.
Furthermore, Dynkin diagrams obey the following rules:
There exist at most \(k-1\) connections (each consisting of 1, 2, or 3 lines) among \(k\) points.
There exist at most 3 lines at any point.
If a simple chain (\(k\) points connected by single lines) in a valid diagram is replaced by a single point, the resulting diagram is also valid.
If a simple chain of length \(p\) is connected by a double line to a simple chain of length \(q\), then \((p-1)(q-1)<2\).
If three simple chains of lengths \(p\), \(q\), and \(r\) meet at a single point (which counts as belonging to all three chains), then \(\frac1p+\frac1q+\frac1r>1\).
Find all Dynkin diagrams that satisfy these constraints.