Given Cartan matrix, find root and Dynkin diagram

Question

Given:

Cartan matrix of $\tilde{\mathfrak{so}}=B_2=\begin{pmatrix}2 & -2 \\ -1 & 2 \end{pmatrix}$

The formula for the components of a Cartan matrix $\textbf{A}$ is $A_{ij}=2\frac{\alpha_i\cdot\alpha_j}{\alpha_i\cdot\alpha_i}$.

I have determined that the length of $\alpha_j$ is (thanks @TravisWillse) $\sqrt2$ that of $\alpha_i$, and the angle between $\alpha_i$ and $\alpha_j$ is 135 degrees.

If a reader struggling would like to know whow I got this let me know and I will Tex it up in the future, but for now, I am the student.

What is my next step in drawing the Dynkin diagram?

What is my next step in drawing the root diagram?

For the root diagram, I've been given the suggestion to use Weyl reflections.

Can anyone provide me some instruction in the next steps I take on the journey from Cartan matrix to Dynkin diagram and root diagram?

Answer 1

You've actually collected all of the information you need from the Cartan matrix in order to construct the Dynkin and root diagrams: The quantities $\langle \alpha_i, \alpha_j \rangle$, $$1 \leq i \leq j .$$

1. You know that that the Dynkin diagram has two nodes (the Cartan matrix is $2 \times 2$), all that remains is to find the valence of the edge connecting them, which you can read off the computation $||\alpha_1||^2 = 2 ||\alpha_2||^2$. The arrow, by convention, points from the long root to the short root.
2. If you apply Weyl reflections to the simple roots (until applying further reflections generates no new roots) generates the root diagram.