How do I find all the roots of a lie algebra and hence its root system diagram given the cartan matrix for that algebra?

Question

If I am given the cartan matrix, I can find the $2\langle αi , αj\rangle/\langle αj , αj\rangle$ of the simple roots where $α$ are the simple roots. But, from this how do I find the $\langle αi , αj\rangle$ themselves. After finding this, how do I find the other roots and their scalar products, and hence construct the root diagram?

Answer 1

The lengths of the roots can be determined from the Cartan matrix $A_{ij}=2\frac{(\alpha_i, \alpha_j)}{(\alpha_j, \alpha_j)}$ up to an overall normalization.

The ratios of the root lengths can be computed from the ratios of the nonvanishing off-diagonal elements:

$\frac{A_{ij}}{A_{ji}} = \frac{(\alpha_i, \alpha_i)}{(\alpha_j, \alpha_j)}$

(Actually the ratios can only assume the values 1,2,3 and their reciprocals).

The overall normalization can be chosen by convention for example by choosing the shortest root length as one. However, it is immaterial as different normalizations give rize to isomorphic Lie algebras.