Highest Roots of Classical Lie Algebra

Question

I want to compute the length of the highest root of the classical lie algebra.

The classical lie algebra are $\mathfrak{sl}(r+1)$, $\mathfrak{so}(2r+1)$, $\mathfrak{sp}(2r)$, and $\mathfrak{so}(2r)$.

I have found highest root in terms of simple root vectors. For example $\mathfrak{sp}(2r)$ has simple root vectors $E_{i,i+1}-E_{r+i+1,r+i}$ and $E_{r,2r}$. If we call the corresponding roots $\alpha_{i}$ I find that $\theta=2(\alpha_{1}+...+\alpha_{r-1})+\alpha_{r}$ is highest root and highest root vector is $E_{1 2r}$.

But now I can't find $||\theta||^{2}$.

I try using killing form ($||\theta||^{2}=\kappa(\theta,\theta)$), but make not progress. Please I appreciate help!

Answer 1

Recalling that $\langle \alpha , \beta \rangle $ must be an integer, you can easy obtain that

$2 ||\alpha||/||\beta||cos \theta_{\alpha \beta} \in \mathbb{Z}$

where $ \theta_{\alpha \beta} $ is the angle between the two roots. The integrality condition forces some precise values for $ \theta_{\alpha \beta} $, determmined univoquely by the type of the root system you are considering.

Now you can easy find the ratio between the lenghts by direct inspection just supposing, for example, that $\beta$ has lenght 1.