A generalized Cartan matrix is a square matrix $ A=(a_{ij})$ with integral entries such that
- For diagonal entries,$a_{ii}=2$.
- For non-diagonal entries, $a_{ij}\leq 0$.
- $a_{ij}=0$ if and only if $a_{ji}=0$
- $A$ can be written as $DS$, where $D$ is a diagonal matrix, and $S$ is a symmetric matrix.
From the Cartan Matrix I can recover a semisimple Lie algebra. What I'm wondering is what goes wrong if the coefficients are non integers but real? I know in Lie theory this cannot happen but I do not really grasp what goes wrong in the definition of the Lie algebra...
The reason is related to the root system of the Lie Algebra. The entries of the Cartan matrix are defined as;
$$a_{ij}=2{(r_i,r_j)\over(r_j,r_j)}$$
Where $r_i$ are the simple roots$\space(\Phi)$ of the Lie algebra. As it turns out the simple roots have the property that if $\alpha,\beta \in \Phi$ then the projection of $\alpha$ onto $\beta$ is always an integer or a half integer multiple of $\beta$.
This will property will obviously always cause $a_{ij}$ to be an integer.