Cartan matrix - Knowlpedia

Let $\mathfrak{g}$ be a complex semisimple Lie algebra, choose a Cartan subalgebra $\mathfrak{h}$ , and let $\Phi\subset \mathfrak{h}^*$ be the associated root system. Fix a set of simple roots $\Delta=\{\alpha_1,\dots,\alpha_\ell\}$ .

Choose any $W$ -invariant inner product on the real span of $\Phi$ .

Definition. The Cartan matrix $A=(a_{ij})_{1\le i,j\le \ell}$ is defined by

a_{ij} \;=\; 2\,\frac{(\alpha_i,\alpha_j)}{(\alpha_i,\alpha_i)}.

Properties

Basic properties.

$a_{ii}=2$ for all $i$ .
For $i\ne j$ , one has $a_{ij}\in \mathbb{Z}_{\le 0}$ , and $a_{ij}=0$ iff $\alpha_i$ is orthogonal to $\alpha_j$ .
The matrix $A$ determines the Dynkin diagram (and conversely): the off-diagonal entries record the angles and relative lengths between simple roots.

Examples

The Killing form induces a standard $W$ -invariant inner product on the real span of the roots.

Remarks

Context. The Cartan matrix is the combinatorial input for the classification of complex simple Lie algebras: connected Dynkin diagrams (or indecomposable Cartan matrices) correspond to simple Lie algebras, while disjoint unions correspond to direct sums.