Cartan matrix

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 $\Phi$ (for example, the one induced by the Killing form ).

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

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.

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.