Cartan matrix

Remarks

The diagonal entries are $a_{ii} = 2$, the other entries lie in $\set{0, -1, -2, -3}$, and $a_{ij} = 0$ if and only if $a_{ji} = 0$. Example: for $B_2$ with $\alpha_1 = e_1 - e_2$ and $\alpha_2 = e_2$, one has $(\alpha_1, \alpha_1) = 2$, $(\alpha_2, \alpha_2) = 1$ and $(\alpha_1, \alpha_2) = -1$. So $a_{12} = 2(-1)/1 = -2$ and $a_{21} = 2(-1)/2 = -1$, and $A = \begin{pmatrix} 2 & -2 \\ -1 & 2 \end{pmatrix}$. Non-example: $\begin{pmatrix} 2 & -4 \\ -1 & 2 \end{pmatrix}$ is not the Cartan matrix of a root system, since $a_{12} a_{21} = 4$ would force $\alpha_1$ and $\alpha_2$ to be proportional, which two simple roots never are. Some authors, for example in the theory of Kac–Moody algebras, use the transpose convention $a_{ij} = \langle \alpha_j, \alpha_i \rangle$. A different reordering of $\Delta$ conjugates $A$ by a permutation matrix, and a different base gives the same matrix up to such a reordering. This notion is not the Cartan matrix of an algebra in the representation theory of finite-dimensional algebras.