Cartan matrix of an algebra

Remarks

Example: $A = K[x]/(x^2)$ has the single indecomposable projective $P_1 = A$, with composition series $0 \subset (x) \subset A$. Both factors are the simple module $K$, so $C = (2)$. Example: for the algebra $T_2(K)$ of upper triangular $2 \times 2$ matrices, $P_1 = K E_{11}$ is simple and $P_2 = K E_{12} + K E_{22}$ has the factors $S_2$ on top and $S_1$ below, so $C = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$. Non-example: the polynomial ring $K[x]$ is not finite-dimensional, and its projective module $K[x]$ has no composition series, so it has no Cartan matrix in this sense. Some authors use the transpose of $C$. If $K$ is a splitting field for $A$, for example algebraically closed, then $c_{ij} = \dim_K \Hom_A(P_j, P_i)$. For the group algebra $KG$ of a finite group over a splitting field of characteristic $p$, Brauer showed that $C = D^{T} D$, where $D$ is the decomposition matrix, so $C$ is symmetric.