Geodesic
Some Special Classes of Cartan Matrices
Canadian journal of mathematics, Tome 36 (1984) no. 5, pp. 800-819

Voir la notice de l'article provenant de la source Cambridge University Press

Let A = (Aij )l≦ij≦l be a Cartan matrix, i.e., Aii = 2 for all i and Aij is an integer ≦ 0 for i ≠ j, with Aij = 0 if Aji = 0. The size l of A is called its rank, for Lie-theoretic reasons, and may be larger than its matrix rank. We associate to A its Dynkin diagram, with vertices 1, 2, ... , l, with AijAji lines joining i to j, and with an arrow pointing from i to j if Aij/Aji < 1, i.e., pointing toward the shorter root (see below). The Cartan matrix A is indecomposable if its diagram is connected, and symmetrizable if there exist positive rational numbers q1 ... , ql with qiAij = qjAji for all i and j.Symmetrizability is automatic if the diagram contains no cycle. 
Ogg, A. P. Some Special Classes of Cartan Matrices. Canadian journal of mathematics, Tome 36 (1984) no. 5, pp. 800-819. doi: 10.4153/CJM-1984-047-2
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