Cartan Matrix

The Cartan matrix of a simple Lie algebra is an integer matrix that encodes the entire structure of the algebra's root system in compact form. Each entry measures the geometric relationship between a pair of simple roots, and from this matrix alone one can reconstruct the full root system, the Dynkin diagram, and the Lie algebra itself. The matrix was introduced by Élie Cartan as a tool for making rigorous the classification that Wilhelm Killing had discovered through more intuitive means.

The Cartan matrix is not merely a bookkeeping device. It determines the generators and relations of the Lie algebra through the Serre Relations, a presentation that makes the algebra explicit in terms of a small set of generators and quadratic-cubic relations. The same matrix structure appears in the representation theory of the corresponding Weyl Group, in the theory of Kac-Moody algebras, and in the study of quantum groups — suggesting that the Cartan matrix is a universal combinatorial invariant, not an artifact of classical Lie theory.