Cartan matrix

Summary

The Cartan matrix of a root system is the square matrix of the integers $\langle \alpha_i, \alpha_j \rangle$ for the simple roots $\alpha_1, \dots, \alpha_l$. It determines the root system up to isomorphism, and so the semisimple Lie algebra over $\C$ up to isomorphism. Its entries appear as coefficients and exponents in the Serre relations, which present the Lie algebra by generators and relations.