Coefficients of non-negative quasi-Cartan matrices, their symmetrizers and Gram matrices

Abstract

Cartan matrices, quasi-Cartan matrices and associated upper triangular Gram matrices control important combinatorial aspects of Lie theory and representation theory of associative algebras. We provide a graph theoretic proof of the fact that the absolute values of the coefficients of a non-negative quasi-Cartan matrix A as well as of its (minimal) symmetrizer D are bounded by 4, and that the analogous bound in case of the associated Gram matrix GˇA is 8. Moreover, we show that D (and GˇA) has at least one diagonal coefficient equal to 1. We describe some other restrictions and interrelations between the coefficients of A, D and GˇA, and the corank and other properties of A relevant in Lie theory. We apply our results to construct an algorithm by which we classify all non-negative quasi-Cartan matrices of small sizes.