Root systems and inflations of non-negative quasi-Cartan matrices

Abstract

Cartan matrices, quasi-Cartan matrices and associated integral quadratic forms and root systems play an important role in such areas like Lie theory, representation theory and algebraic graph theory. We study quasi-Cartan matrices by means of the inflation algorithm, an idea used in Ovsienko's classical proof of the classification of positive definite integral quadratic forms and recently applied in several other classification results. We prove that it is enough to perform linear number of inflations to reduce positive or non-negative principal quasi-Cartan matrix to its canonical form, i.e., to the Cartan matrix of a Dynkin or Euclidean diagram, respectively. Moreover, we show that in the positive case the length of every sequence of inflations has quadratic bound, and that the only other “natural” non-negative class having such universal bound is the class of so-called pos-sincerely principal quasi-Cartan matrices (and in this case the bound is linear). We obtain such low bounds by applying, among others, some new observations on the properties of the reduced root systems (in the sense of Bourbaki) associated with positive quasi-Cartan matrices.