Symmetrizable intersection matrices and their root systems

Abstract

In this paper we study symmetrizable intersection matrices, namely generalized intersection matrices introduced by P. Slodowy such that they are symmetrizable. Every such matrix can be naturally associated with a root basis and a Weyl root system. Using $d$-fold affinization matrices we give a classification, up to braid-equivalence, for all positive semi-definite symmetrizable intersection matrices. We also give an explicit structure of the Weyl root system for each $d$-fold affinization matrix in terms of the root system of the corresponding Cartan matrix and some special null roots.