Root Systems and Cartan Matrices

Abstract

This paper is concerned with two things. The first is a (primarily) geometric axiomatic description for the systems of real roots of Lie algebras arising from (generalized) Cartan matrices. The description is base free and is a natural extension of the well-known axiomatic description of finite root systems. The primary component of our description is an open convex cone which, following Looijenga [3], we call the Tits cone. In fact it was Looijenga's paper that led to this axiomatic formulation. Unlike his construction, the dimension of the Tits cone is not tightly connected to the dimension of the Cartan matrix which it eventually yields. This leads us to the second part of the paper which concerns the construction of Cartan matrices of low row rank. We can show that if we have an l × l Cartan matrix of row rank n, then we can model an axiomatic description of it with a cone of dimension n + 1.