Eigenvalues of Coxeter transformations and the Gel\cprime fand-Kirillov dimension of the preprojective algebras

Abstract

The spectral radius of a Coxeter transformation is shown to be an eigenvalue which can be expressed in terms of lengths of certain positive roots of the corresponding valued graph. This result is used to determine the Gelfand-Kiril- lov dimension of the preprojective algebras: This dimension is equal to 0, 1 or co according to whether the underlying graph is Dynkin, Euclidean or otherwise. Let C = (Cy) be an « X « Cartan matrix, that is c = 2 for 1 d(\)for