Periodic Coxeter matrices

摘要

Let A = kQ/I be a finite dimensional triangular k-algebra. Consider the Cartan matrix C-A and the Coxeter matrix phi(A) = C(A)(-t)CA. Let X-phi(T) = det(Tid - phi(A)) be the Coxeter polynomial of A. We study conditions on Spec phi(A) in order that phi(A) is a periodic matrix. We show that in case phi(A) is periodic then the Euler quadratic form q(A) (x) = xC(A)(-t)x(t) is nonnegative and q(A) > 0 if and only if 1 is not an element of Spec phi(A). (C) 2002 Elsevier Science Inc. All rights reserved. [References: 12]