  Let A be a finite dimensional algebra over an algebraically closed field k.
Assume A is basic connected with n pairwise non-isomorphic simplemodules. We
consider the Coxeter transformation ?A as the automorphism of the Grothendieck
group K0(A) induced by the Auslander-Reiten translation ? in the derived
category Der(modA) of the module category modA of finite dimensional left
A-modules. We say that A is an algebra of cyclotomic type if the characteristic
polynomial ?A of ?A is a product of cyclotomic polynomials. There are many
examples of algebras of cyclotomic type in the representaton theory literature:
hereditary algebras of Dynkin and extended Dynkin types, canonical algebras,
some supercanonical and extended canonical algebras. Among other results, we
show that: (a) algebras satisfying the fractional Calabi-Yau property have
periodic Coxeter transformation and are, therefore, of cyclotomic type, and (b)
algebras whose homological form hA is non-negative are of cyclotomic type. For
an algebra A of cyclotomic type we describe the shape of the Auslander-Reiten
components of Der(modA). 1.
