Algebras Whose Coxeter Polynomials are Products of Cyclotomic Polynomials

Abstract

Let A be a finite dimensional algebra over an algebraically closed field k. Assume A is basic connected with n pairwise non-isomorphic simple modules. We consider the Coxeter transformation ϕ A as the automorphism of the Grothendieck group K 0(A) induced by the Auslander-Reiten translation τ 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 χ A of ϕ 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 h A 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).