On the trace of the Coxeter polynomial of an algebra

Let A=KQ/I be a finite dimensional triangular K-algebra. Let ϕA be the Coxeter matrix of A. We relate homological conditions for A with properties of the traces of the Coxeter transformation ϕA. For instance, a finite dimensional accessible algebra A is strongly accessible if and only if Tr(ϕA)=−1. We say A is of cyclotomic type if the eigenvalues of ϕA lie on the unit circle. Clearly, if A is of cyclotomic type then |Tr(ϕA)k|≤n, for k≥0. We prove that A is of cyclotomic type if |Tr(ϕA)k|≤n holds for 0≤k≤n. We illustrate the results with examples of Nakayama algebras.