$q$-Coxeter matrix and $q$-Cartan matrix for a homogeneous bound quiver

The aim of this paper is to introduce the concept of $q$-Coxeter transformation and $q$-Coxeter matrix for a connected acyclic bound quiver $(Q,I)$ with a homogeneous relations $I$, then to establish the relationship between $q$-Coxeter matrix and $q$-Cartan matrix of $(Q,I)$.