Algebras given by Labelled Quivers

In [3], Bautista and Salmeron explained how, given a finite quiver Z without double arrows or oriented cycles, certain sets of subsets of ZZ (called (P)-generating systems) may, under appropriate conditions, generate algebras whose preprojective components can be embedded in ZZ in a nice way, and which are (P)-algebras [2] whenever they are representation-finite. Here, we define numerical functions on Z, called labels, allowing the construction of all (P)-generating systems, and hence, under a purely combinatorial condition (which we call admissibility) of all the algebras of Bautista and Salmeron. The construction also shows that, for every finite quiver Z without double arrows or oriented cycles, the number of isomorphism classes of (P)-algebras having Z as orbit quiver is finite. As an illustration, we prove that algebras given by an admissible label on a quiver with underlying graph An are tilted [6] because they have a complete slice in their preprojective components. In what follows, k will denote an algebraically closed field. For a finitedimensional /c-algebra A, a module will always mean a finite-dimensional right /1-module. For a quiver Z, Zo will denote its set of vertices, and Zx its set of arrows. If Z is a translation quiver, T will denote its translation, and /c(Z) its mesh category [5].