Séminaires : Séminaire d'Algèbre

We introduce the notion of a lower (upper) bound cluster algebra generated by projective cluster variables. Projective cluster variables are often categorified by projective modules of the corresponding quiver with relations. We show that under an acyclicity assumption, the cluster algebra and the lower bound cluster algebra generated by projective cluster variables coincide. In this case, we use our results to construct a basis for the cluster algebra. We also show that the coincidence between cluster algebra and the lower bound cluster algebra generated by projective cluster variables holds beyond acyclic seeds. Part of this talk is based on joint work with Karin Baur.