Endomorphism algebras of maximal rigid objects in cluster tubes

Abstract

Given a maximal rigid object $T$ of the cluster tube, we determine the objects finitely presented by $T$. We then use the method of Keller and Reiten to show that the endomorphism algebra of $T$ is Gorenstein and of finite representation type, as first shown by Vatne. This algebra turns out to be the Jacobian algebra of a certain quiver with potential, when the characteristic of the base field is not 3. We study how this quiver with potential changes when $T$ is mutated. We also provide a derived equivalence classification for the endomorphism algebras of maximal rigid objects.

28 pages. The way of numbering subsections/propositions/theorems/lemmas/corollaries changed, several references added or updated, a few mistakes and typos corrected, some pictures added. To appear in Comm. Alg