Endomorphism algebras of maximal rigid objects in cluster tubes

TL;DR

This paper characterizes the endomorphism algebras of maximal rigid objects in cluster tubes, showing they are Gorenstein, of finite representation type, and related to Jacobian algebras, with classifications under mutation.

Contribution

It provides a detailed analysis of endomorphism algebras in cluster tubes, including their Gorenstein property, Jacobian algebra structure, and derived equivalence classification, extending prior results.

Findings

Endomorphism algebra is Gorenstein and of finite representation type.

The algebra is a Jacobian algebra of a quiver with potential (char ≠ 3).

Derived equivalence classes are classified under mutation.

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.