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.

Key Words:

• Cluster tube
• Derived equivalence
• Finitely presented object
• Maximal rigid object
• Quiver with potential

2000 Mathematics Subject Classification:

• 18E30
• 16G10
• 16E35

ACKNOWLEDGMENTS

The author gratefully acknowledges financial support from Max—Planck—Institut für Mathematik in Bonn. He thanks Bernhard Keller and Pierre-Guy Plamondon for valuable conversations, and he thanks Bernhard Keller, Pierre-Guy Plamondon, Yann Palu, Yu Zhou, Bin Zhu, and a referee for many helpful remarks on a preliminary version of this article.

Notes

Many thanks to Laurent Demonet for much help in drawing these two graphs.

Communicated by D. Zacharia.