Tubular Jacobian Algebras

TL;DR

This paper demonstrates that endomorphism rings of cluster tilting objects in tubular cluster categories are finite-dimensional tame Jacobian algebras with polynomial growth, linked via quivers with non-degenerate potentials and compatible mutations.

Contribution

It establishes a connection between cluster tilting objects in tubular categories and Jacobian algebras, showing their tameness and mutation compatibility.

Findings

Endomorphism rings are finite-dimensional Jacobian algebras.

These algebras are tame with polynomial growth.

Mutation of cluster tilting objects corresponds to mutation of quivers with potentials.

Abstract

We show that the endomorphism ring of each cluster tilting object in a tubular cluster category is a finite dimensional Jacobian algebra which is tame of polynomial growth. Moreover, these Jacobian algebras are given by a quiver with a non-degenerate potential and mutation of cluster tilting objects is compatible with mutation of QPs.