Finite dimensional 2-cyclic Jacobian algebras

TL;DR

This paper constructs Jacobian-finite potentials on 2-cyclic quivers and demonstrates their categorification of certain generalized cluster algebras using $ au$-rigid modules.

Contribution

It introduces a method to ensure Jacobian-finiteness on 2-cyclic quivers and applies this to categorify specific generalized cluster algebras.

Findings

Existence of Jacobian-finite potentials on 2-cyclic quivers.

Categorification of generalized cluster algebras in three geometric cases.

Application of covering theory and Caldero-Chapoton formula in the proofs.

Abstract

In this paper, we start with a class of quivers that containing only 2-cycles and loops, referred to as 2-cyclic quivers. We prove that there exists a potential on these quivers that ensures the resulting quiver with potential is Jacobian-finite. As an application, we first demonstrate, using covering theory, that a Jacobian-finite potential exists on a class of 2-acyclic quivers. Secondly, by using the 2-cyclic Caldero-Chapoton formula, the $\tau$-rigid modules over the Jacobian algebras of our proven Jacobian-finite 2-cyclic quiver with potential can categorify Paquette-Schiffler's generalized cluster algebras in three specific cases: one for a disk with two marked points and one 3-puncture, one for a sphere with one puncture, one 3-puncture and one orbifold point, and another for a sphere with one puncture and two 3-punctures.