Jacobian algebras with periodic module category and exponential growth

TL;DR

This paper studies Jacobian algebras from surface triangulations, showing they are symmetric, tame, have exponential growth (except for small punctured spheres), and exhibit 2-periodic Auslander-Reiten translation, revealing a new family of such algebras.

Contribution

It introduces a new family of symmetric, tame Jacobian algebras with periodic module categories and exponential growth, expanding understanding of their representation theory.

Findings

Auslander-Reiten translate acts 2-periodically on objects.

Most such algebras have exponential growth, except for spheres with 4 or fewer punctures.

The Auslander-Reiten quiver consists only of stable tubes of rank 1 or 2.

Abstract

The Jacobian algebra associated to a triangulation of a closed surface $S$ with a collection of marked points $M$ is (weakly) symmetric and tame. We show that for these algebras the Auslander-Reiten translate acts 2-periodical on objects. Moreover, we show that excluding only the case of a sphere with $4$ (or less) punctures, these algebras are of exponential growth. These four properties implies that there is a new family of algebras symmetric, tame and with periodic module category. As a consequence of the 2-periodical actions of the Auslander-Reiten translate on objects, we have that the Auslander-Reiten quiver of the generalized cluster category $\cC_{(S,M)}$ consists only of stable tubes of rank $1$ or $2$.