Jacobian algebras with periodic module category and exponential growth
Abstract
The Jacobian algebra associated to a triangulation of a closed surface with a collection of marked points 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 (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 consists only of stable tubes of rank or .
Keywords
Cite
@article{arxiv.1309.2708,
title = {Jacobian algebras with periodic module category and exponential growth},
author = {Yadira Valdivieso-Díaz},
journal= {arXiv preprint arXiv:1309.2708},
year = {2016}
}
Comments
In the previous version I showed that the Jacobian algebras of closed surfaces, excluding the case of the sphere with 4 and 5 punctures, are algebras of exponential growth and it was changed grammar errors. In this new version I change the name of this note and I show that the Jacobian algebras of the sphere with 5 punctures are algebras of exponential growth. This is the final version