Jacobian algebras with periodic module category and exponential growth
| Autor: | Valdivieso-Díaz, Yadira |
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| Rok vydání: | 2013 |
| Předmět: | |
| Zdroj: | Journal of Algebra 449 (2016) 163-174 |
| Druh dokumentu: | Working Paper |
| Popis: | 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 $2$. Comment: 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 |
| Databáze: | arXiv |
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