Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2408.10056 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929603302391808 |
|---|---|
| author | Li, Yiyu Peng, Liangang |
| author_facet | Li, Yiyu Peng, Liangang |
| contents | 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 $τ$-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_10056 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Finite dimensional 2-cyclic Jacobian algebras Li, Yiyu Peng, Liangang Representation Theory Rings and Algebras 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 $τ$-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. |
| title | Finite dimensional 2-cyclic Jacobian algebras |
| topic | Representation Theory Rings and Algebras |
| url | https://arxiv.org/abs/2408.10056 |