Finite-dimensional Jacobian algebras: Finiteness and tameness
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866918149548408832 |
|---|---|
| author | Haerizadeh, Mohamad Yurikusa, Toshiya |
| author_facet | Haerizadeh, Mohamad Yurikusa, Toshiya |
| contents | Finite-dimensional Jacobian algebras are studied from the perspective of representation types. We establish that (like other representation types) the notions of $E$-finiteness and $E$-tameness are invariant under mutations of quivers with potentials. Consequently, by applying our results on laminations on marked surfaces, and the results of Plamondon and the second author, we classify $E$-finite and $E$-tame finite-dimensional Jacobian algebras. More precisely, we demonstrate that (resp., except for a few cases,) a finite-dimensional Jacobian algebra $\mathcal{J}(Q,W)$ is $E$-finite (resp., $E$-tame) if and only if it is $\operatorname{g}$-finite (resp., $\operatorname{g}$-tame), if and only if it is representation-finite (resp., representation-tame), and this holds exactly when $Q$ is of Dynkin type (resp., finite mutation type), as shown by Geiss, Labardini and Schröer. This also proves Demonet's conjecture for finite-dimensional Jacobian algebras. Furthermore, we provide an application of our results in the theory of cluster algebras. More precisely, we establish the converse of Reading's theorem: if the $\operatorname{g}$-fan of the cluster algebra associated with a connected quiver $Q$ is complete, then $Q$ must be of Dynkin type. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_04570 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Finite-dimensional Jacobian algebras: Finiteness and tameness Haerizadeh, Mohamad Yurikusa, Toshiya Representation Theory Rings and Algebras 16G20, 16G60, 13F60 Finite-dimensional Jacobian algebras are studied from the perspective of representation types. We establish that (like other representation types) the notions of $E$-finiteness and $E$-tameness are invariant under mutations of quivers with potentials. Consequently, by applying our results on laminations on marked surfaces, and the results of Plamondon and the second author, we classify $E$-finite and $E$-tame finite-dimensional Jacobian algebras. More precisely, we demonstrate that (resp., except for a few cases,) a finite-dimensional Jacobian algebra $\mathcal{J}(Q,W)$ is $E$-finite (resp., $E$-tame) if and only if it is $\operatorname{g}$-finite (resp., $\operatorname{g}$-tame), if and only if it is representation-finite (resp., representation-tame), and this holds exactly when $Q$ is of Dynkin type (resp., finite mutation type), as shown by Geiss, Labardini and Schröer. This also proves Demonet's conjecture for finite-dimensional Jacobian algebras. Furthermore, we provide an application of our results in the theory of cluster algebras. More precisely, we establish the converse of Reading's theorem: if the $\operatorname{g}$-fan of the cluster algebra associated with a connected quiver $Q$ is complete, then $Q$ must be of Dynkin type. |
| title | Finite-dimensional Jacobian algebras: Finiteness and tameness |
| topic | Representation Theory Rings and Algebras 16G20, 16G60, 13F60 |
| url | https://arxiv.org/abs/2507.04570 |