Finite-dimensional Jacobian algebras: Finiteness and tameness

TL;DR

This paper classifies finite-dimensional Jacobian algebras based on their representation types, showing invariance under mutations and linking algebraic properties to Dynkin and finite mutation types, with applications to cluster algebras.

Contribution

It establishes the invariance of $E$-finiteness and $E$-tameness under mutations and classifies Jacobian algebras by their representation type, confirming Demonet's conjecture.

Findings

$E$-finiteness and $E$-tameness are mutation-invariant.

Classification of Jacobian algebras by Dynkin and finite mutation types.

Complete the converse of Reading's theorem for cluster algebras.

Abstract

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 Schr\"{o}er. 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.