Finite-dimensional Jacobian algebras: Finiteness and tameness

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.