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, we prove that, except for a few cases, a finite-dimensional Jacobian algebra $\mathcal{J}(Q,W)$ is $E$-tame if and only if it is $\operatorname{g}$-tame, if and only if it is representation-tame, and this holds precisely when $Q$ is of finite mutation type. Moreover, our results on laminations on marked surfaces lead to the classification of finite-dimensional $E$-finite Jacobian algebras. Consequently, we show that a finite-dimensional Jacobian algebra $\mathcal{J}(Q,W)$ is $E$-finite if and only if it is $\operatorname{g}$-finite, if and only if it is representation-finite, and this holds exactly when $Q$ is of Dynkin type. Furthermore, we provide an application of this result in the theory of cluster algebras. More precisely, the converse of Reading's theorem is established: if the $\operatorname{g}$-fan of the cluster algebra associated with a connected quiver $Q$ is complete, then $Q$ must be of Dynkin type.