The representation type of Jacobian algebras

TL;DR

This paper investigates the representation type of Jacobian algebras associated with quivers, showing invariance under mutations and characterizing when they are tame, with implications for classification based on quiver mutation types.

Contribution

It establishes the invariance of the representation type of Jacobian algebras under QP-mutations and characterizes when these algebras are of tame type in relation to quiver mutation types.

Findings

Representation type is invariant under QP-mutations.

P(Q,S) is of tame type iff Q has finite mutation type.

Most finite mutation type quivers admit a unique non-degenerate potential.

Abstract

We show that the representation type of the Jacobian algebra P(Q,S) associated to a 2-acyclic quiver Q with non-degenerate potential S is invariant under QP-mutations. We prove that, apart from very few exceptions, P(Q,S) is of tame representation type if and only if Q is of finite mutation type. We also show that most quivers Q of finite mutation type admit only one non-degenerate potential up to weak right equivalence. In this case, the isomorphism class of P(Q,S) depends only on Q and not on S.