The representation type of Jacobian algebras

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.

71 pages. v2: Theorem 1.4(ii) is now more general than in v1. v3: several typos fixed, references updated and not used references deleted. Exposition, mainly in Section 8.2 expanded and improved. Final version, to appear in Adv. Math

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