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arXiv · 2609.11200

Obstructions to Jacobi-Finiteness of Quivers with Potentials

Wen Chang·QuanYu Tang
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Abstract

We show that Jacobi-finite potentials need not exist on finite $2$-acyclic quivers. Our main tool is a matrix-valued Golod--Shafarevich--Vinberg inequality for quotients of completed path algebras by finitely many, possibly nonhomogeneous, topological relations. Applied to cyclic derivatives, it yields a potential-dependent obstruction to the finite-dimensionality of completed Jacobian algebras. We then construct a purely quiver-level criterion excluding every Jacobi-finite potential on a given quiver, and exhibit a family of quivers for which every potential has an infinite-dimensional Jacobian algebra.

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Wen Chang, QuanYu Tang. 2026-09-10. Obstructions to Jacobi-Finiteness of Quivers with Potentials. https://arxiv.org/abs/2609.11200

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