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PreprintCC BY 4.0arXiv

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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Cite

BibTeXRIS
BibTeX
@misc{chang2026obstructions,
  title = {{Obstructions to Jacobi-Finiteness of Quivers with Potentials}},
  author = {Wen Chang and QuanYu Tang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.11200},
  eprint = {2609.11200},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.11200},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Obstructions to Jacobi-Finiteness of Quivers with Potentials
AU  - Wen Chang
AU  - QuanYu Tang
PY  - 2026
DA  - 2026/09/10
PB  - arXiv
DO  - 10.48550/arxiv.2609.11200
UR  - https://arxiv.org/abs/2609.11200
AB  - 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.
ER  -