带势箭图的Jacobi有限性的障碍
Obstructions to Jacobi-Finiteness of Quivers with Potentials
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中文总结 AI 辅助
本文证明有限2-无环箭图上不一定存在Jacobi有限势，利用矩阵值Golod-Shafarevich-Vinberg不等式构造障碍，并给出纯箭图判据及无限维Jacobian代数的箭图族。
中文摘要 AI 辅助
我们证明了在有限的$2$-无环箭图上，Jacobi有限势不一定存在。我们的主要工具是一个矩阵值的Golod--Shafarevich--Vinberg不等式，用于由有限多个（可能非齐次的）拓扑关系生成的完备路径代数的商。将其应用于循环导数，得到一个依赖于势的障碍，用于判断完备Jacobian代数的有限维性。然后我们构造了一个纯箭图层面的判据，排除给定箭图上的所有Jacobi有限势，并展示了一族箭图，其上每个势的Jacobian代数都是无限维的。
英文摘要
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.
发表机构
- School of Mathematics and Statistics, Shaanxi Normal University（陕西师范大学数学与统计学院）
- School of Mathematical Sciences, University of Science and Technology of China（中国科学技术大学数学科学学院）
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