Tame代数具有稠密的$\mathbf{g}$ -向量扇
Tame algebras have dense $\mathbf{g}$-vector fans
摘要：
有限维代数的$\mathbf{g}$ -向量扇是其射线是其$2$ -项预淤对象的$\mathbf{g}$ -向量的扇。证明了Tame代数的$\mathbf{g}$ -向量扇是稠密的.然后，我们应用这一结果，以获得一个近分类的箭图的集群$\mathbf{g}$ -向量扇的封闭是密集的或半空间，使用添加剂分类的集群代数通过雅可比代数。作为另一个应用，我们证明了颤抖与潜在的一次穿刺闭曲面所产生的，稳定性和集群的散射图只不同的墙壁上的跨壁功能包含在一个分离超平面。附录致力于截断扭转函子及其伴随的建设。
英文摘要：
The $\mathbf{g}$-vector fan of a finite-dimensional algebra is a fan whose rays are the $\mathbf{g}$-vectors of its $2$-term presilting objects. We prove that the $\mathbf{g}$-vector fan of a tame algebra is dense. We then apply this result to obtain a near classification of quivers for which the closure of the cluster $\mathbf{g}$-vector fan is dense or is a half-space, using the additive categorification of cluster algebras by means of Jacobian algebras. As another application, we prove that for quivers with potentials arising from once-punctured closed surfaces, the stability and cluster scattering diagrams only differ by wall-crossing functions on the walls contained in a separating hyperplane. The appendix is devoted to the construction of truncated twist functors and their adjoints.
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