发表机构
Aarhus University（奥胡斯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究从Ringel–Hall代数视角，为有限维结合k-代数Λ构造忠实群函子，证明其Picture空间局部CAT(0)的条件，还将Green序列与Picture群正表达式的一一对应结果推广到非遗传k-代数。
AI 中文摘要
设k为代数闭域，Λ为有限维结合k-代数。我们应用Joyce和Brideland提出的Ringel–Hall代数概念，证明关于Picture空间与Picture群的若干结果。例如，我们为Λ构造了一个忠实群函子，即从Λ的τ-簇态射范畴到群胚的忠实函子。由此，根据Hanson–Igusa的结果，若Λ的τ-簇态射范畴容许最后因子的相容性，则Λ的Picture空间是局部CAT(0)的。我们还证明Green序列与Picture群中某些正表达式一一对应，这将Igusa–Todorov的结果推广到非遗传k-代数的情形。
英文摘要
Let $k$ be an algebraically closed field and let $Λ$ be a finite-dimensional associative $k$-algebra. We apply Joyce and Brideland's notion of Ringel--Hall algebra to prove results about picture spaces and picture groups. For instance, we construct a faithful group functor for $Λ$, i.e. a faithful functor from the $τ$-cluster morphism category of $Λ$ into a groupoid. Consequently, by results of Hanson--Igusa, the picture space of $Λ$ is locally CAT(0) provided that the $τ$-cluster morphism category of $Λ$ admits compatibility of last factors. We also show that green sequences are in bijection with certain positive expressions in the picture group, which generalizes a result of Igusa--Todorov beyond hereditary $k$-algebras.
Comments16 pages, comments welcome!
