发表机构
University of Connecticut; Pedagogical and Technological University of Colombia（康涅狄格大学; 哥伦比亚教学技术大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文引入类型 $\mathbb{D}$ 偏序集，证明其socle-投射范畴有限表示型，并通过带孔多边形弧范畴与模范畴的等价给出几何实现，空alien箭头时cluster子代数等于完整cluster代数。
AI 中文摘要
我们引入了类型 $\mathbb{D}$ 的偏序集，这是一族由类型 $D_n$ 的容许Dynkin箭图连同相容的额外箭头（alien箭头）集合所描述的偏序集，并证明了它们的socle-投射表示范畴具有有限表示型。继续沿着为类型 $\mathbb{A}$ 偏序集所开创的同一精神的研究计划 [R. Schiffler, R-J. Serna, A geometric realization of socle-projective categories for posets of type $\mathbb{A}$, J. Pure Appl. Algebra 224 (2020), no. 12, 106436]，我们基于类型 $D_n$ 的cluster-tilted代数的现有几何模型进行构建，该模型的几何组合比类型 $\mathbb{A}$ 更为复杂。我们的主要结果建立了 $\Bbbk$-线性范畴等价 $\Theta$，它介于带孔 $(n+3)$-边形的弧范畴的全子范畴 $(\mathcal C/T)_F$（其对象为特定的 $sp$-弧）与有限生成socle-投射 $\Bbbk\mathscr{P}$-模的范畴之间，其中 $\mathscr{P}$ 为类型 $\mathbb{D}$ 的偏序集。作为推论，当alien箭头集合为空时，我们得出由 $sp$-弧生成的cluster子代数与完整cluster代数一致。
英文摘要
We introduce posets of type $\mathbb{D}$, a family of posets described by an admissible Dynkin quiver of type $D_n$ together with a compatible set of extra arrows (alien arrows), and show that their socle-projective representation category is of finite representation type. Continuing, in the same spirit, a program initiated for posets of type $\mathbb{A}$ [R. Schiffler, R-J. Serna, A geometric realization of socle-projective categories for posets of type $\mathbb{A}$, J. Pure Appl. Algebra 224 (2020), no. 12, 106436], we build on an existing geometric model for cluster-tilted algebras of type $D_n$, whose geometric combinatorics is more intricate than that of type $\mathbb{A}$. Our main result establishes a $\Bbbk$-linear categorical equivalence $Θ$ between a full subcategory $(\mathcal C/T)_F$ of the arc category of a punctured $(n+3)$-gon, whose objects are certain $sp$-arcs, and the category of finitely generated socle-projective $\Bbbk\mathscr{P}$-modules, for $\mathscr{P}$ a poset of type $\mathbb{D}$. As a consequence, when the set of alien arrows is empty, we conclude that the cluster subalgebra generated by the $sp$-arcs coincides with the full cluster algebra.
Comments40 pages, 11 figures
