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A geometric realization of socle-projective categories for posets of type A
Journal of Pure and Applied Algebra ( IF 0.8 ) Pub Date : 2020-12-01 , DOI: 10.1016/j.jpaa.2020.106436
Ralf Schiffler , Robinson-Julian Serna

Abstract This paper establishes a link between the theory of cluster algebras and the theory of representations of partially ordered sets. We introduce a class of posets by requiring avoidance of certain types of peak-subposets and show that these posets can be realized as the posets of quivers of type A with certain additional arrows. This class of posets is therefore called posets of type A . We then give a geometric realization of the category of finitely generated socle-projective modules over the incidence algebra of a poset of type A as a combinatorial category of certain diagonals of a regular polygon. This construction is inspired by the realization of the cluster category of type A as the category of all diagonals by Caldero, Chapoton and the first author [10] . We also study the subalgebra of the cluster algebra generated by those cluster variables that correspond to the socle-projectives under the above construction. We give a sufficient condition for when this subalgebra is equal to the whole cluster algebra.

中文翻译:

A 类偏序集的社会射影范畴的几何实现

摘要 本文建立了簇代数理论和偏序集表示理论之间的联系。我们通过要求避免某些类型的峰值子集来引入一类偏序集,并表明这些偏序集可以实现为具有某些附加箭头的 A 型箭袋的偏序集。因此,这类偏序集称为 A 类型的偏序集。然后,我们在类型 A 的偏序集的关联代数上给出有限生成的 socle-projective 模的范畴的几何实现,作为正多边形的某些对角线的组合范畴。这种构造的灵感来自于 Caldero、Chapoton 和第一作者 [10] 将类型 A 的簇类别实现为所有对角线的类别。我们还研究了由与上述构造下的社会射影对应的那些簇变量生成的簇代数的子代数。当这个子代数等于整个簇代数时,我们给出了一个充分条件。
更新日期:2020-12-01
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