论文标题
2简化志趣相投的收藏II的成对兼容性II：前置代数和半级级别对
Pairwise Compatibility for 2-Simple Minded Collections II: Preprojective Algebras and Semibrick Pairs of Full Rank
论文作者
论文摘要
令$λ$成为一个有限维度的协会代数。一对半纤维对是一组有限的$λ$ - 模型，某些hom-和延伸式消失。如果可以将其放大，则可以完整地完成一对，以便满足生成条件。我们证明，如果$λ$是$τ$ - 最多使用3个简单模块的有限，则可以使用成对模块上的条件来表征半光对的完整性。然后，我们使用弱顺序为$ a_n $类型的前注体代数的半纤维对构建组合模型。从该模型中，我们推断出任何半尺寸的尺寸$ n $都可以满足生成条件，并且任何半核对对的尺寸向量形成了一些$ c $ -matrix的列向量的子集。最后，我们表明，对于具有3个以上顶点的Dynkin图的前代数的完整性代数，不存在“成对”标准。
Let $Λ$ be a finite-dimensional associative algebra over a field. A semibrick pair is a finite set of $Λ$-modules for which certain Hom- and Ext-sets vanish. A semibrick pair is completable if it can be enlarged so that a generating condition is satisfied. We prove that if $Λ$ is $τ$-tilting finite with at most 3 simple modules, then the completability of a semibrick pair can be characterized using conditions on pairs of modules. We then use the weak order to construct a combinatorial model for the semibrick pairs of preprojective algebras of type $A_n$. From this model, we deduce that any semibrick pair of size $n$ satisfies the generating condition, and that the dimension vectors of any semibrick pair form a subset of the column vectors of some $c$-matrix. Finally, we show that no "pairwise" criteria for completability exists for preprojective algebras of Dynkin diagrams with more than 3 vertices.
