关于平方超势代数的非交换几何
On the noncommutative geometry of square superpotential algebras
摘要：
一个超势代数是正方形的，如果它的顶点允许嵌入到一个双环面中，使得它的底层图的图像是一个正方形网格，可能在单位正方形中有对角边;物理学中的二聚体模型提供了例子。这样的嵌入通过我们引入的一种称为印象的方法揭示了代数表示论的大部分内容。设A是平方超势代数，Z是中心，\mathfrak{m}是Spec（Z）原点的极大理想.利用一个印象，给出了直至同构的单A-模的分类，并给出了最大k维单A-模的代数和同调刻画，通过确定超越基和Z-正则序列，证明了Z是3维正规环面整环，Z_{ \mathfrak{m} }是Gorenstein;证明了A_{ \mathfrak{m} }是Z_{ \mathfrak{m} }的一个非交换的收缩分解，从而是一个局部Calabi-Yau代数.本文详细讨论了一类特殊的平方超势代数Y^{p，q}代数。我们证明了中心的Azumaya轨迹和光滑轨迹重合，并提出每个位于奇异轨迹上的分歧极大理想是收缩到零大小的爆破的例外轨迹。
英文摘要：
A superpotential algebra is square if its quiver admits an embedding into a two-torus such that the image of its underlying graph is a square grid, possibly with diagonal edges in the unit squares; examples are provided by dimer models in physics. Such an embedding reveals much of the algebras representation theory through a device we introduce called an impression. Let A be a square superpotential algebra, Z its center, and \mathfrak{m} the maximal ideal at the origin of Spec(Z). Using an impression, we give a classification of all simple A-modules up to isomorphism, and give algebraic and homological characterizations of the simple A-modules of maximal k-dimension; show that Z is a 3-dimensional normal toric domain and Z_{\mathfrak{m}} is Gorenstein, by determining transcendence bases and Z-regular sequences; and show that A_{\mathfrak{m}} is a noncommutative crepant resolution of Z_{\mathfrak{m}}, and thus a local Calabi-Yau algebra. A particular class of square superpotential algebras, the Y^{p,q} algebras, is considered in detail. We show that the Azumaya and smooth loci of the centers coincide, and propose that each ramified maximal ideal sitting over the singular locus is the exceptional locus of a blowup shrunk to zero size.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
