扭曲卡拉比-丘代数的局部史密斯谱
Local Smith Profiles of Twisted Calabi--Yau Algebras
AI总结：
该研究针对扭曲卡拉比-丘代数，利用其对称性细化局部指数，推导约束条件并计算循环斜群代数的局部数据，解决了三维箭图分类的相关问题。
AI中文摘要：
局部有限初等扭曲卡拉比-丘代数的矩阵希尔伯特级数是矩阵多项式的逆。该多项式在x=1处的幂级数环上的史密斯标准型会生成一组有限的局部指数，这些指数是对格尔范德-基里洛夫维数的细化，而格尔范德-基里洛夫维数仅记录其中最大的指数。我们证明，卡拉比-丘对称性使该局部数据具有刚性：该代数会根据 Nakayama 循环上的平均阿廷-谢尔特指数分解为环因子；经局部归一化后，该对称性会在史密斯余核上诱导出非奇异的联结形式，并在其层上诱导出有限阶 Nakayama 作用，从而对指数的重数施加互逆特征值和奇偶性约束。我们计算了循环斜群代数的完整局部数据，推导了三维箭图分类的奇偶筛选法，并将一个四顶点型实现为分次 down-up 代数的迭代 smash 积，而该四顶点型是近期分类工作中未解决的类型。
英文摘要：
The matrix Hilbert series of a locally finite elementary twisted Calabi--Yau algebra is the inverse of a matrix polynomial. The Smith normal form of this polynomial over the power series ring at $x=1$ produces a finite list of local exponents refining the Gelfand--Kirillov dimension, which records only the largest of them. We show that the Calabi--Yau symmetry makes this local data rigid: the algebra decomposes into ring factors according to the average Artin--Schelter index along Nakayama cycles, and after a local normalization the symmetry induces a nonsingular linking form on the Smith cokernel together with a finite-order Nakayama action on its layers, forcing reciprocal-eigenvalue and parity constraints on the multiplicities of the exponents. We compute the complete local data for cyclic skew-group algebras, derive a parity sieve for quiver classifications in dimension three, and realize, as an iterated smash product of a graded down-up algebra, a four-vertex type that a recent classification had left open.
