论文标题
部分可观测时空混沌系统的无模型预测
On Gorenstein algebras of finite Cohen-Macaulay type: dimer tree algebras and their skew group algebras
论文作者
论文摘要
二聚体树代数是Gorenstein Dimension 1的一类非共同的Gorenstein代数。在先前的工作中，我们表明，二聚体树代数的Cohen-Macaulay模块的稳定类别类别是二聚体树代数$ a $ a $ a $ as $ as a $ as a $ as 2 cluster类别的dynkin类型$ \ Mathbb $ \ Mathbb {a a} $ {a} $。在这里，我们表明，如果$ a $具有$ g $带有两个元素的$ g $的可允许的操作，则偏斜组的稳定的cohen-macaulay类别是偏斜组$ ag $ ag $ ag $ ag $ ag ag $ \ mathbb {d} $。这个结果让人联想起Reiten and Riedtmann的结果并启发，他们表明，在类型$ \ Mathbb {a} $的路径代数上接受可允许的$ G $ ATACTION，其结果偏斜组的代数为$ \ Mathbb {D} $。此外，我们提供了$ ag $的Syzygy类别的几何模型，该类别的刺穿Polygon $ \ Mathcal {P} $具有内部装饰的棋盘格模式，使得与与$ AG $ AG $和2-Pivivs的Indecomposopable Syzygies in $ \ MATHCAL {P MATHCAL {p} $相应地相对于$ AG和2-PIVIVS的nimcal {p} $。特别是，二聚体树代数及其偏斜组代数分别是有限的Cohen-Macaulay类型的Gorenstein代数$ \ Mathbb {A} $和$ \ Mathbb {D} $。我们还提供类型的示例$ \ Mathbb {e} _6，\ Mathbb {e} _7，$和$ \ Mathbb {e} _8 $。
Dimer tree algebras are a class of non-commutative Gorenstein algebras of Gorenstein dimension 1. In previous work we showed that the stable category of Cohen-Macaulay modules of a dimer tree algebra $A$ is a 2-cluster category of Dynkin type $\mathbb{A}$. Here we show that, if $A$ has an admissible action by the group $G$ with two elements, then the stable Cohen-Macaulay category of the skew group algebra $AG$ is a 2-cluster category of Dynkin type $\mathbb{D}$. This result is reminiscent of and inspired by a result by Reiten and Riedtmann, who showed that for an admissible $G$-action on the path algebra of type $\mathbb{A}$ the resulting skew group algebra is of type $\mathbb{D}$. Moreover, we provide a geometric model of the syzygy category of $AG$ in terms of a punctured polygon $\mathcal{P}$ with a checkerboard pattern in its interior, such that the 2-arcs in $\mathcal{P}$ correspond to indecomposable syzygies in $AG$ and 2-pivots correspond to morphisms. In particular, the dimer tree algebras and their skew group algebras are Gorenstein algebras of finite Cohen-Macaulay type $\mathbb{A}$ and $\mathbb{D}$ respectively. We also provide examples of types $\mathbb{E}_6,\mathbb{E}_7,$ and $\mathbb{E}_8$.
