论文标题
$ \ mathbb {a} $的离散群集类别的完成
Completions of discrete cluster categories of type $\mathbb{A}$
论文作者
论文摘要
我们通过将这样的离散群集类别嵌入较大的类别，然后使用一定的Verdier商来完成Igusa和Todorov定义的类型$ \ Mathbb {a} $的离散集群类别。最终的类别是一个由离散集群类别作为完整子类别的Hom-Finite Krull-Schmidt三角形类别。即使该类别不是$ 2 $ -CALABI-YAU，并且扩展空间并不总是对称的，但该新类别中的对象和HOM空间也可以用几何描述。我们描述所有倾斜子类别。给定这样的子类别，我们定义了一个群集字符，该字符在无限许多不确定的环中以一个环为单位。我们的群集特征是新的，因为它考虑了无限尺寸的无限尺寸子代理。我们表明，如果所交换的对象满足某些局部的卡拉比野条件，则可以满足乘法公式和交换公式。
We complete the discrete cluster categories of type $\mathbb{A}$ as defined by Igusa and Todorov, by embedding such a discrete cluster category inside a larger one, and then taking a certain Verdier quotient. The resulting category is a Hom-finite Krull-Schmidt triangulated category containing the discrete cluster category as a full subcategory. The objects and Hom-spaces in this new category can be described geometrically, even though the category is not $2$-Calabi-Yau and Ext-spaces are not always symmetric. We describe all cluster-tilting subcategories. Given such a subcategory, we define a cluster character that takes values in a ring with infinitely many indeterminates. Our cluster character is new in that it takes into account infinite dimensional sub-representations of infinite dimensional ones. We show that it satisfies the multiplication formula and also the exchange formula, provided that the objects being exchanged satisfy some local Calabi-Yau conditions.
