论文标题
通过对称表示形式明显的不可定向表面和群集类别
Marked non-orientable surfaces and cluster categories via symmetric representations
论文作者
论文摘要
我们启动了不可取向表面的表示理论的研究。作为寻找杜邦和Palesi的准群集代数相关的标记不可取向表面的添加分类的第一步，我们研究了与未施加的情况下与可定向双重覆盖的群集类别的对象进行的一定修改。更确切地说，我们考虑Derksen-Weyman和Boos-Cerulli Irelli研究的对称表示理论，并将其提升为群集类别。这提供了一种方法，可以考虑违反双重性函数下的“对象的不可分解轨道”。因此，我们可以将曲线分配在不可定向的表面$（\ mathbb {s}，\ mathbb {m}）$上，以不可拆卸的对称对象。此外，我们定义了一个对称扩展的新概念，并表明$（\ Mathbb {s}，\ Mathbb {M}）上的弧和准arcs对应于无象征性自我extension。因此，我们表明$（\ mathbb {s}，\ mathbb {m}）$的准三角剖分对应于集群倾斜对象的对称类似物。
We initiate the investigation of representation theory of non-orientable surfaces. As a first step towards finding an additive categorification of Dupont and Palesi's quasi-cluster algebras associated marked non-orientable surfaces, we study a certain modification on the objects of the cluster category associated to the orientable double covers in the unpunctured case. More precisely, we consider symmetric representation theory studied by Derksen-Weyman and Boos-Cerulli Irelli, and lift it to the cluster category. This gives a way to consider `indecomposable orbits of objects' under a contravariant duality functor. Hence, we can assign curves on a non-orientable surface $(\mathbb{S}, \mathbb{M})$ to indecomposable symmetric objects. Moreover, we define a new notion of symmetric extension, and show that the arcs and quasi-arcs on $(\mathbb{S}, \mathbb{M})$ correspond to the indecomposable symmetric objects without symmetric self-extension. Consequently, we show that quasi-triangulations of $(\mathbb{S}, \mathbb{M})$ correspond to a symmetric analogue of cluster tilting objects.
