论文标题
在群集类别的加权投影线类别上，最多三个权重
On cluster categories of weighted projective lines with at most three weights
论文作者
论文摘要
令$ \ mathbb {x} $为加权投影线，$ \ mathcal {c} _ \ mathbb {x} $相关的群集类别。众所周知，$ \ Mathcal {C} _ \ Mathbb {X} $可以实现为具有潜力的Quiver的广义集群类别。在此注释中，在假设$ \ mathbb {x} $最多具有三个权重或管状类型的假设，我们证明，如果一般的群集类别$ \ MATHCAL {C} _ {（q，q，w）} $的jacobi-finite non-finite nontemente al-degentore al a al ligent a al an $ \ MATHCAL {C} _ \ MATHBB {X} $，然后$ \ MATHCAL {C} _ {（Q，W）} $是三角形，等于$ \ Mathcal {C} _ \ Mathbb {x} $。作为副产品，$ 2 $ -CY的倾斜代数为$ \ Mathcal {C} _ \ Mathbb {X} $，只要$ \ Mathbb {x} $最多具有三个权重。为此，对于任何加权投影线$ \ mathbb {x} $，最多可以使用三个权重，我们还可以通过Buan-Iyama-Reiten-Scott构建$ 2 $ 2 $ - 从prepfropopripementive alge alge algebras构建$ \ Mathcal {C} _ \ Mathbb {x} $。
Let $\mathbb{X}$ be a weighted projective line and $\mathcal{C}_\mathbb{X}$ the associated cluster category. It is known that $\mathcal{C}_\mathbb{X}$ can be realized as a generalized cluster category of quiver with potential. In this note, under the assumption that $\mathbb{X}$ has at most three weights or is of tubular type, we prove that if the generalized cluster category $\mathcal{C}_{(Q,W)}$ of a Jacobi-finite non-degenerate quiver with potential $(Q,W)$ shares a $2$-CY tilted algebra with $\mathcal{C}_\mathbb{X}$, then $\mathcal{C}_{(Q,W)}$ is triangle equivalent to $\mathcal{C}_\mathbb{X}$. As a byproduct, a $2$-CY tilted algebra of $\mathcal{C}_\mathbb{X}$ is determined by its quiver provided that $\mathbb{X}$ has at most three weights. To this end, for any weighted projective line $\mathbb{X}$ with at most three weights, we also obtain a realization of $\mathcal{C}_\mathbb{X}$ via Buan-Iyama-Reiten-Scott's construction of $2$-CY categories arising from preprojective algebras.
