论文标题
分级偏斜式代数的两个几何模型
Two geometric models for graded skew-gentle algebras
论文作者
论文摘要
在第1部分中，我们将（不可解释的）对象分类为完美的派生类别$ \ mathrm {per}λ$的分级偏斜偏度代数$λ$，burban-drozd和deng的结果/结果概括为毕业的设置。我们还使用平常的刺穿标记的表面$ \ Mathbf {s}^λ$带有分级（和完整的正式弧系统）为此分类提供几何模型。 在Part2中，我们引入了一个新的表面$ \ MATHBF {s}^λ_*$，用$ \ Mathbf {s}^λ$ binaries用binaries通过边界组成部分$*_ p $（称为二进制），用一个标记的点来替换每个点点$ p $，并用一个标记的点来$ d _ {*_ p}^2 = \ mathrm {id} $，其中$ d _ {*_ p} $是沿$*_ p $的dehn Twist。 $ \ mathrm {per}λ$中的某些不可分解的对象也可以通过$ \ mathbf {s}^λ_*$上的分级未结弧进行分类。此外，使用这种新的几何模型，我们表明，任何两个未结的弧之间的相互作用提供了相应的弧对象之间形态的基础，即公式$ \ mathrm {int} = \ mathrm {dim} \ mathrm {dim} \ mathrm {hom {hom} $ holds。
In Part 1, we classify (indecomposable) objects in the perfect derived category $\mathrm{per}Λ$ of a graded skew-gentle algebra $Λ$, generalizing technique/results of Burban-Drozd and Deng to the graded setting. We also use the usual punctured marked surface $\mathbf{S}^λ$ with grading (and a full formal arc system) to give a geometric model for this classification. In Part2, we introduce a new surface $\mathbf{S}^λ_*$ with binaries from $\mathbf{S}^λ$ by replacing each puncture $P$ by a boundary component $*_P$ (called a binary) with one marked point, and composing an equivalent relation $D_{*_P}^2=\mathrm{id}$, where $D_{*_p}$ is the Dehn twist along $*_P$. Certain indecomposable objects in $\mathrm{per}Λ$ can be also classified by graded unknotted arcs on $\mathbf{S}^λ_*$. Moreover, using this new geometric model, we show that the intersections between any two unknotted arcs provide a basis of the morphisms between the corresponding arc objects, i.e. formula $\mathrm{Int}=\mathrm{dim}\mathrm{Hom}$ holds.
