Two geometric models for graded skew-gentle algebras

Abstract

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.

Figures (42)

• Figure 1: $\mathrm{D}_{\text{\Cancer}_P}^2$-action
• Figure 2: Clockwise angles from one arc to the other at intersections
• Figure 3: Types of $\mathbf{A}$-polygons
• Figure 4: A full formal open arc system $\mathbf{A}$ and its dual $\mathbf{A}^\ast$.
• Figure 5: A full formal closed arc system $\mathbf{A}^\ast$ and the corresponding datum

Theorems & Definitions (147)

• Definition 1.1
• Definition 1.2
• Example 1.3
• Definition 1.4
• Remark 1.5
• Remark 1.6
• Example 1.7
• Remark 1.8
• Remark 1.10
• Definition 1.11