Two geometric models for graded skew-gentle algebras

Abstract

In Part 1, we classify (indecomposable) objects in the perfect derived category $\mathrm{per}\Lambda$ of a graded skew-gentle algebra $\Lambda$, generalizing technique/results of Burban-Drozd and Deng to the graded setting. We also use the usual punctured marked surface ${\mathbf{S}}^{\lambda }$ with grading (and a full formal arc system) to give a geometric model for this classification. In Part2, we introduce a new surface ${\mathbf{S}}_{\ast }^{\lambda }$ with binaries from ${\mathbf{S}}^{\lambda }$ by replacing each puncture $P$ by a boundary component ${\ast }_P$ (called a binary) with one marked point, and composing an equivalent relation $D_{{\ast }_P}^2=\mathrm{id}$, where $D_{{\ast }_p}$ is the Dehn twist along ${\ast }_P$. Certain indecomposable objects in $\mathrm{per}\Lambda$ can be also classified by graded unknotted arcs on ${\mathbf{S}}_{\ast }^{\lambda }$. 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 $\mathrm{Int}=\dim \mathrm{Hom}$ holds.

101 pages, the non-positive assumption for the intersection-dimension formula in part 2 has been removed, due to an upgrade of the classification theorem in the part 1