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| Main Authors: | , , |
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| Format: | Preprint |
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2022
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| Online Access: | https://arxiv.org/abs/2212.10369 |
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| _version_ | 1866910378166845440 |
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| author | Qiu, Yu Zhang, Chao Zhou, Yu |
| author_facet | Qiu, Yu Zhang, Chao Zhou, Yu |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_10369 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Two geometric models for graded skew-gentle algebras Qiu, Yu Zhang, Chao Zhou, Yu Representation Theory Category Theory Geometric Topology 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. |
| title | Two geometric models for graded skew-gentle algebras |
| topic | Representation Theory Category Theory Geometric Topology |
| url | https://arxiv.org/abs/2212.10369 |