2022/12/20 by Qiu, Yu, Zhang, Chao, Zhou, Yu · 2 citations
#Category Theory (math.CT) #FOS: Mathematics #Geometric Topology (math.GT) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2212.10369
In Part 1, we classify (indecomposable) objects in the perfect derived category perΛ 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 Sλ with grading (and a full formal arc system) to give a geometric model for this classification. In Part2, we introduce a new surface Sλ_* with binaries from Sλ by replacing each puncture P by a boundary component *P (called a binary) with one marked point, and composing an equivalent relation D*P2=id, where D*p is the Dehn twist along *P. Certain indecomposable objects in perΛ can be also classified by graded unknotted arcs on Sλ_*. 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 Int=dimHom holds.