A Geometric Model for the Module Category of a Skew-gentle Algebra
Organizers
Speaker
Time
Monday, November 7, 2022 3:30 PM - 5:00 PM
Venue
1129B
Online
Zoom 537 192 5549
(BIMSA)
Abstract
The main object of this report is to give a geometric model for the module category of a skew-gentle algebra via a partial ideal/tagged triangulation on a puncture marked surface. On the one hand, by using this model, we give a geometrical realization of a class of indecomposable modules and the Auslander-Reiten translations; we also give an intersection-dimension formula which shows that the dimension of a morphism space equals to the intersection number of corresponding curves. In
particular, support τ -tilting modules can be classified. On the other hand, we study the two-term rigid subcategory of a (2-Calabi-Yau) triangulated categories and show that it admits cluster structures. Such a clsuter structure can be geometricall interpreted after we generalizing our model to the so called surface rigid algebra. We give a combinatorial proof to show that the cluster structure is mutation-connected, which implies the support τ -tilting graph of a skew-gentle algebra is connected. Finally, by constructing morphisms explicitly, we also give the fundamental group of the support τ -tilting graph over a gentle algebra.