Derived categories of skew-gentle algebras and orbifolds

Abstract

Skew-gentle algebras are a generalisation of the well-known class of gentle algebras with which they share many common properties. In this work, using non-commutative Gröbner basis theory, we show that these algebras are strong Koszul and that the Koszul dual is again skew-gentle. We give a geometric model of their bounded derived categories in terms of polygonal dissections of surfaces with orbifold points, establishing a correspondence between curves in the orbifold and indecomposable objects. Moreover, we show that the orbifold dissections encode homological properties of skew-gentle algebras such as their singularity categories, their Gorenstein dimensions and derived invariants such as the determinant of their q-Cartan matrices.

Snake Graphs from Triangulated Orbifolds

“…These algebras from triangulated orbifolds will always be gentle. They are studied in [22] and [23].…”

“…Labardini-Fragoso and Mou define a family of gentle algebras associated to triangulated unpunctured orbifolds with all orbifold points of order three [22]. The algebras associated to a polygon with one orbifold point were previously studied in [24].…”

“…• Labardini-Fragoso and Mou show in [22] that the Caldero-Chapoton map gives a bijection between τ -rigid pairs of the algebra KQ T /I T and cluster monomials from the generalized cluster algebra A(Q T gen ).…”

Skew-gentle algebras and orbifolds: Oberwolfach talk, January 2020

“…Originating in cluster theory, geometric surface models have been instrumental in connecting representation theory with other areas of mathematics such as homological mirror mirror symmetry, see for example [9,10] and [7]. In this extended abstract based on [6], we give an orbifold model for the bounded derived category D b (A) of a skew-gentle algebra A which encodes the indecomposable objects in D b (A) in terms of graded curves in the orbifold.…”

“…We are now in a position to state the main result of [6]. (2) Given a curve γ as in (1), set f : γ ∩ G * → Z to be the map such that, denoting the segment connecting two consecutive (in the direction of γ) intersection points x i , x i+1 of γ with G * by γ i , we have…”