Derived equivalences between skew-gentle algebras using orbifolds

TL;DR

This paper establishes a geometric framework linking skew-gentle algebras to dissected orbifolds and their double covers, enabling the study of derived equivalences via orbifold diffeomorphisms and line fields.

Contribution

It introduces a bijection between skew-gentle algebras and dissected orbifolds with double covers, connecting algebraic derived equivalences to geometric transformations.

Findings

Skew-gentle algebras correspond to dissected orbifolds with double covers.

Derived equivalences relate to diffeomorphisms respecting line fields.

The geometric model captures algebraic properties of skew-gentle algebras.

Abstract

Skew-gentle algebras are skew-group algebras of gentle algebras equipped with a certain $\Z_2$-action. Building on the bijective correspondence between gentle algebras and dissected surfaces, we obtain in this paper a bijection between skew-gentle algebras and certain dissected orbifolds that admit a double cover. We prove the compatibility of the $\Z_2$-action on the double cover with the skew-group algebra construction. This allows us to investigate the derived equivalence relation between skew-gentle algebras in geometric terms: We associate to each skew-gentle algebra a line field on the orbifold, and on its double cover, and interpret different kinds of derived equivalences of skew-gentle algebras in terms of diffeomorphisms respecting the homotopy class of the line fields associated to the algebras.