Derived equivalences between skew-gentle algebras using orbifolds (Q2149092)

Summary: Skew-gentle algebras are skew-group algebras of gentle algebras equipped with a certain \(\mathbb{Z}_2\)-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 \(\mathbb{Z}_2\)-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.