Derived Equivalence in Surface Algebras of Genus 0 via Graded Equivalence

TL;DR

This paper investigates derived equivalences among surface algebras of genus 0, constructed from triangulated Riemann surfaces with boundary, identifying conditions under which different internal cuts yield equivalent algebras.

Contribution

It provides a classification of derived equivalences for a class of gentle surface algebras based on triangulation cuts, expanding understanding of their algebraic relationships.

Findings

Derived equivalences depend on specific cuts in triangulations.

Conditions for derived equivalence are explicitly characterized.

Results apply to unpunctured genus 0 surfaces with boundary.

Abstract

We determine some of the derived equivalences of a class of gentle algebras called surface algebras. These algebras are constructed from an unpunctured Riemann surface of genus 0 with boundary and marked points by introducing cuts in internal triangles of an arbitrary triangulation of the surface. In particular, we fix a triangulation of a surface and determine when different cuts produce derived equivalent algebras.