A geometric model for the module category of a gentle algebra

TL;DR

This paper introduces a geometric model for the module category of gentle algebras by representing them as tiling algebras associated with partial triangulations of unpunctured surfaces, extending existing surface algebra concepts.

Contribution

It provides a new geometric framework for understanding gentle algebras as tiling algebras linked to surface triangulations, generalizing Jacobian and surface algebras.

Findings

Geometric model for gentle algebra modules

Extension of surface algebra concepts

Unified framework for algebra and geometry

Abstract

In this article, gentle algebras are realised as tiling algebras, which are associated to partial triangulations of unpunctured surfaces with marked points on the boundary. This notion of tiling algebras generalise the notion of Jacobian algebras of triangulations of surfaces and the notion of surface algebras. We use this description to give a geometric model of the module category of any gentle algebra.