Extensions in Jacobian Algebras and Cluster Categories of Marked Surfaces

TL;DR

This paper explicitly describes extension spaces between indecomposable modules in Jacobian algebras from surface triangulations, relating them to arc crossings and introducing a new snake graph interpretation within cluster categories.

Contribution

It provides a novel method to compute extension spaces in gentle Jacobian algebras using snake graphs and surface crossings, expanding understanding of their module categories.

Findings

Extension dimensions correspond to arc crossings in surfaces.

Explicit bases for extension spaces are constructed in most cases.

The approach links surface topology with algebraic extension structures.

Abstract

In the context of representation theory of finite dimensional algebras, string algebras have been extensively studied and most aspects of their representation theory are well-understood. One exception to this is the classification of extensions between indecomposable modules. In this paper we explicitly describe such extensions for a class of string algebras, namely gentle algebras associated to surface triangulations. These algebras arise as Jacobian algebras of unpunctured surfaces. We relate the extension spaces of indecomposable modules to crossings of arcs in the surface and give explicit bases of the extension spaces for indecomposable modules in almost all cases. We show that the dimensions of these extension spaces are given in terms of crossing arcs in the surface. Our approach is new and consists of interpreting snake graphs as indecomposable modules. In order to show that our basis is a spanning set, we need to work in the associated cluster category where we explicitly calculate the middle terms of extensions and give bases of their extension spaces. We note that not all extensions in the cluster category give rise to extensions for the Jacobian algebra.