Algebras from surfaces without punctures

TL;DR

This paper introduces surface algebras derived from unpunctured Riemann surfaces with boundary, linking them to cluster categories, and provides geometric models and invariants for their module categories.

Contribution

It presents a new class of gentle algebras called surface algebras, constructed from unpunctured surfaces, and explores their connections to cluster categories and geometric models.

Findings

Surface algebras are endomorphism algebras of partial cluster-tilting objects.

Computed the Avella-Alaminos and Geiss invariant for surface algebras.

Provided a geometric model for the module category of surface algebras.

Abstract

We introduce a new class of finite dimensional gentle algebras, the surface algebras, which are constructed from an unpunctured Riemann surface with boundary and marked points by introducing cuts in internal triangles of an arbitrary triangulation of the surface. We show that surface algebras are endomorphism algebras of partial cluster-tilting objects in generalized cluster categories, we compute the invariant of Avella-Alaminos and Geiss for surface algebras and we provide a geometric model for the module category of surface algebras.