A generalization of cancellative dimer algebras to hyperbolic surfaces

TL;DR

This paper introduces geodesic ghor algebras, a new class of quiver algebras on higher genus surfaces, revealing their rich geometric and algebraic structure and contrasting them with known dimer algebras on tori.

Contribution

It generalizes cancellative dimer algebras to higher genus surfaces and explores the relationship between their algebraic properties and surface topology.

Findings

Center of these algebras is a polynomial ring in one variable

Noetherian central localizations are endomorphism rings over their centers

Rich interplay between algebraic center and surface topology

Abstract

We study a new class of quiver algebras on surfaces, called 'geodesic ghor algebras'. These algebras generalize cancellative dimer algebras on a torus to higher genus surfaces, where the relations come from perfect matchings rather than a potential. Although cancellative dimer algebras on a torus are noncommutative crepant resolutions, the center of any dimer algebra on a higher genus surface is just the polynomial ring in one variable, and so the center and surface are unrelated. In contrast, we establish a rich interplay between the central geometry of geodesic ghor algebras and the topology of the surface in which they are embedded. Furthermore, we show that noetherian central localizations of such algebras are endomorphism rings of modules over their centers.