Cyclic contractions of dimer algebras always exist

TL;DR

This paper proves that all nondegenerate dimer algebras on a torus can be cyclically contracted to cancellative dimer algebras, revealing key properties like Calabi-Yau conditions and center dimension.

Contribution

It establishes the existence of cyclic contractions for nondegenerate dimer algebras on a torus, linking algebraic properties to geometric structures.

Findings

Nondegenerate dimer algebras admit cyclic contractions to cancellative dimer algebras

Calabi-Yau property is equivalent to being noetherian for these algebras

Center of the algebra has Krull dimension 3

Abstract

We show that every nondegenerate dimer algebra $A$ on a torus admits a cyclic contraction to a cancellative dimer algebra. This implies, for example, that $A$ is Calabi-Yau if and only if it is noetherian; and that the center of $A$ has Krull dimension $3$.