A geometric model for syzygies over 2-Calabi-Yau tilted algebras

TL;DR

This paper introduces a geometric model using polygons with checkerboard patterns to describe syzygy categories over a class of 2-Calabi-Yau tilted algebras called dimer tree algebras, establishing a connection with cluster categories.

Contribution

It constructs a polygon-based geometric category that models the syzygy category of dimer tree algebras and proves the conjecture in specific cases, linking algebraic and geometric structures.

Findings

The geometric category is conjectured to be equivalent to the syzygy category.

The conjecture is proved when all chordless cycles are triangles.

The syzygy category is shown to be equivalent to the 2-cluster category of type A.

Abstract

In this article, we consider the class of 2-Calabi-Yau tilted algebras that are defined by a quiver with potential whose dual graph is a tree. We call these algebras \emph{dimer tree algebras} because they can also be realized as quotients of dimer algebras on a disc. These algebras are wild in general. For every such algebra $B$, we construct a polygon $\mathcal{S}$ with a checkerboard pattern in its interior that gives rise to a category $\text{Diag}(\mathcal{S})$. The indecomposable objects of $\text{Diag}(\mathcal{S})$ are the 2-diagonals in $\mathcal{S}$, and its morphisms are given by certain pivoting moves between the 2-diagonals. We conjecture that the category $\text{Diag}(\mathcal{S})$ is equivalent to the stable syzygy category over the algebra $B$, such that the rotation of the polygon corresponds to the shift functor on the syzygies. In particular, the number of indecomposable syzygies is finite and the projective resolutions are periodic. We prove the conjecture in the special case where every chordless cycle in the quiver is of length three. As a consequence, we obtain an explicit description of the projective resolutions. Moreover, we show that the syzygy category is equivalent to the 2-cluster category of type $\mathbb{A}$, and we introduce a new derived invariant for the algebra $B$ that can be read off easily from the quiver.