Fractionally Calabi-Yau lattices that tilt to higher Auslander algebras of type A

TL;DR

This paper demonstrates that certain lattices derived from product posets are fractionally Calabi-Yau and are derived equivalent to higher Auslander algebras of type A, linking poset theory to advanced algebraic structures.

Contribution

It establishes the fractional Calabi-Yau property and derived equivalence of these lattices to higher Auslander algebras, confirming a conjecture connecting posets to Fukaya-Seidel categories.

Findings

Lattices of order ideals of product posets are fractionally Calabi-Yau.

These lattices are derived equivalent to higher Auslander algebras of type A.

Supports a conjecture linking posets to Fukaya-Seidel categories.

Abstract

We prove that the bounded derived category of the lattice of order ideals of the product of two ordered chains is fractionally Calabi-Yau. We also show that these lattices are derived equivalent to higher Auslander algebras of type A. The proofs involve the study of intervals of the poset that have resolutions described with antichains having rigid properties. These two results combined corroborate a conjecture by Chapoton linking posets to Fukaya-Seidel Categories.