Lattices of t-structures and thick subcategories for discrete cluster categories

TL;DR

This paper classifies t-structures and thick subcategories in discrete cluster categories of Dynkin type A, revealing their lattice structures and connections to non-crossing partitions, thus advancing the understanding of categorical and combinatorial structures.

Contribution

It provides a complete classification of t-structures and thick subcategories in these categories and establishes their lattice structures and relation to non-crossing partitions.

Findings

The set of t-structures forms a lattice under inclusion.

The lattice of t-structures is isomorphic to the lattice of non-crossing partitions.

Thick subcategories are closely related to non-crossing partitions.

Abstract

We classify t-structures and thick subcategories in discrete cluster categories $\mathcal{C}(\mathcal{Z})$ of Dynkin type $A$, and show that the set of all t-structures on $\mathcal{C}(\mathcal{Z})$ is a lattice under inclusion of aisles, with meet given by their intersection. We show that both the lattice of t-structures on $\mathcal{C}(\mathcal{Z})$ obtained in this way and the lattice of thick subcategories of $\mathcal{C}(\mathcal{Z})$ are intimately related to the lattice of non-crossing partitions of type $A$. In particular, the lattice of equivalence classes of non-degenerate t-structures on such a category is isomorphic to the lattice of non-crossing partitions of a finite linearly ordered set.