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

Abstract

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