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

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.