Abstract
We study the oriented exchange graph EG∘(ΓNQ) of reachable hearts in the finite-dimensional derived category D(ΓNQ) of the CY-N Ginzburg algebra ΓNQΓNQ associated to an acyclic quiver Q . We show that any such heart is induced from some heart in the bounded derived category D(Q)D(Q) via some ‘Lagrangian immersion’ L:D(Q)→D(ΓNQ). We build on this to show that the quotient of EG∘(ΓNQ) by the Seidel–Thomas braid group is the exchange graph CEGN−1(Q)CEGN−1(Q) of cluster tilting sets in the (higher) cluster category CN−1(Q)CN−1(Q). As an application, we interpret Buan–Thomas' coloured quiver for a cluster tilting set in terms of the Ext quiver of any corresponding heart in D(ΓNQ).
| Original language | English |
|---|---|
| Article number | 5143 |
| Pages (from-to) | 1106-1154 |
| Number of pages | 49 |
| Journal | Advances in Mathematics |
| Volume | 285 |
| Early online date | 3 Sept 2015 |
| DOIs | |
| Publication status | Published - 5 Nov 2015 |
Keywords
- Calabi-Yau category
- Exchange graph
- Higher cluster theory
- T-Structure
