2011/09/30 by Alastair King, Yu Qiu · 1 citation
Mathematics · #math.RT
paper · pdf · doi:10.1016/j.aim.2015.08.017
published as Adv. Math. 285 (2015) pp1106-1154 · Final version, to appear in Adv. Math
arxiv created 2015/05/31 · arxiv updated 2015/09/16
We study the oriented exchange graph \textrmEG^∘(ΓN Q) of reachable hearts in the finite-dimensional derived category D(ΓN Q) of the CY-N Ginzburg algebra ΓNQ associated to an acyclic quiver Q. We show that any such heart is induced from some heart in the bounded derived category D(Q) via some `Lagrangian immersion' L:D(Q)\toD(ΓN Q). We build on this to show that the quotient of \textrmEG^∘(ΓN Q) by the Seidel-Thomas braid group is the exchange graph \textrmCEGN-1(Q) of cluster tilting sets in the (higher) cluster category 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(ΓN Q).