Exchange graphs and Ext quivers

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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 languageEnglish
Article number5143
Pages (from-to)1106-1154
Number of pages49
JournalAdvances in Mathematics
Volume285
Early online date3 Sep 2015
DOIs
Publication statusPublished - 5 Nov 2015

Keywords

  • Calabi-Yau category
  • Exchange graph
  • Higher cluster theory
  • T-Structure

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