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Exchange graphs and Ext quivers

Alastair King University of Bath Yu Qiu Tsinghua University

Representation Theory mathscidoc:1906.01002

Advances in Mathematics, 285, 1106–1154, 2015
We study the oriented exchange graph EG(Gamma_N Q) ofreachable hearts in the finite-dimensional derived category D(Gamma_N Q) of the CY-N Ginzburg algebra Gamma_N Q 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)->D(Gamma_N Q). We build on this to show that the quotient of EG(Gamma_N Q) by the Seidel–Thomas braid group is the exchange graph CEG_N−1(Q)of cluster tilting sets in the (higher) cluster category C_N−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(Gamma_N Q).
Exchange graph t-Structure Calabi–Yau category Higher cluster theory
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@inproceedings{alastair2015exchange,
  title={Exchange graphs and Ext quivers},
  author={Alastair King, and Yu Qiu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190625202158654956358},
  booktitle={Advances in Mathematics},
  volume={285},
  pages={1106–1154},
  year={2015},
}
Alastair King, and Yu Qiu. Exchange graphs and Ext quivers. 2015. Vol. 285. In Advances in Mathematics. pp.1106–1154. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190625202158654956358.
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