### 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 Γ_{NQΓNQ} 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 CEG_{N−1(Q)CEGN−1(Q)} of cluster tilting sets in the (higher) cluster category C_{N−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).

Language | English |
---|---|

Article number | 5143 |

Pages | 1106-1154 |

Number of pages | 49 |

Journal | Advances in Mathematics |

Volume | 285 |

Early online date | 3 Sep 2015 |

DOIs | |

Status | Published - 5 Nov 2015 |

### Fingerprint

### Keywords

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

### Cite this

**Exchange graphs and Ext quivers.** / King, Alastair; Qiu, Yu.

Research output: Contribution to journal › Article

*Advances in Mathematics*, vol. 285, 5143, pp. 1106-1154. DOI: 10.1016/j.aim.2015.08.017

}

TY - JOUR

T1 - Exchange graphs and Ext quivers

AU - King,Alastair

AU - Qiu,Yu

PY - 2015/11/5

Y1 - 2015/11/5

N2 - 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).

AB - 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).

KW - Calabi-Yau category

KW - Exchange graph

KW - Higher cluster theory

KW - T-Structure

UR - http://www.scopus.com/inward/record.url?scp=84940823602&partnerID=8YFLogxK

UR - http://dx.doi.org/10.1016/j.aim.2015.08.017

U2 - 10.1016/j.aim.2015.08.017

DO - 10.1016/j.aim.2015.08.017

M3 - Article

VL - 285

SP - 1106

EP - 1154

JO - Advances in Mathematics

T2 - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

M1 - 5143

ER -