### Citations

280 | Stability structures, motivic Donaldson-Thomas invariants and cluster transformations,” 0811.2435
- Kontsevich, Soibelman
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Citation Context ... the mutation µk(Q,W ) is well-defined. Due to the result by Keller and Yang, (Q,W ) and µk(Q,W ) provide the same derived category with different t-structures ([KY, Kelb]). Kontsevich and Soibelman (=-=[KS]-=-) observed that the cluster transformation appears in the transformation formula of non-commutative Donaldson-Thomas invariants under a mutation. In this paper, generalizing their observation, we prov... |

239 |
Cluster algebras I
- Fomin, Zelevinsky
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Citation Context ...±) = { x(k),i,± i 6= k, x(k),k,±(1 + (yk,±) −1) i = k. (0.4) This recovers the results in [KS, pp143]. Composition of cluster transformations Cluster algebras were introduced by Fomin and Zelevinsky (=-=[FZ02]-=-) in their study of dual canonical bases and total positivity in semi-simple groups. Although the initial aim has not been established, it has been discovered that the theory of cluster algebras has m... |

199 | A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on K3 fibrations - Thomas |

181 | Gromov-Witten theory and DonaldsonThomas theory II, - Maulik, Nekrasov, et al. - 2006 |

177 | Quivers with potentials and their representations. - Derksen, Weyman, et al. - 2008 |

144 | A theory of generalized Donaldson-Thomas invariants - Joyce, Song |

135 | Cluster algebras as Hall algebras of quiver representations
- Caldero, Chapoton
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Citation Context ...ariables xk,i,± and yk,i,± on the left hand side of the equations does make sense since we have identified the two tori TQk,± and TQ,±. 5 In the case of a quiver of finite type, Caldero and Chapoton (=-=[CC06]-=-) described a composition of cluster transformations in terms of quiver Grassmannians of the original quiver. This result is generalized by many people (see the references in [Pla] for example). Final... |

106 | Cluster algebras, quiver representations and triangulated categories,” 0807.1960 - Keller |

86 | Derived equivalences from mutations of quivers with potential - Keller, Yang - 2011 |

83 | The Harder–Narashiman system in quantum groups and cohomology of quiver moduli,
- Reineke
- 2003
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Citation Context ... the following powerful method in Donaldson-Thomas theory for 3-Calabi-Yau categories, which originates with Reineke’s computation of the Betti numbers of the spaces of stable quiver representations (=-=[Rei03]-=-): Starting from a simple categorical statement, provide an identity in the motivic Hall algebra. Pushing it out by the integration map, we get a power series identity for the generating functions of ... |

71 |
Donaldson–Thomas invariants via microlocal geometry
- Behrend
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Citation Context .... . 37 Introduction Donaldson-Thomas invariants ([Tho00, MNOP06]) are defined as the topological Euler characteristics (more precisely, the weighted Euler characteristics weighted by Behrend function =-=[Beh09]-=-) of the moduli spaces of sheaves on a CalabiYau 3-fold (more generally, the moduli spaces of objects in a 3-Calabi-Yau category [Sze08, Joy08, KS, JS]). Dominic Joyce introduced the motivic Hall alge... |

61 | Quivers with potentials associated to triangulated surfaces,
- Labardini-Fragoso
- 2009
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Citation Context ...theory of the motivic Hall algebra in the formal setting, we can apply all the arguments in this paper. (2) A typical example of a finite potential is a potential associated to a triangulated surface =-=[LF09]-=-. We will apply the results in this paper for a triangulated surface in [Naga]. 7 (3) It is expected that there is a refinement of the DT theory, which is called the motivic DT theory ([KS, BBS]). Wal... |

49 | Quiver varieties and cluster algebras
- Nakajima
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Citation Context ...and encouragement. In particular, the proof of Theorem 3.4 is due to him; PierreGuy Plamondon who kindly explained the results in his PhD thesis [Pla]; Hiraku Nakajima who explained me his results in =-=[Naka]-=- and encouraged me to promote the result of [KS]; Bernard Leclerc who recommended me to give alternative proofs for the conjectures in [FZ07]; Andrei Zelevinsky who gave me some comments on the prelim... |

43 | Hall algebras and curve-counting invariants, - Bridgeland - 2011 |

43 | Configurations in abelian categories. I. Basic properties and moduli
- Joyce
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Citation Context ...ategories A, Ak, Tk, Tk[−1], T ⊥k and S(r). As we showed in §4.2, we have a Bridgeland’s stability condition (Z,P) on DfdΓ such that P((0, 1]) = A, P((0, φ]) = C for some 0 < φ ≤ 1. By the results in =-=[Joy06]-=-, we get the algebraic moduli stacks MC of objects in C. We put εC := log(1 +MC) := ∑ l≥1 (−1)l l MC ∗ · · · ∗MC ∈ M̂H(C) (6.2) and ε̃C := (L− 1)εC ∈ M̂H(C). Then we have MC = exp(εC) := ∑ l≥1 1 l! εC... |

41 | Motivic degree zero Donaldson-Thomas invariants, - Behrend, Bryan, et al. - 2013 |

37 |
Non-commutative Donaldson-Thomas invariants and the conifold, Geom
- Szendrői
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Citation Context ...have a 3-Calabi-Yau triangulated category (the derived category of Ginzburg’s dg algebra) with a t-structure whose core A is the module category of the Jacobi algebra. It was proposed by B. Szendroi (=-=[Sze08]-=-) to study Donaldson-Thomas theory for the Abelian category A ≃ modJ (non-commutative Donaldson-Thomas theory ). For a vertex i ∈ I, let Pi denote the projective indecomposable J-module corresponding ... |

25 | Cluster characters for cluster categories with infinite-dimensional morphism spaces,
- Plamondon
- 2011
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Citation Context ...dero and Chapoton ([CC06]) described a composition of cluster transformations in terms of quiver Grassmannians of the original quiver. This result is generalized by many people (see the references in =-=[Pla]-=- for example). Finally, Derksen-Weyman-Zelevinsky and Plamondon ([DWZ, Pla]) provided the Caldero-Chapoton type formula for an arbitrary quiver without loops and oriented 2-cycles. In this paper, we p... |

15 | On tropical dualities in cluster algebras
- Nakanishi, Zelevinsky
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Citation Context ...ed as a tropical counterpart of the xvariable, while the c-vector can be viewed as a tropical counterpart of the yvariable. The duality between the g- and the c-vectors is called toropical duality in =-=[NZ]-=- 9. From our view point, the x-variable corresponds to the “projective” Γi and the y-variable corresponds to the simple si, and the toropical duality is a consequence of the duality between {Γi} and {... |

14 |
algebras IV
- Cluster
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Citation Context ... have FZk,i(x) = xk,i · (∑ v e ( Grass(k; i,v) ) · y−v ) . (0.6) where (y)−v = ∏ j(yj) −vj and yj = ∏ i(xi) Q̄(i,j). Application to cluster algebras In [DWZ, Pla], they prove six conjectures given in =-=[FZ07]-=- for cluster algebras associated to quivers 5. In §8.3 and §8.4 we give alternative proofs for them under the assumption that the quiver with principal framing is successively f-mutatable. 6. Let Qpf ... |

14 |
Periodicities in cluster algebras and dilogarithm identities, in Representations of algebras and related topics
- Nakanishi
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Citation Context ...k,i◦ = −tgi,k i = k0, tgi,k +Q(i, k0) · tgk0,k i 6= k0, ε(0) = −, tgi,k +Q(k0, i) · tgk0,k i 6= k0, ε(0) = +. Proof. This is a consequence of (2.3) and Corollary 8.6. 9The duality has proved in =-=[Nakb]-=- for skewsymmetric matrices. For skewsymmetrizable matrices, it is still a conjecture. 36 8.4 g-vectors determine F -polynomials We define ζ′ : MR → Stab(D′Γ). in the same way as §4.1. For θ ∈MR, let ... |

13 | Wall-crossing of the motivic Donaldson–Thomas invariants, arXiv:1103.2922, - Nagao - 2011 |

11 | conditions on triangulated categories - Stability |

11 | Non-commutative Donaldson-Thomas theory and vertex operators,” arXiv:0910.5477 [math.AG - Nagao |

1 |
Donaldson-Thomas theory for triangulated surfaces
- Nagao
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Citation Context ... arguments in this paper. (2) A typical example of a finite potential is a potential associated to a triangulated surface [LF09]. We will apply the results in this paper for a triangulated surface in =-=[Naga]-=-. 7 (3) It is expected that there is a refinement of the DT theory, which is called the motivic DT theory ([KS, BBS]). Wall-crossing phenomena of the motivic DT theory has been studied in [KS, Nagc]. ... |