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## THE HALL ALGEBRA OF A SPHERICAL OBJECT

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Citations: | 19 - 2 self |

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1165 |
Symmetric functions and Hall polynomials
- Macdonald
- 1995
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Citation Context ...presentations over the Jordan quiver. Then the desired result follows from [20, Proposition 7.1] and the classical result on the Hall algebra of the above hereditary abelian category (cf. for example =-=[16]-=-), with zi,j representing Σ−iMj, where Mj is the indecomposable nilpotent representation of the Jordan quiver of dimension j. (iv) In this case, the triangulated category D is equivalent to the bounde... |

515 | Symmetric functions and Hall polynomials, 2nd ed., With contributions by A. Zelevinsky - Macdonald - 1995 |

189 | On differential graded categories - Keller |

151 | Calabi-Yau algebras
- Ginzburg
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Citation Context ... extension spaces have dimensions encoded by the quiver Q. Alternatively, it may be described as the derived category of dg modules with finite-dimensional total homology over the Ginzburg dg algebra =-=[5]-=- associated with Q and a generic potential. In this note, we consider the case where Q is reduced to a single vertex without any arrows. For this simplest non empty quiver, we classify the objects of ... |

122 | Deriving DG categories
- Keller
- 1994
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Citation Context ... category D(Γ) containing Γ and by Dfd(Γ) the finite dimensional derived category, i.e. the full triangulated subcategory consisting of the dg Γ-modules whose homology has finite total dimension (cf. =-=[7]-=-). The triangulated category Dfd(Γ) is Hom-finite and 3-Calabi-Yau (cf. [8]). Notice that a triangulated category classically generated by a 3sperical object is triangle equivalent to Dfd(Γ). Let [1] ... |

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

94 | Cluster ensembles, quantization and the dilogarithm
- Fock, Goncharov
(Show Context)
Citation Context ...Let us recall the context: To a finite quiver Q without loops and without 2-cycles, one can associate the cluster algebra AQ and the cluster variety XQ (endowed with a Poisson structure), cf. [4] and =-=[3]-=-. If Q does not have oriented cycles, we have at our disposal a very good categorical model for the combinatorics of the cluster algebra AQ, cf. the surveys [1] [11] [12] [9]. In contrast, for the mom... |

86 | Deriving DG categories, Ann - Keller - 1994 |

35 | Some remarks concerning tilting modules and tilted algebras.
- Ringel
- 2007
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Citation Context ...oisson structure), cf. [4] and [3]. If Q does not have oriented cycles, we have at our disposal a very good categorical model for the combinatorics of the cluster algebra AQ, cf. the surveys [1] [11] =-=[12]-=- [9]. In contrast, for the moment, there is no corresponding theory for the cluster variety XQ. Ongoing work by Kontsevich-Soibelman [10], Bridgeland [2] and others shows that there is a close link be... |

30 |
Cluster-tilting theory, Trends in representation theory of algebras and related topics
- Buan, Marsh
- 2006
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Citation Context ... with a Poisson structure), cf. [4] and [3]. If Q does not have oriented cycles, we have at our disposal a very good categorical model for the combinatorics of the cluster algebra AQ, cf. the surveys =-=[1]-=- [11] [12] [9]. In contrast, for the moment, there is no corresponding theory for the cluster variety XQ. Ongoing work by Kontsevich-Soibelman [10], Bridgeland [2] and others shows that there is a clo... |

24 |
Auslander-Reiten theory over topological spaces
- Jørgensen
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Citation Context ... a single spherical object. We first show that this category is indeed well-determined up to a triangle equivalence (Theorem 2.1). Then we classify the objects of TQ (Theorem 4.1, due to P. Jørgensen =-=[7]-=-), compute the Hall algebra of TQ (Theorem 5.1) and establish the link with the cluster variety, which in this case is just a one-dimensional torus (section 6). The Hall algebra of the algebraic trian... |

24 |
les A∞-catégories. Thèse de doctorat, Université Denis Diderot – Paris 7
- Sur
- 2003
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Citation Context ...taking direct factors and containing G. Theorem 2.1. If G classically generates T , there is a triangle equivalence from T to the perfect derived category of B. Proof. According to theorem 7.6.0.6 of =-=[15]-=-, there is a triangle equivalence between T and the perfect derived category of a minimal strictly unital A∞algebra whose underlying graded algebra is B. This A∞-structure is given by linear maps mp :... |

23 |
Stability structures, Donaldson–Thomas invariants and cluster transformations
- Kontsevich, Soibelman
(Show Context)
Citation Context ...binatorics of the cluster algebra AQ, cf. the surveys [1] [11] [12] [9]. In contrast, for the moment, there is no corresponding theory for the cluster variety XQ. Ongoing work by Kontsevich-Soibelman =-=[10]-=-, Bridgeland [2] and others shows that there is a close link between the quantized version [3] of XQ and the Hall algebra [13] of a certain triangulated 3-Calabi-Yau category TQ associated with Q. The... |

23 |
algebras associated to triangulated categories
- Xiao, Xu, et al.
(Show Context)
Citation Context ...of the the Ginzburg dg algebra of type A1. We begin with some reminders on Hall algebras of triangulated categories.THE HALL ALGEBRA OF A SPHERICAL OBJECT 7 4.1. The Hall algebra. We follow [13] and =-=[14]-=-. Let q be a prime power and let Fq be the finite field with q elements. Let C be a Hom-finite triangulated Fq-category with suspension functor Σ, such that for all objects X and Y of C, the space of ... |

20 | Lectures on Hall algebras - Schiffmann |

19 |
algebras, quiver representations and triangulated categories. In Triangulated categories, volume 375
- Cluster
- 2010
(Show Context)
Citation Context ...n structure), cf. [4] and [3]. If Q does not have oriented cycles, we have at our disposal a very good categorical model for the combinatorics of the cluster algebra AQ, cf. the surveys [1] [11] [12] =-=[9]-=-. In contrast, for the moment, there is no corresponding theory for the cluster variety XQ. Ongoing work by Kontsevich-Soibelman [10], Bridgeland [2] and others shows that there is a close link betwee... |

17 |
differential graded categories
- On
(Show Context)
Citation Context ...r Jørgensen for pointing out references [7] and [8]. 2. The triangulated category generated by a spherical object Let k be a field and T a k-linear algebraic triangulated category (cf. section 3.6 of =-=[12]-=- for this terminology). We write Σ for the suspension functor of T . We assume that T is idempotent complete, i.e. each idempotent endomorphism of an object of T comes from a direct sum decomposition.... |

11 |
Tilting theory and cluster algebras, preprint available at www.institut.math.jussieu.fr/ ˜ keller/ictp2006/lecturenotes/reiten.pdf
- Reiten
(Show Context)
Citation Context ...h a Poisson structure), cf. [4] and [3]. If Q does not have oriented cycles, we have at our disposal a very good categorical model for the combinatorics of the cluster algebra AQ, cf. the surveys [1] =-=[11]-=- [12] [9]. In contrast, for the moment, there is no corresponding theory for the cluster variety XQ. Ongoing work by Kontsevich-Soibelman [10], Bridgeland [2] and others shows that there is a close li... |

11 | Degeneration for derived categories
- Jensen, Su, et al.
(Show Context)
Citation Context ...y the surjectivity of φ, the set of products {[M][N]|M, N ∈ Dfd(Γ), H odd M = H even N = 0} generates the Q-vector space H. It remains to prove that these products are linearly independent. Following =-=[6]-=-, we define a partial order ≤∆ on the set of isoclasses of objects in Dfd(Γ) as follows: if X and Y are two objects of Dfd(Γ), then [Y ] ≤∆ [X] if there exists an object Z of Dfd(Γ) and a triangle in ... |

11 | Categorification of acyclic cluster algebras: an introduction, to appear in
- Keller
(Show Context)
Citation Context ...e. the full triangulated subcategory consisting of the dg Γ-modules whose homology is of finite total dimension (cf. [11]). The triangulated category Dfd(Γ) is Hom-finite and 3-Calabi-Yau (cf. [7] or =-=[9]-=-), classically generated by the simple dg Γ-module S = Γ/(tΓ) concentrated in degree 0, which is a spherical object of dimension 3. Let [1] denote the shift functor of the category grmod(Γ) of finitel... |

7 | The Auslander-Reiten quiver of a Poincaré duality space - Jørgensen |

5 |
Xiuping Su and Alexander Zimmermann, Degeneration-like orders for triangulated categories,
- Jensen
- 2005
(Show Context)
Citation Context ...y the surjectivity of φ, the set of products {[M][N]|M, N ∈ Dfd(Γ), H odd M = H even N = 0} generates the Q-vector space H. It remains to prove that these products are linearly independent. Following =-=[6]-=-, we define a partial order ≤∆ on the set of isoclasses of objects in Dfd(Γ) as follows: if X and Y are two objects of Dfd(Γ), then [Y ] ≤∆ [X] if there exists an object Z of Dfd(Γ) and a triangle in ... |

5 | Abgeleitete Kategorien und Matrixprobleme - Burban |

4 |
Degeneration-like orders in triangulated categories
- Jensen, Su, et al.
(Show Context)
Citation Context ...y the surjectivity of φ, the set of products {[M][N]|M, N ∈ Dfd(Γ), H odd M = H even N = 0} generates the Q-vector space H. It remains to prove that these products are linearly independent. Following =-=[8]-=-, cf. also [7], we define a partial order ≤∆ on the set of isoclasses of objects in Dfd(Γ) as follows: if X and Y are two objects of Dfd(Γ), then [Y ] ≤∆ [X] if there exists an object Z of Dfd(Γ) and ... |

1 |
Cluster mutations and Donaldson-Thomas invariants, talk at the workshop on algebraic methods in geometry and physics
- Bridgeland
- 2008
(Show Context)
Citation Context ...r of the perfect derived category has the following shape S[3] S[1] S[−1] S[−3] S[−5] �� �� �� �� �� �� �� �� �� P ◦ ◦ P [3] ◦ ◦ ◦ ◦ P [−2] �� �� ��� �� ��� ��� �� ��� ��� �� ��� ��� ��� ��� ��� �� P =-=[2]-=- �� ◦ ◦ ◦ P [1] ◦ ◦ ◦ P ◦ ��� ��� ��� ��� ��� ��� ��� �� ��� ��� �� ��� �� �� ��� �� P [4] �� ◦ ◦ ◦ ◦ P [−1] ◦ ◦ P [2] ◦ �� �� �� �� �� �� �� �� �� �� �� S[4] S[2] S S[−2] S[−4] S[2] S where the pictu... |

1 | On the center of the derived category, preprint available at http://www.math.rwth-aachen.de∼Matthias.Kuenzer/manuscripts.html - Künzer |

1 | On the center of the derived category - Künzer |

1 |
ensembles, quantization and the dilogarithm,” arXiv:math.AG/0311245
- Fock, Goncharov, et al.
- 2003
(Show Context)
Citation Context ...Let us recall the context: To a finite quiver Q without loops and without 2-cycles, one can associate the cluster algebra AQ and the cluster variety XQ (endowed with a Poisson structure), cf. [5] and =-=[4]-=-. If Q does not have oriented cycles, we have at our disposal a very good categorical model for the combinatorics of the cluster algebra AQ, cf. the surveys [1] [19] [20] [12]. In contrast, for the mo... |