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## DERIVED INVARIANTS FOR SURFACE ALGEBRAS

Citations: | 1 - 0 self |

### Citations

521 |
Triangulated categories in the representation theory of finite dimensional algebras
- Happel
- 1988
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Citation Context ...lgebra. In [6] the authors give an interpretation of the AG-invariant of a gentle algebra Λ in terms of the Auslander-Reiten quiver of the category mod Λ̂ where Λ̂ is the repetitive algebra of Λ (see =-=[15]-=- for definition). Roughly speaking AG(Λ) describe the characteristic components of the AR-quiver, that is, the connected components which are tubes (coming from string modules) and of type ZA∞. More p... |

185 | On triangulated orbit categories
- Keller
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Citation Context ... := S[−2]. We say that Λ is τ2-finite if the functor H0(S2) : mod Λ→ D b(Λ)→ mod Λ is nilpotent. For Λ τ2-finite, a cluster category CΛ is defined in [3]. It is the triangulated hull (in the sense of =-=[16]-=-) of the orbit category Db(Λ)/S2, so it comes naturally with a triangle functor. π : Db(Λ)։ Db(Λ)/S2 →֒ CΛ. Moreover the object π(Λ) is a cluster-tilting object. Furthermore two τ2-finite algebras are... |

178 | Quivers with potentials and their representations
- Derksen, Weyman, et al.
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Citation Context ... obtain the following result. Proposition 3.2. Let (S,M) be a surface as in section 2, and ∆ an ideal triangulation of (S,M). For each arc i of ∆ there is a (graded) right equivalence in the sense of =-=[13]-=- (see [2] for the graded version) µLi (Q ∆,W∆, d) ∼ (Qµi(∆),W µi(∆), µLi (d)), where µLi (Q ∆,W∆, d) is the mutation of the graded QP defined in [13] (and in [2] for the graded version). For a graded ... |

132 |
Cluster algebras and triangulated surfaces
- Fomin, Shapiro, et al.
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Citation Context ...ngulation in section 2.1. We give the definition of weight in section 2.2, and in section 2.3 we look at how mutation affects the weight. 2.1. Ideal triangulations. We recall here some definitions of =-=[14]-=-. Let (S,M) be a connected oriented marked surface of genus g with non-empty boundary, which is not a disc. The boundary ∂S = B1 ∪ B2 ∪ . . . Bb is the disjoint union of b components Bi each of which ... |

120 | Cluster categories for algebras of global dimension 2 and quivers with potential - Amiot |

86 | Derived equivalences from mutations of quivers with potential
- Keller, Yang
- 2011
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Citation Context ...nsion ≤ 2. A cluster category C(Q,W ) was associated to an arbitrary Jacobi-fintie quiver with potential in [3]. For a surface (S,M) the category C(Q∆,W∆) does not depend on the ideal triangulation ∆ =-=[19, 18]-=-. The cluster category of a marked surface without punctures has been studied in [8], where the Auslander-Reiten structure of C(S,M) was described using a geometric model. In [3] a cluster category is... |

47 | Gentle algebras arising from surface triangulations
- Assem, Brüstle, et al.
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Citation Context ... S be an oriented Riemann surface with non empty boundary, andM be a set of marked points on the boundary of S. A quiver with potential (Q∆,W∆) is associated to each ideal triangulation ∆ of (S,M) in =-=[5]-=-. The potential consists of the sum of the oriented three cycles in the quiver which correspond to internal triangles of the triangulation. The Jacobian algebra of (Q∆,W∆), denoted Jac(Q∆,W∆) is a fin... |

45 | Topology and geometry, volume 139 of Graduate Texts in Mathematics - Bredon - 1993 |

35 | Deformed Calabi-Yau completions - Keller |

21 | Cluster equivalence and graded derived equivalence
- Amiot, Oppermann
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Citation Context ... wǫ(d) up to homeomorphism of the surface. Surface algebras are gentle and of global dimension ≤ 2, and any surface algebras coming from the same surface (S,M) are cluster equivalent, in the sense of =-=[2]-=-. To prove that the weight is a derived invariant we strongly use results about cluster equivalent algebras from [2]. Furthermore we also show that for surface algebras the invariant defined for gentl... |

9 | On the cluster category of a marked surface without punctures’, Alg. and Num. Theory 4
- Brüstle, Zhang
- 2011
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Citation Context ... with potential in [3]. For a surface (S,M) the category C(Q∆,W∆) does not depend on the ideal triangulation ∆ [19, 18]. The cluster category of a marked surface without punctures has been studied in =-=[8]-=-, where the Auslander-Reiten structure of C(S,M) was described using a geometric model. In [3] a cluster category is also associated to any finite dimensional algebra of global dimension 2 satisfying ... |

8 | Algebras from surfaces without punctures
- David-Roesler, Schiffler
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Citation Context ...DERIVED INVARIANTS FOR SURFACE ALGEBRAS CLAIRE AMIOT AND YVONNE GRIMELAND Abstract. In this paper we study the derived equivalences between surface algebras, introduced by David-Roesler and Schiffler =-=[12]-=-. Each surface algebra arises from a cut of an ideal triangulation of an unpunctured marked Riemann surface with boundary. A cut can be regarded as a grading on the Jacobian algebra of the quiver with... |

6 |
with potentials associated to triangulated surfaces
- Quivers
- 2009
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Citation Context ...nsion ≤ 2. A cluster category C(Q,W ) was associated to an arbitrary Jacobi-fintie quiver with potential in [3]. For a surface (S,M) the category C(Q∆,W∆) does not depend on the ideal triangulation ∆ =-=[19, 18]-=-. The cluster category of a marked surface without punctures has been studied in [8], where the Auslander-Reiten structure of C(S,M) was described using a geometric model. In [3] a cluster category is... |

5 |
Avella-Alaminos and Christof Geiss, Combinatorial derived invariants for gentle algebras
- Diana
(Show Context)
Citation Context ...ariant we strongly use results about cluster equivalent algebras from [2]. Furthermore we also show that for surface algebras the invariant defined for gentle algebras by Avella-Alaminos and Geiss in =-=[6]-=-, is determined by the weight. 1. Introduction In this paper we study derived equivalence for the class of algebras called surface algebras. Surface algebras were introduced by David-Roesler and Schif... |

2 | Algebras of acyclic cluster type: tree type and type A
- Amiot, Oppermann
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Citation Context ...mutation σ ∈ Σb with pσi = pi for any i = 1, . . . , b with d ′(c̄i) = d(c̄σi). Remark 4.3. In the case where g = 0 and b = 2, that is when (S,M) is an annulus, this result has already been proved in =-=[4]-=-. Indeed one easily checks that the weight w(d) defined in [4, Def. 6.16] coincide with d(c1). Moreover by Lemma 2.9 we have d(c1) + d(c2) = 0. By Corollary 4.2 the derived equivalence class of Λ is d... |

1 |
The derived category of surface algebras: genus one case
- Amiot
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Citation Context ...e grading d ′. One direction of this result has been already proved by [10]. Remark 5.8. When g ≥ 1 there are possibly surface algebras with the same AGinvariant which are not derived equivalent (see =-=[1]-=-). 5.2. Auslander-Reiten quiver of the derived category of a surface algebra. In [6] the authors give an interpretation of the AG-invariant of a gentle algebra Λ in terms of the Auslander-Reiten quive... |

1 | Derived equivalence in surface algebras of genus 0 via graded equivalence
- David-Roesler
- 2011
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Citation Context ...hat pσi = pi and with ℓσi = ℓ ′ i where ℓi is the number of cuts on Bi corresponding to the grading d and ℓ′i corresponding to the grading d ′. One direction of this result has been already proved by =-=[10]-=-. Remark 5.8. When g ≥ 1 there are possibly surface algebras with the same AGinvariant which are not derived equivalent (see [1]). 5.2. Auslander-Reiten quiver of the derived category of a surface alg... |

1 |
Algebras From Surfaces and Their Derived Equivalences
- David-Roesler
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Citation Context ...) Db(Λ) ≃ Db(Λ′). (2) There exists an orientation preserving homeomorphism Φ : S → S with wǫ(∆, d) = wǫ(Φ−1(∆′), d′ ◦ Φ) (or equivalently wǫ(∆, d) = wΦ(ǫ)(∆′, d′)). Our result generalizes a result in =-=[11]-=-, in which the author studies the case where ∆ = ∆′ and the genus of S is zero. As mentioned above surface algebras are gentle, see [12]. Avella-Alaminos and Geiss (abbreviated AG) introduced a derive... |