## Cluster mutation via quiver representations

Venue: | Comment. Math. Helv |

Citations: | 45 - 15 self |

### BibTeX

@ARTICLE{Buan_clustermutation,

author = {Aslak Bakke Buan and Robert J. Marsh and Idun Reiten},

title = {Cluster mutation via quiver representations},

journal = {Comment. Math. Helv},

year = {},

pages = {2008}

}

### OpenURL

### Abstract

Abstract. Matrix mutation appears in the definition of cluster algebras of Fomin and Zelevinsky. We give a representation theoretic interpretation of matrix mutation, using tilting theory in cluster categories of hereditary algebras. Using this, we obtain a representation theoretic interpretation of cluster mutation in case of acyclic cluster algebras.

### Citations

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Citation Context ...odules with no projective direct summands. This functor has the property that τC = A when there is an almost split sequence 0 → A → B → C → 0. We have the AR-formula Hom(X, τY ) ≃ D Ext 1 (Y, X), see =-=[AR]-=-, valid for any module Y with no projective non-zero direct summands. Here D denotes the ordinary duality for finite-dimensional algebras. The bounded derived category of Λ, denoted D b (mod Λ) is a t... |

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Citation Context ...unctor mentioned above. We have almost split triangles A → B → C → in D b (mod H), where τC = A, for each indecomposable C in Db (mod H). We also have the formula HomD(X, τY ) ≃ D Ext 1 D (Y, X), see =-=[H]-=-. Here D denotes the ordinary duality for finite-dimensional algebras. Let H be a hereditary finite-dimensional algebra. Then a module T in modH is called a tilting module if Ext 1 H(T, T) = 0 and T h... |

186 |
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Citation Context ...ilting module if Ext 1 H(T, T) = 0 and T has,4 BUAN, MARSH, AND REITEN up to isomorphism, n indecomposable direct summands. The endomorphism ring EndH(T) op is called a tilted algebra. See [ARS] and =-=[R]-=- for further information on the representation theory of finite dimensional algebras and almost split sequences. 1.2. Approximations. Let E be an additive category, and X a full subcategory. Let E be ... |

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Citation Context ... category, and M is a thick subcategory of T . Then, for any map f in T we have LM(f) = 0 if and only if f factors through an object in M. We will need the following result of Verdier [V, Ch. 2, 5-3],=-=[V2]-=-: Proposition 2.2. Let T be a triangulated category with thick subcategory M, and let TM be the quotient category with quotient functor LM : T → TM. Fix an object Y of T . Then every morphism from an ... |

91 | From triangulated categories to cluster algebras - Caldero, Keller |

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Citation Context ...s. Also new links between this field and other fields of mathematics can be expected, having in mind the influence of cluster algebras on other areas.CLUSTER MUTATION VIA QUIVER REPRESENTATIONS 3 In =-=[CCS1]-=- an alternative description of the cluster category is given for type A. The cluster category was also the motivation for a Hall-algebra type definition of a cluster algebra of finite type [CC, CK]. T... |

80 | On triangulated orbit categories
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Citation Context ...e dimensional path algebra KQ. The corresponding cluster category C is defined in [BMRRT] as a certain quotient of the bounded derived category of KQ, which is shown to be canonically triangulated by =-=[K]-=-. In [BMRRT] (cluster-)tilting theory is developed in C, with emphasis on connections to cluster algebras. The analogs of clusters are (cluster-)tilting objects, and the analogs of cluster variables a... |

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Citation Context ... subcategory of T closed under taking direct summands. When M is a thick subcategory of T , one can form a new triangulated category TM = T / M, and there is a canonical exact functor LM: T → TM. See =-=[Ric]-=- and [V] for details. For every M ′ in M, we have LM(M ′ ) = 0, and LM is universal with respect to this property. We also have the following. Lemma 2.1. Assume T is a triangulated category, and M is ... |

69 | Tilting in abelian categories and quasitilted algebras - Happel, Reiten, et al. - 1996 |

64 |
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Citation Context ...called a tilting module if Ext 1 H(T, T) = 0 and T has,4 BUAN, MARSH, AND REITEN up to isomorphism, n indecomposable direct summands. The endomorphism ring EndH(T) op is called a tilted algebra. See =-=[ARS]-=- and [R] for further information on the representation theory of finite dimensional algebras and almost split sequences. 1.2. Approximations. Let E be an additive category, and X a full subcategory. L... |

64 | algebras I: Foundations
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Citation Context ...utation in case of acyclic cluster algebras. Introduction This paper was motivated by the interplay between the recent development of the theory of cluster algebras defined by Fomin and Zelevinsky in =-=[FZ1]-=- (see [Z] for an introduction) and the subsequent theory of cluster categories and cluster-tilted algebras [BMRRT, BMR]. Our main results can be considered to be interpretations within cluster categor... |

60 | Cluster algebras as Hall algebras of quiver representations - Caldero, Chapoton |

59 | Y -systems and generalized associahedra
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Citation Context ...ations of the Stasheff polytopes (associahedra) to other Dynkin types [CFZ, FZ3]; consequently there are likely to be interesting links with operad theory. They have been used to provide the solution =-=[FZ3]-=- of a conjecture of Zamolodchikov concerning Y -systems, a class of functional relations important in the theory of the thermodynamic Bethe Ansatz, as well as solution [FZ4] of various recurrence prob... |

56 | Cluster-tilted algebras are Gorenstein and stably
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Citation Context ...path-algebras of such quivers by non-zero admissible ideals are never cluster-tilted. We omit our original proof of this fact since it is a consequence of the more general (recently proven) fact from =-=[KR]-=- that any cluster-tilted algebra is either hereditary or of infinite global dimension. Hence, since Qrst has no oriented cycles, it follows that if it is the quiver of a cluster-tilted algebra, there ... |

53 |
Large modules over Artin algebras
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Citation Context ... If there is an object X in X, and a map f : X → E, such that for every object X ′ in X and every map g: X ′ → E, there is a map h: X ′ → X, such that g = fh, then f is called a right X-approximation =-=[AS]-=-. The approximation map f : X → E is called minimal if no non-zero direct summand of X is mapped to 0. The concept of (minimal) left X-approximations is defined dually. If there is a field K, such tha... |

49 | Cluster ensembles, quantization and the dilogarithm
- Fock, Goncharov
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Citation Context ... of the distinguished generators of a cluster algebra play an important role. Cluster algebras have also been related to Poisson geometry [GSV1], Teichmüller spaces [GSV2], positive spaces and stacks =-=[FG]-=-, dual braid monoids [BES], ad-nilpotent ideals of a Borel subalgebra of a simple Lie algebra [P] as well as representation theory, see amongst others [BMRRT, BMR, CC, CCS1, CCS2, MRZ]. A cluster alge... |

47 |
The dual braid monoid, Ann
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Citation Context ...erators of a cluster algebra play an important role. Cluster algebras have also been related to Poisson geometry [GSV1], Teichmüller spaces [GSV2], positive spaces and stacks [FG], dual braid monoids =-=[BES]-=-, ad-nilpotent ideals of a Borel subalgebra of a simple Lie algebra [P] as well as representation theory, see amongst others [BMRRT, BMR, CC, CCS1, CCS2, MRZ]. A cluster algebra (without coefficients)... |

46 |
Catégories dérivées, état 0
- Verdier
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Citation Context ...ry of T closed under taking direct summands. When M is a thick subcategory of T , one can form a new triangulated category TM = T / M, and there is a canonical exact functor LM: T → TM. See [Ric] and =-=[V]-=- for details. For every M ′ in M, we have LM(M ′ ) = 0, and LM is universal with respect to this property. We also have the following. Lemma 2.1. Assume T is a triangulated category, and M is a thick ... |

45 | Polytopal realizations of generalized associahedra - Chapoton, Fomin, et al. |

37 | Cluster algebras and Poisson geometry
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Citation Context ...alised Somos sequences. Here the remarkable Laurent properties of the distinguished generators of a cluster algebra play an important role. Cluster algebras have also been related to Poisson geometry =-=[GSV1]-=-, Teichmüller spaces [GSV2], positive spaces and stacks [FG], dual braid monoids [BES], ad-nilpotent ideals of a Borel subalgebra of a simple Lie algebra [P] as well as representation theory, see amon... |

34 | Cluster algebras and Weil-Petersson forms
- Gekhtman, Shapiro, et al.
(Show Context)
Citation Context ...e the remarkable Laurent properties of the distinguished generators of a cluster algebra play an important role. Cluster algebras have also been related to Poisson geometry [GSV1], Teichmüller spaces =-=[GSV2]-=-, positive spaces and stacks [FG], dual braid monoids [BES], ad-nilpotent ideals of a Borel subalgebra of a simple Lie algebra [P] as well as representation theory, see amongst others [BMRRT, BMR, CC,... |

27 |
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Citation Context ...ed to provide the solution [FZ3] of a conjecture of Zamolodchikov concerning Y -systems, a class of functional relations important in the theory of the thermodynamic Bethe Ansatz, as well as solution =-=[FZ4]-=- of various recurrence problems involving Laurent polynomials, including a conjecture of Gale and Robinson on the integrality of generalised Somos sequences. Here the remarkable Laurent properties of ... |

26 | A characterization of hereditary categories with tilting object - Happel |

25 |
Ad-nilpotent ideals of a Borel subalgebra: generators and duality
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Citation Context ...e also been related to Poisson geometry [GSV1], Teichmüller spaces [GSV2], positive spaces and stacks [FG], dual braid monoids [BES], ad-nilpotent ideals of a Borel subalgebra of a simple Lie algebra =-=[P]-=- as well as representation theory, see amongst others [BMRRT, BMR, CC, CCS1, CCS2, MRZ]. A cluster algebra (without coefficients) is defined via a choice of free generating set x = {x1, . . .,xn} in t... |

23 |
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Citation Context ...wing properties: B1) E is a complement of M (that is, E ∐ M is a tilting module). B2) For any module X in modH, we have that Ext 1 H (M, X) = 0 implies also Ext 1 H(E, X) = 0. This is due to Bongartz =-=[B]-=-, and the module E is sometimes called the Bongartzcomplement of M. For a module X in modH, we denote by X ⊥ the full subcategory (X, Y ) = 0. If T is a tilting module, of mod H with objects Y satisfy... |

21 |
Zelevinsky A., Generalized associahedra via quiver representations
- Marsh, Reineke
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Citation Context ...T]. An alternative description in type A was given in [CCS1]. They were motivated by the theory of cluster algebras of Fomin and Zelevinsky [FZ1, FZ2], and their connections to quiver representations =-=[MRZ]-=-. In [BMRRT] a tilting theory in cluster categories was developed (actually in the more general context of cluster categories for hereditary categories with a tilting object), and the associated algeb... |

19 |
Todorov G. Tilting theory and cluster combinatorics, preprint math.RT/0402054
- Buan, Marsh, et al.
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Citation Context ...d of the matrix mutation and multiplication exchange rule for cluster variables. For the case of acyclic cluster variables so-called cluster categories were introduced as a candidate for such a model =-=[BMRRT]-=-. Skew-symmetric matrices are in one-one correspondence with finite quivers with no loops or cycles of length two, and the corresponding cluster algebra is called acyclic if there is a seed (x, B) suc... |

16 |
Zelevinsky: Cluster algebras. III. Upper bounds and double Bruhat cells
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Citation Context ...sztig and total positivity for algebraic groups. It was also expected that cluster algebras should model the classical and quantised coordinate rings of varieties associated to algebraic groups — see =-=[BFZ]-=- for an example of this phenomenon (double Bruhat cells). Cluster algebras have been used to define generalisations of the Stasheff polytopes (associahedra) to other Dynkin types [CFZ, FZ3]; consequen... |

11 | On the set of tilting objects in hereditary categories. Representations of algebras and related topics - Happel, Unger - 2005 |

8 |
Tame algebras and integral quadratic forms, Lect
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Citation Context ...t 1 (T, T) = 0 and T has, up to isomorphism, n indecomposable direct summands. The endomorphism ring EndH(T) op is called a tilted algebra.CLUSTER MUTATION VIA QUIVER REPRESENTATIONS 3 See [ARS] and =-=[R]-=- for further information on the representation theory of finite dimensional algebras and almost split sequences. 1.2. Approximations. Let E be an additive category, and X a full subcategory. Let E be ... |

7 |
Piecewise hereditary algebras
- Happel, Rickard, et al.
- 1988
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Citation Context ...T ⊥ = Fac T, where FacT is the full subcategory of all modules that are factors of objects in addT. Note that B2) can be reformulated as M ⊥ = (M ∐ E) ⊥ . The following result can be found in [H] and =-=[HRS]-=-. Proposition 2.3. (a) Assume M is an indecomposable non-projective H-module with Ext 1 H(M, M) = 0, and let E be the complement as above. Then the endomorphism ring H ′ = EndH(E) op is hereditary, an... |

5 |
Representations of wild quivers, in Representation Theory of Algebras and Related
- Kerner
- 1996
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Citation Context ...n-zero. This contradicts HomH(U, τ2U) = 0. Now assume Ext 1 H(U, U) = 0. Then by [Ho], U is quasi-simple. Thus, there is an almost split sequence 0 → τU → V → U → 0, where V is indecomposable, and by =-=[Ke2]-=- we have EndH(V ) ≃ K, while Ext 1 H(V, V ) ̸= 0. Applying HomH(U, ) to the almost split sequence, we obtain the exact sequence HomH(U, τU) → HomH(U, V ) → HomH(U, U) → Ext 1 H(U, τU) Since HomH(U, U)... |

4 |
Andrei Cluster algebras: notes for the CDM-03 conference. Current developments in mathematics, 2003, 1 – 34, Int
- Fomin, Zelevinsky
- 1965
(Show Context)
Citation Context ...w Theorem 5.1 gives such a connection. In order to formulate our result we first need to give a short introduction to a special type of cluster algebras [FZ1], relevant to our setting [BFZ]. See also =-=[FZ2]-=- for an overview of the theory of cluster algebras. Let F = Q(u1, . . . , un) be the field of rational functions in indeterminates u1, . . . , un, let x = {x1, . . .,xn} ⊂ F be a transcendence basis o... |

3 |
Coxeter functors without diagrams Trans
- Auslander, Platzeck, et al.
- 1979
(Show Context)
Citation Context ...to H, where H is the hereditary algebra considered as a tilting object in CH. If k is a source or a sink in the quiver of a hereditary algebra, then mutation at k coincides with so-called APR-tilting =-=[APR]-=- (see [BMR]), and the quiver of the mutated algebra δk(H) is obtained by reversing all arrows ending or starting in k. Lemma 3.1. The cluster-tilted algebras of rank at most 2 are hereditary. Proof. T... |

3 |
Tilted algebras Trans
- Happel, Ringel
- 1982
(Show Context)
Citation Context ...HomH(U, τ 2 U) ̸= 0. Proof. Assume first Ext 1 H (U, U) ̸= 0. By the AR-formula, then also HomH(U, τU) ̸= 0. Assume now HomH(U, τ2U) = 0. Then also Ext 1 H (τU, U) = 0 and, by the Happel-Ringel lemma =-=[HR]-=-, a non-zero map f : U → τU is either surjective or injective. In either case, g = τ(f) ◦ f : U → τ2U is non-zero. This contradicts HomH(U, τ2U) = 0. Now assume Ext 1 H(U, U) = 0. Then by [Ho], U is q... |

3 |
Noetherian hereditary categories satisfying Serre duality
- Reiten, Bergh
(Show Context)
Citation Context ...he next lemma, since it has been generalised by Keller, with a simpler proof [K]. Note that the existence of minimal left almost split maps is equivalent to the existence of a left Serre functor G by =-=[RV]-=-, and that G = τ −1 [−1]. Lemma 2.14. Let X be an indecomposable object in D0 ⊂ D. Then ˜ X is indecomposable and ˜ τ −1 D −1 X ≃ τD ′ ˜ X. Let Tx be an indecomposable direct summand in T, not isomorp... |

3 | Cluster algebras from cluster categories, preprint math.RT/0410187
- Caldero, Chapoton
- 2004
(Show Context)
Citation Context ...ster-)tilting object in CH, called cluster-tilted algebras, were investigated [BMR]. The cluster category has also been used to give a Hall-algebra type definition of a cluster algebra of finite type =-=[CC]-=-. Given a cluster-tilted algebra Γ = EndCH(T) op , with T = T1 ∐ · · · ∐ Tn a direct sum of n nonisomorphic indecomposable objects Ti in CH, there is a unique indecomposable object T ∗ i ̸≃ Ti in CH, ... |

2 |
Cluster algebras associated with extended Dynkin quivers, preprint math.RT/0507113
- Buan, Reiten
- 2005
(Show Context)
Citation Context ...algebras arising from a given cluster category are exactly the quivers corresponding to the exchange matrices of the associated cluster algebra. This has further applications to cluster algebras (see =-=[BR]-=-). Another main result of this paper is an interpretation within cluster categories of the exchange multiplication rule of an (acyclic) cluster algebra. So, together with the results from [BMRRT], all... |

2 |
with relations and cluster-tilted algebras, preprint math.RT/0411238
- Caldero, Chapoton, et al.
- 2004
(Show Context)
Citation Context ...h the connections with cluster algebras mentioned in the introduction follow. An independent proof of Theorem 1.3 in the case of finite representation type is given by Caldero, Chapoton and Schiffler =-=[CCS2]-=-. 2. Factors of cluster-tilted algebras In this section, our main result is that for any cluster-tilted algebra Γ, and any primitive idempotent e, the factor-algebra Γ/ΓeΓ is in a natural way also a c... |

2 |
Modules without self-extensions and Nakayama’s conjecture
- Hoshino
- 1984
(Show Context)
Citation Context ...l lemma [HR], a non-zero map f : U → τU is either surjective or injective. In either case, g = τ(f) ◦ f : U → τ2U is non-zero. This contradicts HomH(U, τ2U) = 0. Now assume Ext 1 H(U, U) = 0. Then by =-=[Ho]-=-, U is quasi-simple. Thus, there is an almost split sequence 0 → τU → V → U → 0, where V is indecomposable, and by [Ke2] we have EndH(V ) ≃ K, while Ext 1 H(V, V ) ̸= 0. Applying HomH(U, ) to the almo... |

2 |
Exceptional components of wild hereditary algebras
- Kerner
- 1992
(Show Context)
Citation Context ...f. In section 3 some consequences of this are given. In section 4 we prepare for the proof of our main result. This involves studying cluster-tilted algebras of rank 3, and a crucial result of Kerner =-=[Ke]-=- on hereditary algebras. The main result is proved in section 5, while section 6 deals with the connection to cluster algebras, including necessary background. The results of this paper have been pres... |

2 |
algebras: notes for 2004 IMCC (Chonju, Korea, August 2004), preprint math.RT/0407414, (2004) Institutt for matematiske fag, Norges teknisk-naturvitenskapelige universitet, N7491
- Cluster
(Show Context)
Citation Context ...case of acyclic cluster algebras. Introduction This paper was motivated by the interplay between the recent development of the theory of cluster algebras defined by Fomin and Zelevinsky in [FZ1] (see =-=[Z]-=- for an introduction) and the subsequent theory of cluster categories and cluster-tilted algebras [BMRRT, BMR]. Our main results can be considered to be interpretations within cluster categories of im... |