Derived equivalences from mutations of quivers with potential

Keller, Bernhard; Yang, Dong

doi:10.1016/j.aim.2010.09.019

Cited by 203 publications

(266 citation statements)

References 43 publications

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“…For M = 1 these transformation rules reduce to the usual mutation tranformation rules discussed in [8,[27][28][29]. If we further restrict to quivers with generic superpotentials where the Ω S become y-independent, then (3.9) shows that the Ω S are also y-independent.…”

Section: T)mentioning

confidence: 99%

The Coulomb Branch Formula for Quiver Moduli Spaces

Manschot¹,

Pioline²,

Sen³

2017

Confluentes Mathematici

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In recent series of works, by translating properties of multi-centered supersymmetric black holes into the language of quiver representations, we proposed a formula that expresses the Hodge numbers of the moduli space of semi-stable representations of quivers with generic superpotential in terms of a set of invariants associated to `single-centered' or `pure-Higgs' states. The distinguishing feature of these invariants is that they are independent of the choice of stability condition. Furthermore they are uniquely determined by the $\chi_y$-genus of the moduli space. Here, we provide a self-contained summary of the Coulomb branch formula, spelling out mathematical details but leaving out proofs and physical motivations.Comment: 24 pages. v2: final version; minor changes, including a new diagra

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Section: T)mentioning

confidence: 99%

The Coulomb Branch Formula for Quiver Moduli Spaces

Manschot¹,

Pioline²,

Sen³

2017

Confluentes Mathematici

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“…As T ′ = ϕ(T), the associated quivers with potential are right equivalent (in particular, the associated quivers are isomorphic in a canonical way), which induces an isomorphism Γ T → Γ ′ T , see [19,Proposition 3.5] for such type of isomorphisms. Consider the corresponding derived equivalence i ϕ : per Γ T → per Γ T ′ which induces an equivalence E ϕ : C (T) ∼ = C (T ′ ).…”

Section: Lemma 22 There Is a Canonical Bijectionmentioning

confidence: 99%

Tagged mapping class groups: Auslander–Reiten translation

Brüstle

Qiu

2015

Math. Z.

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Abstract. We give a geometric realization, the tagged rotation, of the AR-translation on the generalized cluster category associated to a surface S with marked points and non-empty boundary, which generalizes Brüstle-Zhang's result for the puncture free case.As an application, we show that the intersection of the shifts in the 3-Calabi-Yau derived category D(ΓS) associated to the surface and the corresponding Seidel-Thomas braid group of D(ΓS) is empty, unless S is a polygon with at most one puncture (i.e. of type A or D).

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“…(b) According to [73,Theorem A.17], the dg algebra Γ is (topologically) homolog- Example. Let Q be the quiver 1 α G G 2 of type A 2 .…”

Section: Quivers With Potentialmentioning

confidence: 99%

Ordered exchange graphs

Brüstle¹,

Yang²

2014

Advances in Representation Theory of Algebras

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Abstract. The exchange graph of a cluster algebra encodes the combinatorics of mutations of clusters. Through the recent "categorifications" of cluster algebras using representation theory one obtains a whole variety of exchange graphs associated with objects such as a finite-dimensional algebra or a differential graded algebra concentrated in nonpositive degrees. These constructions often come from variations of the concept of tilting, the vertices of the exchange graph being torsion pairs, t-structures, silting objects, support τ -tilting modules and so on. All these exchange graphs stemming from representation theory have the additional feature that they are the Hasse quiver of a partial order which is naturally defined for the objects. In this sense, the exchange graphs studied in this article can be considered as a generalization or as a completion of the poset of tilting modules which has been studied by Happel and Unger. The goal of this article is to axiomatize the thus obtained structure of an ordered exchange graph, to present the various constructions of ordered exchange graphs and to relate them among each other.

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Derived equivalences from mutations of quivers with potential

Cited by 203 publications

References 43 publications

The Coulomb Branch Formula for Quiver Moduli Spaces

The Coulomb Branch Formula for Quiver Moduli Spaces

Tagged mapping class groups: Auslander–Reiten translation

Ordered exchange graphs

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