Derived equivalences from mutations of quivers with potential

By B. Keller, Dong Ping Yang. 2009

265 citations recorded by the source corpus.

Abstract We show that Derksen–Weyman–Zelevinsky's mutations of quivers with potential yield equivalences of suitable 3-Calabi–Yau triangulated categories. Our approach is related to that of Iyama–Reiten and ‘Koszul dual’ to that of Kontsevich–Soibelman. It improves on previous work by Vitoria. In Appendix A, the first-named author studies pseudo-compact derived categories of certain pseudo-compact dg algebras.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3719761818b6d05f8389e389ca37b9e09a3a7efe

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Citations in the corpus

Quadratic differentials as stability conditions (2013, background): “Gluing the stability conditions obtained from all these chambers using the Keller-Yang equivalences Φ± referred to above eventually leads to a proof of Theorem 1.2.”
Cluster categories for marked surfaces: punctured case (2013, background): “of the morphisms factoring through addTΓ (see [1, Theorem 3.5] and [20, §2.1, Proposition (c)]). 4 QIU AND ZHOU For a vertex i of Q, let µi(Q,W) be the mutation of (Q,W) at i in the sense of [10]. By [21], there exists a canonical triangulated equivalence (1) µei : C(Γ(µi(Q,W))) ≃ C(Γ(Q,W)). 2.2. Qu”
Linear independence of cluster monomials for skew-symmetric cluster algebras (2012, background): “Recall from [25] the functor Φi : Cμi(Q,W ) → CQ,W which sends the direct factor (Γμi(Q,W ))j of Γμi(Q,W ) to Γj if i 6= j and to a cone of the natural”
Triangulated Categories: Cluster algebras, quiver representations and triangulated categories (2008, unknown): “es of length ≤2. Let i be a vertex of Q and Γ′ the Ginzburg algebra of the mutated quiver with potential µi(Q,W). The following theorem improves on Vito´ria’s [126], cf. also [108] [18]. Theorem 9.2 ([95]). There is an equivalence of derived categories F : D(Γ′) →D(Γ) which takes the dg modules P′ j”
Generic bases for cluster algebras from the cluster category (2011, background): “On the other hand, there is a derived equivalence μ−i : CQ,W → Cμi(Q,W ) [36] sending the object Γ to the Ginzburg dg algebra Γ of μi(Q,W ).”
Generic bases for cluster algebras and the Chamber Ansatz (2010, methodology): “ Amiot [A] we have the generalized cluster category as the triangulated quotient CT := Dperf(GT )/Df.d.(GT ). It follows from [ART] that Cw ∼= CV as triangulated categories, and then from [BIRSm] and [KY] that CV ∼= CT. Next, denote by F ⊆ Dperf(GT ) the subcategory which consists of the cones of ma”
Discreteness of silting objects and t ‐structures in triangulated categories (2017, background): “Let Γ̂(Q,W ) be the complete Ginzburg dg algebra associated with the quiver with potential (Q,W ), see [33, 49].”
Weight structures and simple dg modules for positive dg algebras (2010, background): “Important examples of these are the Ginzburg dg algebras associated to quivers with potential [14] [22].”
Silting objects, simple-minded collections, $t$-structures and co-$t$-structures for finite-dimensional algebras (2012, background): “We follow [22, 4, 34], where the dg algebra is not necessarily finite-dimensional.”
Representation theory of strongly locally finite quivers (2013, background): “The latter in turn is useful in the study of the Jacobian algebra of a quiver with potentials, which is endowed with a natural structure of a topological algebra; see [25, 41].”
Quiver algebras as Fukaya categories (2013, background): “It follows from general results of Keller and Yang [26] that there exist distinguished k-linear triangulated equivalences Φ± : D(T1, ǫ) ∼= D(T2, ǫ).”
Representations of Algebras and Related Topics (2011, background): “We refer to [46] for more details on the setup which we now describe.”