Fomin-Zelevinsky mutation and tilting modules over Calabi-Yau algebras

By Osamu Iyama, I. Reiten. 2006

201 citations recorded by the source corpus.

We say that an algebra $\Lambda$ over a commutative noetherian ring $R$ is Calabi-Yau of dimension $d$ ($d$-CY) if the shift functor $[d]$ gives a Serre functor on the bounded derived category of the finite length $\Lambda$-modules. We show that when $R$ is $d$-dimensional local Gorenstein the $d$-CY algebras are exactly the symmetric $R$-orders of global dimension $d$. We give a complete description of all tilting modules of projective dimension at most one for 2-CY algebras, and show that they are in bijection with elements of affine Weyl groups, preserving various natural partial orders. We show that there is a close connection between tilting theory for 3-CY algebras and the Fomin-Zelevinsky mutation of quivers (or matrices). We prove a conjecture of Van den Bergh on derived equivalence of noncommutative crepant resolutions.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/7c2f3a819abb8371747c121fc45b52f21e608da4

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Citations in the corpus

Maximal modifications and Auslander–Reiten duality for non-isolated singularities (2010, methodology): “− if the above functorial isomorphism holds for any X∈Db(ﬂΛ) and Y ∈Kb(projΛ). We know [IR08, 3.1(7)] that Λ is n-CY if and only if it is n-CY− and gl.dimΛ <∞. The next two results can be found in [IR08]; we include both here since we use them extensively. Proposition 2.16. [IR08, 3.10] Let Rbe a”
Cluster algebras and derived categories (2012, background): “• commutative and non commutative algebraic geometry and in particular the study of stability conditions in the sense of Bridgeland [11], Calabi-Yau algebras [64] [75] , Donaldson-Thomas invariants in geometry [76] [89] [88] [124] [137] .”
Poincaré-Birkhoff-Witt deformations of Calabi-Yau algebras (2006, background): “Hochschild cohomology, deformations of algebras, Calabi–Yau algebras, quiver algebras.”
Silting mutation in triangulated categories (2010, background): “Their beautiful theory of ‘tilting mutation’ has a lot of important applications, especially in the study of cluster categories and preprojective algebras [10, 11, 13, 22, 40].”
Cluster structures for 2-Calabi–Yau categories and unipotent groups (2007, methodology): “ of ﬁnite length modules over the completion of the preprojective algebra of a non-Dynkin quiver with no loops. Our main tool is to extend the tilting theory developed for Λ in the noetherian case in [IR]. This turns out to give a large class of 2-CY categories associated with elements in the corres”
Affine Mirković-Vilonen polytopes (2011, result): “Amplifying their previous work [31], Iyama and Reiten, in a joint paper [14] with Buan and Scott, show that Ii is a tilting Λ-module of projective dimension at most one and that EndΛ(Ii) ∼= Λ (at least, when no connected component of (I,E) is of Dynkin type).”
Almost complete cluster tilting objects in generalized higher cluster categories (2010, background): “The author was motivated by the construction of tilting complexes in Section 4 of [17].”
On the Combinatorics of Rigid Objects in 2–Calabi–Yau Categories (2007, methodology): “In this approach, a central rˆole is played by certain triangulated 2-Calabi-Yau categories and by combinatorial invariants associated with their rigid objects (we refer to [ 14 ] [5] for dierent approaches).”
Quivers with potentials and their representations I: Mutations (2007, methodology): “The Jacobian algebras also play an important role in the recent work on Calabi-Yau algebras [4, 25, 26 , 27].”
Calabi–Yau triangulated categories (2008, methodology): “... has as its simples the representable modules associated with the vertices of Q. The heart is in fact equivalent to the category of finite-dimensional modules over the Jacobi algebra associated with (Q,W ). The mutations of the quiver potential (Q,W ) (in the sense of [19]) can be interpreted as ”
Cluster tilting for one-dimensional hypersurface singularities (2007, methodology): “It is interesting to compare with results in [ IR ], where two-dimensional (2-Calabi-Yau) algebras Γ are treated and a bijection between elements in an affine Weyl group and tilting Γ-modules of projective dimension at most one is given.”