Mutation of cluster-tilting objects and potentials

By Aslak Bakke Buan, Osamu Iyama, I. Reiten, D. Smith. 2008

151 citations recorded by the source corpus.

We prove that mutation of cluster-tilting objects in triangulated 2-Calabi-Yau categories is closely connected with mutation of quivers with potentials. This gives a close connection between 2-CY-tilted algebras and Jacobian algebras associated with quivers with potentials. We show that cluster-tilted algebras are Jacobian and also that they are determined by their quivers. There are similar results when dealing with tilting modules over 3-CY algebras. The nearly Morita equivalence for 2-CY-tilted algebras is shown to hold for the finite length modules over Jacobian algebras.

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

View Paper: https://www.semanticscholar.org/paper/c72e0df190ed91e8993feb42aa034c0f2d2bdd28

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Citations in the corpus

Dimer models and cluster categories of Grassmannians (2013, methodology): “In the boundary case, the idea of only considering F-term relations for internal arrows has arisen independently in work of Franco [ 11 , § 6.1] and Buan-Iyama-Reiten-Smith [ 4 , Defn.”
Derived equivalences from mutations of quivers with potential (2009, background): “Keywords: Derived equivalence; Quiver mutation; Cluster category; Quiver with potential; Pseudo-compact algebra”
On cluster algebras with coefficients and 2-Calabi-Yau categories (2007, background): “Large classes of examples of Frobenius categories where the cluster-tilting objects deﬁne a cluster-structure are obtained in [23] and [3]; cf. the survey [24] and Example 5.3 below.”
Quivers with potentials and their representations II: Applications to cluster algebras (2009, result): “It was already established in [3, Theorem 7.1] but the present proof seems to be much simpler.”
Representations of Algebras and Related Topics (2011, background): “s Mw. For any acyclic quiver Q (except An with the linear orientation), there is a triangle equivalence SubΠw ≃CQ when w = c2, where c is the Coxeter element associated with the orientation of Q. See [23], [5] for further results on these categories. This construction and the constructions in Sectio”