Mutation of cluster-tilting objects and potentials

Abstract

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.

41 pages. In the previous version, there was a mistake in the proof of old Theorem 5.1 asserting compatibility of mutation of QPs and cluster-tilting mutation. In the present version, the proof is fixed by adding assumptions (see new Theorems 5.3 and 5.3), which are satisfied by large classes of examples, in particular those investigated in section 6

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• Calabi-Yau algebras
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• Cluster algebra structures and semicanonical bases for unipotent groups
• Deformed Calabi-Yau Completions
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Cited by in corpus (6)

• Deformed Calabi-Yau Completions
• Derived equivalences from mutations of quivers with potential
• On cluster algebras with coefficients and 2-Calabi-Yau categories
• The GL(2) McKay Correspondence
• Endomorphism algebras of maximal rigid objects in cluster tubes
• Cluster tilted algebras with a cyclically oriented quiver