Cluster structures for 2-Calabi-Yau categories and unipotent groups

Abstract

We investigate cluster tilting objects (and subcategories) in triangulated 2-Calabi-Yau categories and related categories. In particular we construct a new class of such categories related to preprojective algebras of non Dynkin quivers associated with elements in the Coxeter group. This class of 2-Calabi-Yau categories contains the cluster categories and the stable categories of preprojective algebras of Dynkin graphs as special cases. For these 2-Calabi-Yau categories we construct cluster tilting objects associated with each reduced expression. The associated quiver is described in terms of the reduced expression. Motivated by the theory of cluster algebras, we formulate the notions of (weak) cluster structure and substructure, and give several illustrations of these concepts. We give applications to cluster algebras and subcluster algebras related to unipotent groups, both in the Dynkin and non Dynkin case.

49 pages. For the third version the presentation is revised, especially Chapter III replaces the old Chapter III and IV

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• Internally Calabi-Yau algebras and cluster-tilting objects
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• Associahedra for finite type cluster algebras and minimal relations between $\mathbf{g}$-vectors
• Homological Classification of 4d $\mathcal{N}=2$ QFT. Part I: Rank-1 revisited
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