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Register a new userLifting to cluster-tilting objects in higher cluster categories
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Pin Liu
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In this note, we consider the $d$-cluster-tilted algebras, the endomorphism algebras of $d$-cluster-tilting objects in $d$-cluster categories. We show that a tilting module over such an algebra lifts to a $d$-cluster-tilting object in this $d$-cluster category.
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We show that a tilting module over the endomorphism algebra of a cluster-tilting object in a 2-Calabi-Yau triangulated category lifts to a cluster-tilting object in this 2-Calabi-Yau triangulated category. This generalizes a recent work of D. Smith for cluster categories.
As a generalization of acyclic 2-Calabi-Yau categories, we consider 2-Calabi-Yau categories with a directed cluster-tilting subcategory; we study their cluster-tilting subcategories and the cluster combinatorics that they encode. We show that such categories have a cluster structure. Triangulated 2-Calabi-Yau categories with a directed cluster-tilting subcategory are closely related to representations of certain semi-hereditary categories, more specifically to representations of thread quivers. Thread quivers are a tool to classify and study certain semi-hereditary categories using both quivers and linearly ordered sets (threads). We study the case where the thread quiver consists of a single thread (so that representations of this thread quiver correspond to representations of some linearly ordered set), and show that, similar to the case of a Dynkin quiver of type $A$, the cluster-tilting subcategories can be understood via triangulations of an associated cyclically ordered set. In this way, we gain insight into the structure of the cluster-tilting subcategories of 2-Calabi-Yau categories with a directed cluster-tilting subcategory. As an application, we show that every 2-Calabi-Yau category which admits a directed cluster-tilting subcategory with countably many isomorphism classes of indecomposable objects has a cluster-tilting subcategory $mathcal{V}$ with the following property: any rigid object in the cluster category can be reached from $mathcal{V}$ by finitely many mutations. This implies that there is a cluster map which is defined on all rigid objects, and thus that there is a cluster algebra whose cluster variables are exactly given by the rigid indecomposable objects.
Building on work by Geiss-Leclerc-Schroer and by Buan-Iyama-Reiten-Scott we investigate the link between certain cluster algebras with coefficients and suitable 2-Calabi-Yau categories. These include the cluster-categories associated with acyclic quivers and certain Frobenius subcategories of module categories over preprojective algebras. Our motivation comes from the conjectures formulated by Fomin and Zelevinsky in `Cluster algebras IV: Coefficients. We provide new evidence for Conjectures 5.4, 6.10, 7.2, 7.10 and 7.12 and show by an example that the statement of Conjecture 7.17 does not always hold.
We generalize the construction of (higher) cluster categories by Claire Amiot and by Lingyan Guo to the relative context. We prove the existence of an $ n $-cluster tilting object in a Frobenius extriangulated category which is stably $ n $-Calabi--Yau and Hom-finite, arising from a left $ (n+1) $-Calabi--Yau morphism. Our results apply in particular to relative Ginzburg dg algebras coming from ice quivers with potential and higher Auslander algebras associated to $ n $-representation-finite algebras.
Given a triangulated 2-Calabi-Yau category C and a cluster-tilting subcategory T, the index of an object X of C is a certain element of the Grothendieck group of the additive category T. In this note, we show that a rigid object of C is determined by its index, that the indices of the indecomposables of a cluster-tilting subcategory T form a basis of the Grothendieck group of T and that, if T and T are related by a mutation, then the indices with respect to T and T are related by a certain piecewise linear transformation introduced by Fomin and Zelevinsky in their study of cluster algebras with coefficients. This allows us to give a combinatorial construction of the indices of all rigid objects reachable from the given cluster-tilting subcategory T. Conjecturally, these indices coincide with Fomin-Zelevinskys g-vectors.
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