Related papers: On the combinatorics of rigid objects in 2-Calabi-…
Let $\mathscr{C}$ be a 2-Calabi-Yau triangulated category, and let $\mathscr{T}$ be a cluster tilting subcategory of $\mathscr{C}$. An important result from Dehy and Keller tells us that a rigid object $c \in \mathscr{C}$ is uniquely…
We prove that indecomposable transjective modules over cluster-tilted algebras are uniquely determined by their dimension vectors. Similarly, we prove that for cluster-concealed algebras, rigid modules lifting to rigid objects in the…
For a given cluster-tilted algebra $A$ of tame type, it is proved that different indecomposable $\tau$-rigid $A$-modules have different dimension vectors. This is motivated by Fomin-Zelevinsky's denominator conjecture for cluster algebras.…
We compute the Grothendieck group of certain 2-Calabi--Yau triangulated categories appearing naturally in the study of the link between quiver representations and Fomin--Zelevinsky's cluster algebras. In this setup, we also prove a…
For a Calabi-Yau triangulated category $\mathcal{C}$ of Calabi-Yau dimension $d$ with a $d-$cluster tilting subcategory $\mathcal{T}$, it is proved that the decomposition of $\mathcal{C}$ is determined by the special decomposition of…
Cluster algebras are categorified by cluster categories, and $g$-vectors are categorified by the classic index with respect to cluster tilting subcategories. However, the recently introduced completed discrete cluster categories of Dynkin…
Palu defined the index with respect to a cluster tilting object in a suitable triangulated category, in order to better understand the Caldero-Chapoton map that exhibits the connection between cluster algebras and representation theory. We…
Let $\mathscr{C}$ be a $2$-Calabi-Yau triangulated category with two cluster tilting subcategories $\mathscr{T}$ and $\mathscr{U}$. Results by Demonet-Iyama-Jasso and J{\o}rgensen-Yakimov known as tropical duality says that the index with…
We develop a general framework for $c$-vectors of 2-Calabi--Yau categories, which deals with cluster tilting subcategories that are not reachable from each other and contain infinitely many indecomposable objects. It does not rely on…
We study the cluster algebras arising from cluster tubes with rank bigger than $1$. Cluster tubes are $2-$Calabi-Yau triangulated categories which contain no cluster tilting objects, but maximal rigid objects. Fix a certain maximal rigid…
Starting from an arbitrary cluster-tilting object $T$ in a 2-Calabi--Yau category over an algebraically closed field, as in the setting of Keller and Reiten, we define, for each object $L$, a fraction $X(T,L)$ using a formula proposed by…
We prove that in a 2-Calabi-Yau triangulated category, each cluster tilting subcategory is Gorenstein with all its finitely generated projectives of injective dimension at most one. We show that the stable category of its Cohen-Macaulay…
We develop a general theory of cluster categories, applying to a 2-Calabi-Yau extriangulated category $\mathcal{C}$ and cluster-tilting subcategory $\mathcal{T}$ satisfying only mild finiteness conditions. We show that the structure theory…
Assume that $\D$ is a Krull-Schmidt, Hom-finite triangulated category with a Serre functor and a cluster-tilting object $T$. We introduce the notion of relative cluster tilting objects, and $T[1]$-cluster tilting objects in $\D$, which are…
Let $\CC$ be a Hom-finite triangulated 2-Calabi-Yau category with a cluster-tilting object $T$. Under a constructibility condition we prove the existence of a set $\mathcal G^T(\CC)$ of generic values of the cluster character associated to…
We study the maximal rigid subcategories in $2-$CY triangulated categories and their endomorphism algebras. Cluster tilting subcategories are obviously maximal rigid; we prove that the converse is true if the $2-$CY triangulated categories…
We study the cluster combinatorics of $d-$cluster tilting objects in $d-$cluster categories. By using mutations of maximal rigid objects in $d-$cluster categories which are defined similarly for $d-$cluster tilting objects, we prove the…
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…
We introduce a new cluster character with coefficients for a cluster category $\mathcal{C}$ and rather than using a Frobenius $2$-Calabi-Yau realization to incorporate coefficients into the representation-theoretic model for a cluster…
Let $k$ be a field and $\mathcal{C}$ a $k$-linear, Hom-finite triangulated category with split idempotents. In this paper, we show that under suitable circumstances, the Grothendieck group of $\mathcal{C}$, denoted $K_0(\mathcal{C})$, can…