Related papers: Cluster tilting objects in generalized higher clus…

Let $\mathcal A$ be a Hom-finite abelian category with enough projectives. In this note, we show that any covariantly finite $\tau$-rigid subcategory is contained in a support $\tau$-tilting subcategory. We also show that support…

We establish the foundations of categorical weave calculus, developing the diagrammatic calculus of weaves and braid varieties within the study of Calabi-Yau triangulated categories and cluster tilting theory. This is achieved by…

In this article, we give a definition and a classification of 'higher' simple-minded systems in triangulated categories generated by spherical objects with negative Calabi-Yau dimension. We also study mutations of this class of objects and…

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…

This paper investigates a certain 2-Calabi-Yau triangulated category D whose Auslander-Reiten quiver is ZA_{\infty}. We show that the cluster tilting subcategories of D form a so-called cluster structure, and we classify these subcategories…

We show that in case a cluster algebra coincides with its upper cluster algebra and the cluster algebra admits a grading with finite dimensional homogeneous components, the corresponding Berenstein-Zelevinsky quantum cluster algebra can be…

For any valued quiver, by using BGP-reflection functors, an injection from the set of preprojective objects in the cluster category to the set of cluster variables of the corresponding cluster algebra is given, the images are called…

We give the first example of a non-trivial cluster tilting module in a local finite dimensional algebra. To do this, we give an explicit calculation of the corresponding higher Auslander algebra by quiver and relations using the GAP-package…

Cluster-tilted algebras are trivial extensions of tilted algebras. This correspondence induces a surjective map from tilted algebras to cluster-tilted algebras. If B is a cluster-tilted algebra, we use the fibre of B under this map to study…

In this work we study the connection between iterated tilted algebras and m-cluster tilted algebras. We show that an iterated tilted algebra induces an m-cluster tilted algebra. This m-cluster tilted algebra can be seen as a trivial…

In extended hearts of bounded $t$-structures on a triangulated category, we provide a Happel-Reiten-Smalo tilting theorem and a characterization for $s$-torsion pairs. Applying these to $m$-extended module categories, we characterize…

In this paper, we study ideal quotients of triangulated categories by higher cluster tilting subcategories. Koenig and Zhu proved that the ideal quotient by a $2$-cluster tilting subcategory is an abelian category; moreover, by Morita's…

We consider three categories arising from the higher Auslander algebras of type $A$ in relation to $d$-dimensional cluster combinatorics: $d$-exact subcategory of the module category of $A^d_{n+1}$ generated by the $d$-cluster-tilting…

We show the existence of tilting objects in the singularity category $\mathsf{D}_{\mathsf{ Sg}}^{\mathsf{ gr}}(eAe)$ associated to certain noetherian AS-regular algebras $A$ and idempotents $e$. This gives a triangle equivalence between…

We show that the $m$-cluster category of type $A_{n-1}$ is equivalent to a certain geometrically-defined category of diagonals of a regular $nm+2$-gon. This generalises a result of Caldero, Chapoton and Schiffler for $m=1$. The approach…

We show that every higher Auslander algebra $A_{n+1}^d$ of type $\mathbb{A}$ such that $\gcd(n,d)=1$ is derived equivalent to a certain replicated algebra $B=B_0^{(n+d)}$. Moreover ${\rm{gldim}} B = nd$ and $B$ admits an $nd$-cluster…

We give a complete classification of (co)torsion pairs in finite $2$-Calabi-Yau triangulated categories with maximal rigid objects which are not cluster tilting. These finite $2$-Calabi-Yau triangulated categories are divided into two main…

We show how a cluster-tilted algebra of finite representation type is related to the corresponding tilted algebra, in the case of algebras defined over an algebraically closed field.

Let $\mathbb{X}$ be a weighted projective line and $\mathcal{C}_\mathbb{X}$ the associated cluster category. It is known that $\mathcal{C}_\mathbb{X}$ can be realized as a generalized cluster category of quiver with potential. In this note,…

Over any field of positive characteristic we construct 2-CY-tilted algebras that are not Jacobian algebras of quivers with potentials. As a remedy, we propose an extension of the notion of a potential, called hyperpotential, that allows to…