相关论文: Positive Fuss-Catalan numbers and Simple-minded sy…

Let $Q$ be an acyclic quiver and $w \geq 1$ be an integer. Let $\mathsf{C}_{-w} (\mathbf{k} Q)$ be the $(-w)$-cluster category of $\mathbf{k} Q$. We show that there is a bijection between simple-minded collections in $\mathsf{D}^b…

This paper is devoted to studying two important classes of objects in triangulated categories; silting objects and $d$-cluster tilting objects, and their correspondences. First, we introduce the notion of $d$-silting objects as a…

We develop the basic properties of $w$-simple-minded systems in $(-w)$-Calabi-Yau triangulated categories for $w \geq 1$. The main result is a reduction technique for negative Calabi-Yau triangulated categories. We show that the theory of…

Silting and Calabi-Yau reductions are important process in representation theory to construct new triangulated categories from given one, which are similar to Verdier quotient. In this paper, first we introduce a new reduction process of…

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…

Fomin and Zelevinsky's definition of cluster algebras laid the foundation for cluster theory. The various categorifications and generalisations of the original definition led to Iyama and Yoshino's generalised cluster categories…

Let W be a finite crystallographic reflection group. The generalized Catalan number of W coincides both with the number of clusters in the cluster algebra associated to W, and with the number of noncrossing partitions for W. Natural…

We count the number of compatible pairs of indexing systems for the cyclic group $C_{p^n}$. Building on work of Balchin--Barnes--Roitzheim, we show that this sequence of natural numbers is another family of Fuss--Catalan numbers. We count…

We present a restricted model for the bounded derived category of gentle algebras that encodes the indecomposable objects and positive extensions between them. The model is then used to count the number of $d$-term silting objects for…

We describe what it means for an algebra to be internally d-Calabi-Yau with respect to an idempotent. This definition abstracts properties of endomorphism algebras of (d-1)-cluster-tilting objects in certain stably (d-1)-Calabi-Yau…

In type A, the q,t-Fuss-Catalan numbers can be defined as a bigraded Hilbert series of a module associated to the symmetric group. We generalize this construction to (finite) complex reflection groups and, based on computer experiments, we…

We prove that for $d \geq 2$, an algebraic $d$-Calabi-Yau triangulated category endowed with a $d$-cluster tilting subcategory is the stable category of a DG category which is perfectly $(d+1)$-Calabi-Yau and carries a non degenerate…

Let $d$ be a positive integer. In a previous article we established a bijective correspondence between the following classes of objects, considered up to the appropriate notion of equivalence: differential graded algebras with…

For $d\geq 1$, we define semibricks and wide subcategories in the $d$-extended hearts of bounded $t$-structures on a triangulated category. We show that these semibricks are in bijection with finite-length wide subcategories. When the…

We establish a bijective correspondence between isomorphism classes of basic silting objects of $\mathsf{per}(A)$ and algebraic $t$-structures of $\mathsf{D}_{\rm fd}(A)$ for locally finite non-positive dg algebra $A$ over a field $k$ (more…

We study higher cluster tilting objects in generalized higher cluster categories arising from dg algebras of higher Calabi-Yau dimension. Taking advantage of silting mutations of Aihara-Iyama, we obtain a class of $m$-cluster tilting…

We introduce a new category C, which we call the cluster category, obtained as a quotient of the bounded derived category D of the module category of a finite-dimensional hereditary algebra H over a field. We show that, in the simply-laced…

Let $U_q'(\mathfrak{g})$ be a quantum affine algebra of untwisted affine $ADE$ type, and $\mathcal{C}_{\mathfrak{g}}^0$ the Hernandez-Leclerc category of finite-dimensional $U_q'(\mathfrak{g})$-modules. For a suitable infinite sequence…

For any given symmetrizable Cartan matrix $C$ with a symmetrizer $D$, Gei\ss~ et al. (2016) introduced a generalized preprojective algebra $\Pi(C, D)$. We study tilting modules and support $\tau$-tilting modules for the generalized…

This paper shows a new phenomenon in higher cluster tilting theory. For each positive integer d, we exhibit a triangulated category C with the following properties. On the one hand, the d-cluster tilting subcategories of C have very simple…