Related papers: Generalised friezes and a modified Caldero-Chapoto…
A generalized topology in a set $X$ is a collection $\text{Cov}_X$ of families of subsets of $X$ such that the triple $(X,\bigcup \text{Cov}_X,\text{Cov}_X)$ is a generalized topological space in the sense of Delfs and Knebusch. In this…
A frieze on a polygon is a map from the diagonals of the polygon to an integral domain which respects the Ptolemy relation. Conway and Coxeter previously studied positive friezes over $\mathbb{Z}$ and showed that they are in bijection with…
The Key map is an important tool in the determination of the Demazure crystals associated to Kac-Moody algebras. In finite type A, it can be computed in the tableau realization of crystals by a simple combinatorial procedure due to Lascoux…
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…
The cluster-tilted algebras have been introduced by Buan, Marsh and Reiten, they are the endomorphism rings of cluster-tilting objects $T$ in cluster categories; we call such an algebra cluster-concealed in case $T$ is obtained from a…
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…
In this paper, we give a method for relating the generalized category $\mathcal{O}$ defined by the author and collaborators to explicit finitely presented algebras, and apply this to quiver varieties. This allows us to describe…
We complete the discrete cluster categories of type $\mathbb{A}$ as defined by Igusa and Todorov, by embedding such a discrete cluster category inside a larger one, and then taking a certain Verdier quotient. The resulting category is a…
We construct frieze patterns of type D_N with entries which are numbers of matchings between vertices and triangles of corresponding triangulations of a punctured disc. For triangulations corresponding to orientations of the Dynkin diagram…
Generalized Cluster Algebras (GCA) are generalizations of Cluster Algebras (CA) with higher-order exchange relations. Previously, Chekhov-Shapiro conjectured that every GCA can be embedded into a CA. In this paper, we prove a modified…
A frieze is an array of numbers obeying the unimodular rule. Coxeter showed that a frieze with integer entries corresponds to a triangulation. Recently, Holm and J{\o}rgenson introduced friezes of type $\Lambda_p$ which correspond to…
This paper investigates a non simply-laced version of cluster structures for 2-Calabi-Yau or stably 2-Calabi-Yau categories over arbitrary fields. It results that 2-Calabi-Yau or stably 2-Calabi-Yau categories having a cluster tilting…
For an endofunctor $F\colon\mathcal{C}\to\mathcal{C}$ on an ($\infty$-)category $\mathcal{C}$ we define the $\infty$-category $\operatorname{Cart}(\mathcal{C},F)$ of generalized Cartier modules as the lax equalizer of $F$ and the identity.…
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…
In this paper we give a geometric-combinatorial description of the cluster categories of type E. In particular, we give an explicit geometric description of all cluster tilting objects in the cluster category of type E_6. The model we…
Y. Palu has generalized the cluster multiplication formulas to 2-Calabi-Yau categories with cluster tilting objects (\cite{Palu2}). The aim of this note is to construct a variant of Y. Palu's formula and deduce a new version of the cluster…
Let f: A\to B be a ring homomorphism between Noetherian normal integral domains. We establish a general criterion for f to induce a homomorphism Cl(f): Cl(A)\to Cl(B) on divisor class groups. For instance, this criterion applies whenever f…
We show that the quotient of the continuous cluster category $\mathcal C_\pi$ modulo the additive subcategory generated by any cluster is an abelian category and we show that it is isomorphic to the category of infinite length modules over…
Let $Q$ be a finite acyclic valued quiver. We define a bialgebra structure and an integration map on the Hall algebra associated to the morphism category of projective representations of $Q$. As an application, we recover the surjective…
Polynomial functors are useful in the theory of data types, where they are often called containers. They are also useful in algebra, combinatorics, topology, and higher category theory, and in this broader perspective the polynomial aspect…