Related papers: From frieze patterns to cluster categories
Let $A$ be the path algebra of a finite acyclic quiver $Q$ over a finite field. We realize the quantum cluster algebra with principal coefficients associated to $Q$ as a sub-quotient of a certain Hall algebra involving the category of…
We initiate a systematic study of the cohomology of cluster varieties. We introduce the Louise property for cluster algebras that holds for all acyclic cluster algebras, and for most cluster algebras arising from marked surfaces. For…
In this paper, we will present the author's interpretation and embellishment of five lectures on cluster theory given by Kiyoshi Igusa during the Spring semester of 2022 at Brandeis University. They are meant to be used as an introduction…
Conway and Coxeter introduced frieze patterns in 1973 and classified them via triangulated polygons. The determinant of the matrix associated to a frieze table was computed explicitly by Broline, Crowe and Isaacs in 1974, a result…
A classic result of Conway and Coxeter on frieze patterns has been generalized to a bijection between $p$-angulations of regular polygons and frieze patterns of type $\Lambda_p$. One of the features of Conway-Coxeter theory is a…
From the viewpoint of mutation, we will give a brief survey of tilting theory and cluster-tilting theory together with a motivation from cluster algebras. Then we will give an introdution to \tau-tilting theory which was recently developed…
In 2017, Michael Cuntz gave a definition of reducibility of quiddity cycles of frieze patterns: It is reducible if it can be written as a sum of two other quiddity cycles. We discuss the commutativity and associativity of this sum operator…
We study factoriality and the class groups of locally acyclic cluster algebras. To do so, we introduce a new class of rings called finite Laurent intersection rings (FLIRs), which includes locally acyclic cluster algebras, full-rank upper…
We introduce admissible group actions on cluster algebras, cluster categories and quivers with potential and study the resulting orbit spaces. The orbit space of the cluster algebra has the structure of a generalized cluster algebra. This…
Recently, Ramos and Whiting showed that any generalized cluster algebra of geometric type is isomorphic to a quotient of a subalgebra of a certain cluster algebra. Based on their idea and method, we show that the same property holds for any…
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…
In \cite{CK2005} and \cite{Hubery2005}, the authors proved the cluster multiplication theorems for finite type and affine type. We generalize their results and prove the cluster multiplication theorem for arbitrary type by using the…
In this article, we introduce the notion of cluster automorphism of a given cluster algebra as a $\ZZ$-automorphism of the cluster algebra that sends a cluster to another and commutes with mutations. We study the group of cluster…
We give a brief introduction to (upper) cluster algebras and their quantization using examples. Then we present several important families of bases for these algebras using topological models. We also discuss tropical properties of these…
The article gives a ring theoretic perspective on cluster algebras. Gei{\ss}-Leclerc-Schr\"oer prove that all cluster variables in a cluster algebra are irreducible elements. Furthermore, they provide two necessary conditions for a cluster…
We study the canonical orbit category of the bounded derived category of finite dimensional representations of the quiver of type $D_{\infty}$. We prove that this orbit category is a cluster category, that is, its cluster-tilting…
This paper continues the study of cluster algebras initiated in math.RT/0104151. Its main result is the complete classification of the cluster algebras of finite type, i.e., those with finitely many clusters. This classification turns out…
We give an explicit subword complex description of the generators of the type cone of the g-vector fan of a finite type cluster algebra with acyclic initial seed. This yields in particular a description of the Newton polytopes of the…
We generalize the construction of reflection functors from classical representation theory of quivers to arbitrary small categories with freely attached sinks or sources. These reflection morphisms are shown to induce equivalences between…
Cyclic poset are generalizations of cyclically ordered sets. In this paper we show that any cyclic poset gives rise to a Frobenius category over any discrete valuation ring R. The continuous cluster categories of arXiv:1209.1879 are…