Related papers: Cluster categories for marked surfaces: punctured …
We introduce a new class of algebras, which we call cluster-tilted. They are by definition the endomorphism algebras of tilting objects in a cluster category. We show that their representation theory is very close to the representation…
We consider $m$-cluster tilted algebras arising from quivers of Euclidean type and we give necessary and sufficient conditions for those algebras to be representation finite. For the case $\widetilde{A}$, using the geometric realization, we…
In this mostly expository paper, we present recent progress on infinite (weak) cluster categories that are related to triangulations of the disk, with and without a puncture. First we recall the notion of a cluster category. Then we move to…
We initiate the investigation of representation theory of non-orientable surfaces. As a first step towards finding an additive categorification of Dupont and Palesi's quasi-cluster algebras associated marked non-orientable surfaces, we…
The aim of this note is to answer several open problems arising from the geometric description of the $m$-cluster categories of type $A_n$ and their realization in terms of the $m$-th power of a translation quiver. In particular, we give a…
We construct and study cluster algebra structures in rings of invariants of the special linear group action on collections of three-dimensional vectors, covectors, and matrices. The construction uses Kuperberg's calculus of webs on marked…
We describe a framework for encoding cluster combinatorics using categorical methods. We give a definition of an abstract cluster structure, which captures the essence of cluster mutation at a tropical level and show that cluster algebras,…
This essay builds on the idea of grouping the polar curves of 2-variable function germs into polar clusters. In the topological category, one obtains a bijective correspondence between certain partitions of the polar quotients of two…
We give a geometric model for a tube category in terms of homotopy classes of oriented arcs in an annulus with marked points on its boundary. In particular, we interpret the dimensions of extension groups of degree 1 between indecomposable…
In this paper, we associate an algebra A(T) to a triangulation T of a surface S with a set of boundary marking points. This algebra A(T) is gentle and Gorenstein of dimension one. We also prove that A(T) is cluster-tilted if and only if it…
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…
The exchange graph of a cluster algebra encodes the combinatorics of mutations of clusters. Through the recent "categorifications" of cluster algebras using representation theory one obtains a whole variety of exchange graphs associated…
For any graded commutative noetherian ring, where the grading group is abelian and where commutativity is allowed to hold in a quite general sense, we establish an inclusion-preserving bijection between, on the one hand, the twist-closed…
In the acyclic case, we establish a one-to-one correspondence between the tilting objects of the cluster category and the clusters of the associated cluster algebra. This correspondence enables us to solve conjectures on cluster algebras.…
In this article, we study the geometric realizations of $m$-cluster categories of Dynkin types A, D, $\tilde{A}$ and $\tilde{D}$. We show, in those four cases, that there is a bijection between $(m+2)$-angulations and isoclasses of basic…
We present a graded mutation rule for quivers of cluster-tilted algebras. Furthermore, we give a technique to recover a cluster-tilting object from its graded quiver in the cluster category of coh $\mathbb{X}$.
We exhibit gluing properties of cluster tilting subcategories in exact $\infty$-categories within the framework of perverse schobers on surfaces with boundary. These results are based on a study of the restriction functors from global…
In this paper, we study the distribution of the genuses of cluster quivers of finite mutation type. First, we prove that in the $11$ exceptional cases, the distribution of genuses is $0$ or $1$. Next, we consider the relationship between…
We introduce a new class of finite dimensional gentle algebras, the surface algebras, which are constructed from an unpunctured Riemann surface with boundary and marked points by introducing cuts in internal triangles of an arbitrary…
Traditional clustering identifies groups of objects that share certain qualities. Tangles do the converse: they identify groups of qualities that often occur together. They can thereby identify and discover 'types': of behaviour, views,…