Related papers: An introduction to higher cluster categories
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…
The paper describes clustering problems from the combinatorial viewpoint. A brief systemic survey is presented including the following: (i) basic clustering problems (e.g., classification, clustering, sorting, clustering with an order over…
We give an introduction to the theory of cluster categories and cluster tilted algebras. We include some background on the theory of cluster algebras, and discuss the interplay with cluster categories and cluster tilted algebras.
The article contains a preliminary glance at balanced clustering problems. Basic balanced structures and combinatorial balanced problems are briefly described. A special attention is targeted to various balance/unbalance indices (including…
Higher cluster categories were recently introduced as a generalization of cluster categories. This paper shows that in Dynkin types A and D, half of all higher cluster categories are actually just quotients of cluster categories. The other…
This is an introductory survey on cluster algebras and their (additive) categorification using derived categories of Ginzburg algebras. After a gentle introduction to cluster combinatorics, we review important examples of coordinate rings…
We consider triangulated orbit categories, with the motivating example of cluster categories, in their usual context of algebraic triangulated categories, then present them from another perspective in the framework of topological…
We survey interactions between the topology and the combinatorics of complex hyperplane arrangements. Without claiming to be exhaustive, we examine in this setting combinatorial aspects of fundamental groups, associated graded Lie algebras,…
We define mutation on coloured quivers associated to tilting objects in higher cluster categories. We show that this operation is compatible with the mutation operation on the tilting objects. This gives a combinatorial approach to tilting…
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,…
Mixture models extend the toolbox of clustering methods available to the data analyst. They allow for an explicit definition of the cluster shapes and structure within a probabilistic framework and exploit estimation and inference…
In the cluster algebra literature, the notion of a graded cluster algebra has been implicit since the origin of the subject. In this work, we wish to bring this aspect of cluster algebra theory to the foreground and promote its study. We…
We consider limits over categories of extensions and show how certain well-known functors on the category of groups turn out as such limits. We also discuss higher (or derived) limits over categories of extensions.
We introduce a category of cluster algebras with fixed initial seeds. This category has countable coproducts, which can be constructed combinatorially, but no products. We characterise isomorphisms and monomorphisms in this category and…
The explosion in the amount of data available for analysis often necessitates a transition from batch to incremental clustering methods, which process one element at a time and typically store only a small subset of the data. In this paper,…
We introduce a notion of globular multicategory with homomorphism types. These structures arise when organizing collections of "higher category-like" objects such as type theories with identity types. We show how these globular…
We construct the intermediate coverings of cluster-tilted algebras by defining the generalized cluster categories. These generalized cluster categories are Calabi-Yau triangulated categories with fraction CY-dimension and have also cluster…
The cluster category is a triangulated category introduced for its combinatorial similarities with cluster algebras. We prove that a cluster algebra A of finite type can be realized as a Hall algebra, called the exceptional Hall algebra, of…
This is a book on higher-categorical diagrams, including pasting diagrams. It aims to provide a thorough and modern reference on the subject, collecting, revisiting and expanding results scattered across the literature, informed by recent…
We present a geometric interpretation of the integration-by-parts formula on an arbitrary vector bundle. As an application we give a new geometric formulation of higher-order variational calculus.