Related papers: Introduction to Cluster Algebras. Chapter 7

This is a concise introduction to Fomin-Zelevinsky's cluster algebras and their links with the representation theory of quivers in the acyclic case. We review the definition of cluster algebras (geometric, without coefficients), construct…

I show how to associate a Clifford algebra to a graph. I describe the structure of these Clifford graph algebras and provide many examples and pictures. I describe which graphs correspond to isomorphic Clifford algebras and also discuss…

This book has eleven chapters. Chapter one describes all types of natural class of intervals and the arithmetic operations on them. Chapter two introduces the semigroup of natural class of intervals using R or Zn and study the properties…

Expanded lecture notes. Preliminary version, comments are welcome.

A family of quantum cluster algebras is introduced and studied. In general, these algebras are new, but subclasses have been studied previously by other authors. The algebras are indexed by double partitions or double flag varieties.…

In this paper, we prove that relation-extensions of quasi-tilted algebras are 2-Calabi-Yau tilted. With the objective of describing the module category of a cluster-tilted algebra of euclidean type, we define the notion of reflection so…

Explainable clustering by axis-aligned decision trees was introduced by Moshkovitz et al. (2020) and has gained considerable interest. Prior work has focused on minimizing the price of explainability for specific clustering objectives,…

We discuss some recent progress in constructing analytic approximations to the galaxy clustering. We show that successful models can be constructed for the clustering of both dark matter and dark matter haloes. Our understanding of galaxy…

We present here two detailed examples of additive categorifications of the cluster algebra structure of a coordinate ring of a maximal unipotent subgroup of a simple Lie group. The first one is of simply-laced type ($A_3$) and relies on an…

This is the author's PhD-thesis, which was written in 2006. The version posted here is identical to the printed one. Instead of an abstract, the short list of contents: Preface 5 1 Introduction 9 2 K-theory and cyclic type homology theories…

We apply our previous work on cluster characters for Hom-infinite cluster categories to the theory of cluster algebras. We give a new proof of Conjectures 5.4, 6.13, 7.2, 7.10 and 7.12 of Fomin and Zelevinsky's Cluster algebras IV for…

We propose a novel graph clustering method guided by additional information on the underlying structure of the clusters (or communities). The problem is formulated as the matching of a graph to a template with smaller dimension, hence…

We develop a general approach to finding combinatorial models for cluster algebras. The approach is to construct a labeled graph called a framework. When a framework is constructed with certain properties, the result is a model…

This paper studies clustering algorithms for dynamically evolving graphs $\{G_t\}$, in which new edges (and potential new vertices) are added into a graph, and the underlying cluster structure of the graph can gradually change. The paper…

This article explains basic constructions and results on group algebras and their cohomology, starting from the point of view of commutative algebra. It provides the background necessary for a novice in this subject to begin reading Dave…

The chapter provides an introduction to the basic concepts of Algebraic Topology with an emphasis on motivation from applications in the physical sciences. It finishes with a brief review of computational work in algebraic topology,…

We classify mutation-finite cluster algebras with arbitrary coefficients of geometric type.

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…

The article concerns the existence and uniqueness of quantisations of cluster algebras. We prove that cluster algebras with an initial exchange matrix of full rank admit a quantisation in the sense of Berenstein-Zelevinsky and give an…

We present a new algorithm to compute initial seeds for cluster structures on categories associated with coordinate rings of open Richardson varieties. This allows us to explicitely determine seeds first considered in Leclerc's 2016…