Related papers: Introduction to Cluster Algebras. Chapters 4-5
This is a preliminary draft of Chapter 6 of our forthcoming textbook "Introduction to Cluster Algebras." Chapters 1-3 have been posted as arXiv:1608.05735. Chapters 4-5 have been posted as arXiv:1707.07190. This installment contains:…
This is a preliminary draft of Chapters 1-3 of our forthcoming textbook "Introduction to Cluster Algebras." This installment contains: Chapter 1. Total positivity Chapter 2. Mutations of quivers and matrices Chapter 3. Clusters and seeds
This is a preliminary draft of Chapter 7 of our forthcoming textbook "Introduction to Cluster Algebras." Chapters 1-3 have been posted as arXiv:1608.05735. Chapters 4-5 have been posted as arXiv:1707.07190. Chapter 6 has been posted as…
This is a first step guide to the theory of cluster algebras. We especially focus on basic notions, techniques, and results concerning seeds, cluster patterns, and cluster algebras.
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…
This is an expanded version of the notes of our lectures given at the conference "Current Developments in Mathematics 2003" held at Harvard University on November 21--22, 2003. We present an overview of the main definitions, results and…
We compute the list of all minimal 2-infinite diagrams, which are cluster algebraic analogues of extended Dynkin graphs.
We provide a complete classification of the singularities of cluster algebras of finite cluster type. This extends our previous work about the case of trivial coefficients. Additionally, we classify the singularities of cluster algebras for…
We classify mutation-finite cluster algebras with arbitrary coefficients of geometric type.
This is an appendix to the Handbook of Tilting Theory, edited by Angeleri-Huegel, Happel and Krause, to be published soon. Part 1 of the appendix provides an outline of the core of tilting theory. Part 2 is devoted to topics where tilting…
Cluster algebras have recently become an important player in mathematics and physics. In this work, we investigate them through the lens of modern data science, specifically with techniques from network science and machine learning. Network…
These are expanded notes of four introductory talks on A-infinity algebras, their modules and their derived categories.
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…
This survey article is intended as an introduction to the recent categorical classification theorems of the three authors, restricting to the special case of the category of modules for a finite group.
Cluster algebras are a recent topic of study and have been shown to be a useful tool to characterize structures in several knowledge fields. An important problem is to establish whether or not a given cluster algebra is of finite type.…
We complete classification of mutation-finite cluster algebras by extending the technique derived by Fomin, Shapiro, and Thurston to skew-symmetrizable case. We show that for every mutation-finite skew-symmetrizable matrix a diagram…
This is Addendum to ``Structure of seeds in generalized cluster algebras'', Pacific J. Math. {277} (2015), 201--218. We extend the class of generalized cluster algebras studied therein to embrace examples in some applications.
Cluster algebras were introduced by Fomin-Zelevinsky in 2002 in order to give a combinatorial framework for phenomena occurring in the context of algebraic groups. Cluster algebras also have links to a wide range of other subjects,…
In this paper, we prove Conjecture 4.8 of "Cluster algebras IV" by S. Fomin and A. Zelevinsky, stating that the mutation classes of rectangular matrices associated with cluster algebras of finite type are precisely those classes which are…
Cluster algebras are a class of commutative algebras whose generators are defined by a recursive process called mutation. We give a brief introduction to cluster algebras, and explain how discrete integrable systems can appear in the…