Related papers: Introduction to Cluster Algebras. Chapter 7
This paper develops techniques for producing presentations of upper cluster algebras. These techniques are suited to computer implementation, and will always succeed when the upper cluster algebra is totally coprime and finitely generated.…
This is an introduction to cluster algebras and their common triangular bases. These bases are Kazhdan-Lusztig-type and serve as the canonical bases of cluster algebras from the representation-theoretic point of view. We review seeds…
These lecture notes are intended to give a modest impulse to anyone willing to start or pursue a journey into the theory of Vertex Algebras by reading one of Kac's or Lepowsky-Li's books. Therefore, the primary goal is to provide required…
In an attempt to create an algebraic framework for dual canonical bases and total positivity in semisimple groups, we initiate the study of a new class of commutative algebras.
These are expanded notes of four introductory talks on A-infinity algebras, their modules and their derived categories.
In this paper, we introduce some particular families of graphicable algebras obtained by following a relatively new line of research, initiated previously by some of the authors. It consists of the use of certain objects of Discrete…
A bibliographic database will be constructed with the purpose to be a general tool for searching references for galaxy clusters. The structure of the database will be completely different from the available now databases as NED, SIMBAD,…
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…
We provide multiple combinatorial expansion formulas - in terms of snake graphs, labelled posets, matrices, and $T$-walks - for elements in generalized cluster algebras associated to arcs on punctured orbifolds and illustrate their…
We characterize mutation-finite cluster algebras of rank at least 3 using positive semi-definite quadratic forms. In particular, we associate with every unpunctured bordered surface a positive semi-definite quadratic space $V$, and with…
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…
This is an extended abstract of my talk at the Oberwolfach Workshop ''Cluster Algebras and Related Topics'' (December 8 - 14, 2013). It is based on a joint work with A. Zelevinsky (arXiv:1306.3495).
We clarify the natural cluster algebra of type A that exists in a residual and tropical form in the kinematical space as suggested in 1711.09102 by the use of triangulations, mutations and associahedron on the definition of scattering…
This is an extended abstract of my talk at the Oberwolfach-Workshop "Representation Theory of Finite-Dimensional Algebras" (February 6 - 12, 2005). It gives self-contained and simplified definitions of quantum cluster algebras introduced…
This paper is a chapter in the forthcoming Handbook of Cluster Analysis, Hennig et al. (2015). For definitions of basic clustering methods and some further methodology, other chapters of the Handbook are referred to. To read this version of…
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…
We continue the work started in parts (I) and (II). In this part we classify which continuous type A quivers are derived equivalent and introduce the new continuous cluster category with E-clusters, which are a generalization of clusters.…
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…
Among the mutation finite cluster algebras the tubular ones are a particularly interesting class. We show that all tubular (simply laced) cluster algebras are of exponential growth by two different methods: first by studying the…
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…