Related papers: Cluster categories
This is a brief and informal introduction to cluster algebras. It roughly follows the historical path of their discovery, made jointly with A.Zelevinsky. Total positivity serves as the main motivation.
We continue our investigation on denominator conjecture of Fomin and Zelevinsky for cluster algebras via geometric models initialed in \cite{FG22}. In this paper, we confirm the denominator conjecture for cluster algebras of finite type.…
This is a preliminary draft of Chapters 4-5 of our forthcoming textbook "Introduction to Cluster Algebras." Chapters 1-3 have been posted as arXiv:1608.05735. This installment contains: Chapter 4. New patterns from old Chapter 5. Finite…
Cluster algebras, introduced by Fomin and Zelevinsky through the process of quiver mutation, have become central objects in modern algebra and geometry, linking combinatorial constructions with diverse mathematical domains such as…
Let Q be a finite quiver without oriented cycles, and let $\Lambda$ be the associated preprojective algebra. To each terminal representation M of Q (these are certain preinjective representations), we attach a natural subcategory $C_M$ of…
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…
Originally studied by Conway and Coxeter, friezes appeared in various recreational mathematics publications in the 1970s. More recently, in 2015, Baur, Parsons, and Tschabold constructed periodic infinite friezes and related them to…
In 2007, Fomin and Zelevinsky introduced the bipartite belt, a sequence of bipartite mutations whose exchange relations form a discrete dynamical system. Periodicity of this system is known as Zamolodchikov periodicity. In our previous work…
In this note we explain how to obtain cluster algebras from triangulations of (punctured) discs following the approach of S. Fomin, M. Shapiro and D. Thurston. Furthermore, we give a description of m-cluster categories via diagonals (arcs)…
We introduce a new cluster character with coefficients for a cluster category $\mathcal{C}$ and rather than using a Frobenius $2$-Calabi-Yau realization to incorporate coefficients into the representation-theoretic model for a cluster…
These are notes for a series of lectures presented at the ASIDE conference 2016. The definition of a cluster algebra is motivated through several examples, namely Markov triples, the Grassmannians $Gr_2(\mathbb{C})$, and the appearance of…
Cylindric algebras, or concept algebras in another name, form an interface between algebra, geometry and logic; they were invented by Alfred Tarski around 1947. We prove that there are 2 to the alpha many varieties of geometric (i.e.,…
In this paper, we introduce a notion of unistructural cluster algebras, for which the set of cluster variables uniquely determines the clusters. We prove that cluster algebras of Dynkin type and cluster algebras of rank 2 are unistructural,…
In recent decades, identities similar to the one in the Ptolemy's theorem started to pop up in many fields in connection to the notion of cluster algebras introduced and studied since 2000 by Fomin and Zelevinsky. In this brief note we will…
Let C be the category of finite-dimensional representations of a quantum affine algebra of simply-laced type. We introduce certain monoidal subcategories C_l (l integer) of C and we study their Grothendieck rings using cluster algebras.
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…
The paper is motivated by an analogy between cluster algebras and Kac-Moody algebras: both theories share the same classification of finite type objects by familiar Cartan-Killing types. However the underlying combinatorics beyond the two…
The cluster morphism category of an hereditary algebra was introduced in [5] to show that the picture space of an hereditary algebra of finite representation type is a $K(\pi,1)$ for the associated picture group, thereby allowing for the…
Berenstein and Zelevinsky introduced quantum cluster algebras [Adv. Math, 2005] and the triangular bases [IMRN, 2014]. The support conjecture by Lee-Li-Rupel-Zelevinsky [PNAS, 2014] asserts that the support of a triangular basis element for…
We show that a certain orbit category considerd by Keller encodes the combinatorics of the $m$-clusters of Fomin and Reading in a fashion similar to the way the cluster category of Buan, Marsh, Reineke, Reiten, and Todorov encodes the…