Related papers: Quivers with relations arising from clusters (A_n …
Let $\mathcal{O}$ be the category of representations of the Borel subalgebra of a quantum affine algebra introduced by Jimbo and the first author. We show that the Grothendieck ring of a certain monoidal subcategory of $\mathcal{O}$ has the…
We show that cluster algebras do not contain non-trivial units and that all cluster variables are irreducible elements. Both statements follow from Fomin and Zelevinsky's Laurent phenomenon. As an application we give a criterion for a…
These are expanded notes from three survey lectures given at the 14th International Conference on Representations of Algebras (ICRA XIV) held in Tokyo in August 2010. We first study identities between products of quantum dilogarithm series…
We study generalized cluster algebras introduced by Chekhov and Shapiro. When the coefficients satisfy the normalization and quasi-reciprocity conditions, one can naturally extend the structure theory of seeds in the ordinary cluster…
The cluster-tilted algebras have been introduced by Buan, Marsh and Reiten, they are the endomorphism rings of cluster-tilting objects $T$ in cluster categories; we call such an algebra cluster-concealed in case $T$ is obtained from a…
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.…
In this paper we propose the notion of cluster superalgebras which is a supersymmetric version of the classical cluster algebras introduced by Fomin and Zelevinsky. We show that the symplectic-orthogonal supergroup $SpO(2|1)$ admits a…
Matrix mutation appears in the definition of cluster algebras of Fomin and Zelevinsky. We give a representation theoretic interpretation of matrix mutation, using tilting theory in cluster categories of hereditary algebras. Using this, we…
Associated to any acyclic cluster algebra is a corresponding triangulated category known as the cluster category. It is known that there is a one-to-one correspondence between cluster variables in the cluster algebra and exceptional…
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,…
We introduce a new function on the set of pairs of cluster variables via $f$-vectors, which we call it the compatibility degree (of cluster complexes). The compatibility degree is a natural generalization of the classical compatibility…
Quiver mutation plays a crucial role in the definition of cluster algebras by Fomin and Zelevinsky. It induces an equivalence relation on the set of all quivers without loops and two-cycles. A quiver is called mutation-acyclic if it is…
We describe a new way to relate an acyclic, skew-symmetrizable cluster algebra to the representation theory of a finite dimensional hereditary algebra. This approach is designed to explain the c-vectors of the cluster algebra. We obtain a…
We calculate the cluster modular groups of affine and doubly extended typecluster algebras in a uniform way by introducing a new family of quivers. We use this uniformdescription to construct a natural finite quotient of the cluster complex…
The generalized cluster complex was introduced by Fomin and Reading, as a natural extension of the Fomin-Zelevinsky cluster complex coming from finite type cluster algebras. In this work, to each face of this complex we associate a…
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…
Motivated by a recent conjecture by Hernandez and Leclerc [arXiv:0903.1452], we embed a Fomin-Zelevinsky cluster algebra [arXiv:math/0104151] into the Grothendieck ring R of the category of representations of quantum loop algebras U_q(Lg)…
Building on work by Geiss-Leclerc-Schroer and by Buan-Iyama-Reiten-Scott we investigate the link between certain cluster algebras with coefficients and suitable 2-Calabi-Yau categories. These include the cluster-categories associated with…
We introduce a family of cluster algebras of infinite rank associated with root systems of type $A$, $D$, $E$. We show that suitable completions of these cluster algebras are isomorphic to the Grothendieck rings of the 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.