Related papers: Symplectic Maps from Cluster Algebras

We study iteration maps of recurrence relations arising from mutation periodic quivers of arbitrary period. Combining tools from cluster algebra theory and (pre)symplectic geometry, we show that these cluster iteration maps can be reduced…

We consider nonlinear recurrences generated from cluster mutations applied to quivers that have the property of being cluster mutation-periodic with period 1. Such quivers were completely classified by Fordy and Marsh, who characterised…

We consider quivers/skew-symmetric matrices under the action of mutation (in the cluster algebra sense). We classify those which are isomorphic to their own mutation via a cycle permuting all the vertices, and give families of quivers which…

One of the remarkable properties of cluster algebras is that any cluster, obtained from a sequence of mutations from an initial cluster, can be written as a Laurent polynomial in the initial cluster (known as the "Laurent phenomenon").…

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…

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…

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…

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…

A cluster algebra is a commutative algebra whose structure is decided by a skew-symmetrizable matrix or a quiver. When a skew-symmetrizable matrix is invariant under an action of a finite group and this action is admissible, the folded…

We introduce a class of non-commutative algebras that carry a non-commutative (geometric) cluster structure which are generated by identical copies of generalized Weyl algebras. Equivalent conditions for the finiteness of the set of the…

A cluster algebra is an algebraic structure generated by operations of a quiver (a directed graph) called the mutations and their associated simple birational mappings. By using a graph-combinatorial approach, we present a systematic way to…

We define cluster $R$-matrices as sequences of mutations in triangular grid quivers on a cylinder, and show that the affine geometric $R$-matrix of symmetric power representations for the quantum affine algebra…

We consider deformations of sequences of cluster mutations in finite type cluster algebras, which destroy the Laurent property but preserve the presymplectic structure defined by the exchange matrix. The simplest example is the Lyness…

We describe a framework for encoding cluster combinatorics using categorical methods. We give a definition of an abstract cluster structure, which captures the essence of cluster mutation at a tropical level and show that cluster algebras,…

For a rooted cluster algebra $\mathcal{A}(Q)$ over a valued quiver $Q$, a \emph{symmetric cluster variable} is any cluster variable belonging to a cluster associated with a quiver $\sigma (Q)$, for some permutation $\sigma$. The subalgebra…

We consider discrete dynamical systems obtained as deformations of mutations in cluster algebras associated with finite-dimensional simple Lie algebras. The original (undeformed) dynamical systems provide the simplest examples of…

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 if the final paper in the seriesContinuous Quivers of Type $A$. In this part, we generalize existing geometric models of type $A$ cluster structures to the new $\mathbf E$-clusters introduced in part (III). We also introduce an…

In this paper, we give a description of the skew-symmetrizable matrices and their mutation classes which are determined by the generalized Cartan matrices of affine type.

We consider a class of map, recently derived in the context of cluster mutation. In this paper we start with a brief review of the quiver context, but then move onto a discussion of a related Poisson bracket, along with the Poisson algebra…