Related papers: Cluster algebras and triangulated orbifolds
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 present a combinatorial model for cluster algebras of type $D_n$ in terms of centrally symmetric pseudotriangulations of a regular $2n$-gon with a small disk in the centre. This model provides convenient and uniform interpretations for…
In 2003, Fomin and Zelevinsky obtained Cartan-Killing type classification of all cluster algebras of finite type, i.e. cluster algebras having only finitely many distinct cluster variables. A wider class of cluster algebras is formed by…
In this paper we give a graph theoretic combinatorial interpretation for the cluster variables that arise in most cluster algebras of finite type. In particular, we provide a family of graphs such that a weighted enumeration of their…
We express cluster variables of type $B_n$ and $C_n$ in terms of cluster variables of type $A_n$. Then we associate a cluster tilted bound symmetric quiver $Q$ of type $A_{2n-1}$ to any seed of a cluster algebra of type $B_n$ and $C_n$.…
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…
We apply our previous work on cluster characters for Hom-infinite cluster categories to the theory of cluster algebras. We give a new proof of Conjectures 5.4, 6.13, 7.2, 7.10 and 7.12 of Fomin and Zelevinsky's Cluster algebras IV for…
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…
We generalise surface cluster algebras to the case of infinite surfaces where the surface contains finitely many accumulation points of boundary marked points. To connect different triangulations of an infinite surface, we consider infinite…
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…
Cluster-tilted algebras are trivial extensions of tilted algebras. This correspondence induces a surjective map from tilted algebras to cluster-tilted algebras. If B is a cluster-tilted algebra, we use the fibre of B under this map to study…
In the acyclic case, we establish a one-to-one correspondence between the tilting objects of the cluster category and the clusters of the associated cluster algebra. This correspondence enables us to solve conjectures on cluster algebras.…
We study quantum cluster algebras from unpunctured surfaces with arbitrary coefficients and quantization. We first give a new proof of the Laurent expansion formulas for commutative cluster algebras from unpunctured surfaces, we then give…
We show the existence of cluster $\mathcal{A}$-structures and cluster Poisson structures on any braid variety, for any simple Lie group. The construction is achieved via weave calculus and a tropicalization of Lusztig's coordinates. Several…
With any non necessarily orientable unpunctured marked surface (S,M) we associate a commutative algebra, called quasi-cluster algebra, equipped with a distinguished set of generators, called quasi-cluster variables, in bijection with the…
In a cluster algebra, a subset of initial cluster variables can be specialised in such a way that all elements of the resulting algebra become polynomial in the remaining variables.
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 study the deformation theory of the Stanley-Reisner rings associated to cluster complexes for skew-symmetrizable cluster algebras of geometric and finite cluster type. In particular, we show that in the skew-symmetric case, these cluster…
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 construct a representation of the braid groups in a cluster C*-algebra coming from a triangulation of the Riemann surface S with one or two cusps. It is shown that the Laurent polynomials attached to the K-theory of such an algebra are…