Related papers: On cluster algebras arising from unpunctured surfa…
We show that in case a cluster algebra coincides with its upper cluster algebra and the cluster algebra admits a grading with finite dimensional homogeneous components, the corresponding Berenstein-Zelevinsky quantum cluster algebra can be…
Considered as commutative algebras, cluster algebras can be very unpleasant objects. However, the first author introduced a condition known as "local acyclicity" which implies that cluster algebras behave reasonably. One of the earliest and…
In this paper, we study combinatorial properties of quasi-Cartan companions defined by the c-vectors of acyclic skew-symmetrizable cluster algebras. In particular, we show that the diagram of any skew-symmetrizable matrix associated with an…
We establish a combinatorial realization of continued fractions as quotients of cardinalities of sets. These sets are sets of perfect matchings of certain graphs, the snake graphs, that appear naturally in the theory of cluster algebras. To…
All algebras in a very large, axiomatically defined class of quantum nilpotent algebras are proved to possess quantum cluster algebra structures under mild conditions. Furthermore, it is shown that these quantum cluster algebras always…
This paper demonstrates that the homogeneous coordinate ring of the Grassmannian $\Bbb{G}(k,n)$ is a {\it cluster algebra of geometric type} - as defined by S. Fomin and A. Zelevinsky. Grassmannians having {\it finite cluster type} are…
We introduce a class of commutative superalgebras generalizing cluster algebras. A cluster superalgebra is defined by a hypergraph called an "extended quiver", and transformations called mutations. We prove the super analog of the "Laurent…
The cluster multiplication formulas for a generalized quantum cluster algebra of Kronecker type are explicitly given. Furthermore, a positive bar-invariant $\mathbb{Z}[q^{\pm\frac{1}{2}}]$-basis of this algebra is constructed.
To each tagged triangulation of a surface with marked points and non-empty boundary we associate a quiver with potential, in such a way that whenever we apply a flip to a tagged triangulation, the Jacobian algebra of the QP associated to…
LP algebras, introduced by Lam and Pylyavskyy, are a generalization of cluster algebras. These algebras are known to have the Laurent phenomenon, but positivity remains conjectural. Graph LP algebras are finite LP algebras encoded by a…
Generalized quantum cluster algebras introduced in [1] are quantum deformation of generalized cluster algebras of geometric types. In this paper, we prove that the Laurent phenomenon holds in these generalized quantum cluster algebras. We…
Motivated by a construction in the theory of cluster algebras (Fomin and Zelevinsky), one associates to each acyclic directed graph a family of sequences of natural integers, one for each vertex; this construction is called a {\em frieze};…
The subject considered in this paper has, at least, three points of interest. Suppose that we have a sequence of one-dimensional analytic varieties in a domain in $\Bbb C^n$. The cluster of this sequence consists from all points in the…
We describe an infinite family of non-Pl\"ucker cluster variables inside the double Bruhat cell cluster algebra defined by Berenstein, Fomin, and Zelevinsky. These cluster variables occur in a family of subalgebras of the double Bruhat cell…
In [LP] we introduced Laurent phenomenon algebras, a generalization of cluster algebras. Here we give an explicit description of Laurent phenomenon algebras with a linear initial seed arising from a graph. In particular, any graph…
This paper is a representation-theoretic extension of Part I. It has been inspired by three recent developments: surface cluster algebras studied by Fomin-Shapiro-Thurston, the mutation theory of quivers with potentials initiated by…
In earlier work, the author introduced a method for constructing a Frobenius categorification of a cluster algebra with frozen variables by starting from the data of an internally Calabi-Yau algebra, which becomes the endomorphism algebra…
Lots of research focuses on the combinatorics behind various bases of cluster algebras. This paper studies the natural basis of a type A cluster algebra, which consists of all cluster monomials. We introduce a new kind of combinatorial…
We prove a conjecture of Kontsevich regarding the solutions of rank two recursion relations for non-commutative variables which, in the commutative case, reduce to rank two cluster algebras of affine type. The conjecture states that…
Combinatorial methods are developed to find the cluster coordinates for moduli space of flat connections which is describing the Coulomb branch of higher rank N=2 theories derived by compactifying six dimensional (2,0) theory on a punctured…