Related papers: Some conjectures on generalized cluster algebras v…

In connection with generalized cluster algebras we introduce a certain generalization of the celebrated Rogers dilogarithm, which we call the Rogers dilogarithms of higher degree. We show that there is an identity of these generalized…

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,…

We develop a general approach to finding combinatorial models for cluster algebras. The approach is to construct a labeled graph called a framework. When a framework is constructed with certain properties, the result is a model…

A cluster algebra is unistructural if the set of its cluster variables determines its clusters and seeds. It is conjectured that all cluster algebras are unistructural. In this paper, we show that any cluster algebra arising from a…

We construct the intermediate coverings of cluster-tilted algebras by defining the generalized cluster categories. These generalized cluster categories are Calabi-Yau triangulated categories with fraction CY-dimension and have also cluster…

We show the polynomial property of $F$-polynomials for generalized quantum cluster algebras and obtain the associated separation formulas under a mild condition. Along the way, we obtain Gupta's formulas of $F$-polynomials for generalized…

We introduce the notion of a lower bound cluster algebra generated by projective cluster variables as a polynomial ring over the initial cluster variables and the so-called projective cluster variables. We show that under an acyclicity…

A major direction in the theory of cluster algebras is to construct (quantum) cluster algebra structures on the (quantized) coordinate rings of various families of varieties arising in Lie theory. We prove that all algebras in a very large…

We consider two algebras of curves associated to an oriented surface of finite type - the cluster algebra from combinatorial algebra, and the skein algebra from quantum topology. We focus on generalizations of cluster algebras and…

Mixture models extend the toolbox of clustering methods available to the data analyst. They allow for an explicit definition of the cluster shapes and structure within a probabilistic framework and exploit estimation and inference…

We construct two bases for each cluster algebra coming from a triangulated surface without punctures. We work in the context of a coefficient system coming from a full-rank exchange matrix, for example, principal coefficients.

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…

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.…

Inspired by recent work of Geiss-Leclerc-Schroer, we use Hom-finite cluster categories to give a good candidate set for a basis of (upper) cluster algebras with coefficients arising from quivers. This set consists of generic values taken by…

In this research, a general theoretical framework for clustering is proposed over specific partial algebraic systems by the present author. Her theory helps in isolating minimal assumptions necessary for different concepts of clustering…

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…

Recent articles have shown the connection between representation theory of quivers and the theory of cluster algebras. In this article, we prove that some cluster algebras of type ADE can be recovered from the data of the corresponding…

This is Addendum to ``Structure of seeds in generalized cluster algebras'', Pacific J. Math. {277} (2015), 201--218. We extend the class of generalized cluster algebras studied therein to embrace examples in some applications.

We develop an elementary formula for certain non-trivial elements of upper cluster algebras. These elements have positive coefficients. We show that when the cluster algebra is acyclic these elements form a basis. Using this formula, we…

We prove a multiplication theorem for quantum cluster algebras of acyclic quivers. The theorem generalizes the multiplication formula for quantum cluster variables in \cite{fanqin}. We apply the formula to construct some $\mathbb{ZP}$-bases…