Related papers: Some conjectures on generalized cluster algebras v…
We define generalized cluster states based on finite group algebras in analogy to the generalization of the toric code to the Kitaev quantum double models. We do this by showing a general correspondence between systems with CSS structure…
We construct common triangular bases for almost all the known (quantum) cluster algebras from Lie theory. These bases provide analogs of the dual canonical bases, long anticipated in cluster theory. In cases where the generalized Cartan…
This is a survey paper of the theory of crystal bases, global bases and the cluster algebra structure on the quantum coordinate rings.
Let $\CC$ be a Hom-finite triangulated 2-Calabi-Yau category with a cluster-tilting object $T$. Under a constructibility condition we prove the existence of a set $\mathcal G^T(\CC)$ of generic values of the cluster character associated to…
We review clustering as an analysis tool and the underlying concepts from an introductory perspective. What is clustering and how can clusterings be realised programmatically? How can data be represented and prepared for a clustering task?…
We present a categorification of four mutation finite cluster algebras by the cluster category of the category of coherent sheaves over a weighted projective line of tubular weight type. Each of these cluster algebras which we call tubular…
We extend the construction of canonical bases for cluster algebras from unpunctured surfaces to the case where the number of marked points is one, and we show that the cluster algebra is equal to the upper cluster algebra in this case.
This article is part of an ongoing investigation of the two-dimensional Jacobian conjecture. In the first paper of this series, we proved the generalized Magnus' formula. In this paper, inspired by cluster algebras, we introduce a sequence…
For each simple Lie algebra $\mathfrak{g}$, we construct an algebra embedding of the quantum group $U_q(\mathfrak{g})$ into certain quantum torus algebra $D_\mathfrak{g}$ via the positive representations of split real quantum group. The…
Clustering is a powerful and extensively used data science tool. While clustering is generally thought of as an unsupervised learning technique, there are also supervised variations such as Spath's clusterwise regression that attempt to…
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 conjecture a characterization of a cluster automorphism as an algebra homomorphism from the cluster algebra to itself that restricts to a bijection between two clusters. This formulation does not require that the map commutes with…
Principal component method is used to study galactic globular clusters in 7- and 8-axis space of structural parameters. It is shown that the manifold properties of clusters with this set of parameters is determined mainly by two independent…
As a kind of basic machine learning method, clustering algorithms group data points into different categories based on their similarity or distribution. We present a clustering algorithm by finding hyper-planes to distinguish the data…
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…
Based on a fact that complex Clifford algebras of even dimension are isomorphic to the matrix ones, we consider bundles in Clifford algebras whose structure group is a general linear group acting on a Clifford algebra by left…
Standard Monte Carlo cluster algorithms have proven to be very effective for many different spin models, however they fail for frustrated spin systems. Recently a generalized cluster algorithm was introduced that works extremely well for…
This paper develops techniques for producing presentations of upper cluster algebras. These techniques are suited to computer implementation, and will always succeed when the upper cluster algebra is totally coprime and finitely generated.…
We give a geometric interpretation of cluster varieties in terms of blowups of toric varieties. This enables us to provide, among other results, an elementary geometric proof of the Laurent phenomenon for cluster algebras (of geometric…
We consider a class of generalized binomials emerging in fractional calculus. After establishing some general properties, we focus on a particular yet relevant case, for which we provide several ready-for-use combinatorial identities,…