Related papers: Cluster algebras of finite type and positive symme…
We give a characterization of radical square zero bound quiver algebras $\mathbf{k} Q/\mathcal{J}^2$ that admit $n$-cluster tilting subcategories and $n\mathbb{Z}$-cluster tilting subcategories in terms of $Q$. We also show that if $Q$ is…
A family of quantum cluster algebras is introduced and studied. In general, these algebras are new, but subclasses have been studied previously by other authors. The algebras are indexed by double partitions or double flag varieties.…
In the present paper we examine the relationship between several type $A$ cluster theories and structures. We define a 2D geometric model of a cluster theory, which generalizes cluster algebras from surfaces, and encode several existing…
We study local algebras, which are structures similar to $\mathbb{Z}$-graded algebras concentrated in degrees $-1,0,1$, but without a product defined for pairs of elements at the same degree $\pm1$. To any triple consisting of a Kac-Moody…
Cluster varieties are geometric objects that have recently found applications in several areas of mathematics and mathematical physics. This thesis studies the geometry of a large class of cluster varieties associated to compact oriented…
Clustering is a popular form of unsupervised learning for geometric data. Unfortunately, many clustering algorithms lead to cluster assignments that are hard to explain, partially because they depend on all the features of the data in 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…
In this note we give explicit isomorphisms of 2-categories between various versions of the categorified quantum group associated to a simply-laced Kac-Moody algebra. These isomorphisms are convenient when working with the categorified…
Simple representations of KLR algebras can be used to realize the infinity crystal for the corresponding symmetrizable Kac-Moody algebra. It was recently shown that, in finite and affine types, certain sub-categories of cuspidal…
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…
Using the unfolding method given in \cite{HL}, we prove the conjectures on sign-coherence and a recurrence formula respectively of ${\bf g}$-vectors for acyclic sign-skew-symmetric cluster algebras. As a following consequence, the…
Matrices are two-dimensional data structures allowing one to conceptually organize information. For example, adjacency matrices are useful to store the links of a network; correlation matrices are simple ways to arrange gene co-expression…
We describe a ring whose category of Cohen-Macaulay modules provides an additive categorification of the cluster algebra structure on the homogeneous coordinate ring of the Grassmannian of k-planes in n-space. More precisely, there is a…
Classification of cluster variables in cluster algebras (in particular, Grassmannian cluster algebras) is an important problem, which has direct application to computations of scattering amplitudes in physics. In this paper, we apply the…
Plabic graphs are intimately connected to the positroid stratification of the positive Grassmannian. The duals to these graphs are quivers, and it is possible to associate to them cluster algebras. For the top-cell graph of $Gr_{+}(k,n)$,…
For a finite group action on a finite EI quiver, we construct its `orbifold' quotient EI quiver. The free EI category associated to the quotient EI quiver is equivalent to the skew group category with respect to the given group action.…
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…
We prove that mutation of cluster-tilting objects in triangulated 2-Calabi-Yau categories is closely connected with mutation of quivers with potentials. This gives a close connection between 2-CY-tilted algebras and Jacobian algebras…
We extend the classification of finite Weyl groupoids of rank two. Then we generalize these Weyl groupoids to `reflection groupoids' by admitting non-integral entries of the Cartan matrices. This leads to the unexpected observation that the…
That finite-dimensional simple Lie algebras over the complex numbers can be classified by means of purely combinatorial and geometric objects such as Coxeter-Dynkin diagrams and indecomposable irreducible root systems, is arguably one of…