Related papers: Polygons of Finite Mutation Type
Finite-dimensional Jacobian algebras are studied from the perspective of representation types. We establish that (like other representation types) the notions of $E$-finiteness and $E$-tameness are invariant under mutations of quivers with…
We compute the class group of a full rank upper cluster algebra in terms of its exchange polynomials. As a corollary, we recover a theorem by Cao, Keller, and Qin from 2023 characterizing the UFDs among these algebras. Furthermore, under…
We investigate the cluster-tilted algebras of finite representation type over an algebraically closed field. We give an explicit description of the relations for the quivers for finite representation type. As a consequence we show that a…
In this article we explore compactifications of cluster varieties of finite type in complex dimension two. Cluster varieties can be viewed as the spec of a ring generated by theta functions and a compactification of such varieties can be…
We classify Fano threefolds with only terminal singularities whose canonical class is Cartier and divisible by 2, and satisfying an additional assumption that the $G$-invariant part of the Weil divisor class group is of rank 1 with respect…
We observe that over an algebraically closed field, any finite-dimensional algebra is the endomorphism algebra of an m-cluster-tilting object in a triangulated m-Calabi-Yau category, where m is any integer greater than 2.
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 give a complete description of the cluster-mutation classes of diagrams of Dynkin types \mathbb{A},\mathbb{B},\mathbb{D} and of affine Dynkin types \mathbb{B}^{(1)},\mathbb{C}^{(1)},\mathbb{D}^{(1)} via certain families of diagrams.
Determining when the birational automorphism group of a Fano variety is finite is an interesting and difficult problem. The main technique for studying this problem is by the Noether-Fano method. This method has been effective in studying…
We consider several distinct characterizations of finite implication algebras. One of these leads to a new characterization of Boolean polymatroids.
It is pretty well-known that toric Fano varieties of dimension k with terminal singularities correspond to convex lattice polytopes P in R^k of positive finite volume, such that intersection of P and Z^k consists of the point 0 and vertices…
We show that the algebraic local fundamental group of any klt singularity as well as the algebraic fundamental group of the smooth locus of any log Fano variety are finite.
For Fano varieties of various singularities such as canonical and terminal, we construct examples with large Fano index. By low-dimensional evidence, we conjecture that our examples have the largest Fano index for all dimensions.
Automorphisms of finite order and real forms of "smooth" affine Kac-Moody algebras are studied, i.e. of 2-dimensional extensions of the algebra of smooth loops in a simple Lie algebra. It is shown that they can be parametrized by certain…
We describe recent progress in a program to understand the classification of three-dimensional Fano varieties with $\mathbb{Q}$-factorial terminal singularities using mirror symmetry. As part of this we give an improved and more conceptual…
We classify three-dimensional nodal Fano varieties that are double covers of smooth quadrics branched over intersections with quartics acted on by finite simple non-abelian groups, and study their rationality.
We classify three-dimensional Fano varieties with canonical Gorenstein singularities of degree bigger than 64.
Based on the construction of polytope functions and several results about them in [LP], we take a deep look on their mutation behaviors to find a link between a face of a polytope and a sub-cluster algebra of the corresponding cluster…
We show that the mutation class of a finite quiver without oriented cycles is finite if and only is the quiver is either Dynkin, extended Dynkin or has at most two vertices.
We compute the list of all minimal 2-infinite diagrams, which are cluster algebraic analogues of extended Dynkin graphs.