Related papers: Polygons of Finite Mutation Type

In this paper, we define and classify the sign-equivalent exchange matrices. We give a Diophantine explanation for the differences between rank 2 cluster algebras of finite type and affine type based on \cite{CL24}. We classify the positive…

Associative algebras with involution over a field of zero characteristic are considered. It is proved that in this case for any finitely generated associative algebra with involution there exists a finite dimensional algebra with involution…

We provide a complete classification of the singularities of cluster algebras of finite type with trivial coefficients. Alongside, we develop a constructive desingularization of these singularities via blowups in regular centers over fields…

We classify the polarized symplectic automorphisms of Fano varieties of smooth cubic fourfolds (equipped with the Pl\"ucker polarization) and study the fixed loci.

We define nodal finite dimensional algebras and describe their structure over an algebraically closed field. For a special class of such algebras (type A) we find a criterion of tameness.

Fixed point subalgebras of finite dimensional factor algebras of algebras of polynomials in n indeterminates over the finite field $\mathbb F_2$ (with respect to all $\mathbb F_2$-algebra automorphisms) are fully described.

Among the mutation finite cluster algebras the tubular ones are a particularly interesting class. We show that all tubular (simply laced) cluster algebras are of exponential growth by two different methods: first by studying the…

We study a subclass of K\"ahler-Einstein Fano polygons and how they behave under mutation. The polygons of interest are K\"ahler-Einstein Fano triangles and symmetric Fano polygons. In particular, we find an explicit bound for the number of…

Let C be the category of finite-dimensional representations of a quantum affine algebra of simply-laced type. We introduce certain monoidal subcategories C_l (l integer) of C and we study their Grothendieck rings using cluster algebras.

We consider the general notion of coloured quiver mutation and show that the mutation class of a coloured quiver $Q$, arising from an $m$-cluster tilting object associated with $H$, is finite if and only if $H$ is of finite or tame…

We study finite morphisms of varieties and the link between their top multiplicity loci under certain assumptions. More precisely, we focus on how to determine that link in terms of the spaces of arcs of the varieties.

We classify toric Fano threefolds having at worst terminal singularities such that a rank of a $G$-invariant part of a class group equals one, where $G$ is a group acting on the variety by automorphisms.

This paper investigates the finite generation of cluster automorphism groups. By applying the pseudo $\mathbb{N}$-grading introduced in our previous work, we establish a sufficient condition for a cluster automorphism group to be finitely…

We define a class of finite-dimensional Jacobian algebras, which are called (simple) polygon-tree algebras, as a generalization of cluster-tilted algebras of type $\D$. They are $2$-CY-tilted algebras. Using a suitable process of mutations…

We classify simple representations of Khovanov-Lauda-Rouquier algebras in finite type. The classification is in terms of a standard family of representations that is shown to yield the dual PBW basis in the Grothendieck group. Finally, we…

We define a new finite type invariant for stably homeomorphic class of curves on compact oriented surfaces without boundaries and extend to a regular homotopy invariant for spherical curves.

We classify some special classes of non-rational Fano threefolds with terminal singularities. In particular, all such hyperelliptic and trigonal varieties are found.

The fundamental property of Fano varieties with mild singularities is that they have a finite polyhedral Mori cone. Thus, it is very interesting to ask: What we can say about algebraic varieties with a finite polyhedral Mori cone? I give a…

In the algebraic theory of K-stability, one of the most challenging problems is to show the graded algebra associated with certain higher rank quasi-monomial valuations are finitely generated. In the global case of Fano varieties and local…

We prove that finite groups have the same complex character tables iff the group algebras are twisted forms of each other as Drinfel'd quasi-bialgebras or iff there is non-associative bi-Galois algebra over these groups. The interpretations…