相关论文: Webs on surfaces, rings of invariants, and cluster…

The rings of SL(V) invariants of configurations of vectors and linear forms in a finite-dimensional complex vector space V were explicitly described by Hermann Weyl in the 1930s. We show that when V is 3-dimensional, each of these rings…

We establish basic properties of cluster algebras associated with oriented bordered surfaces with marked points. In particular, we show that the underlying cluster complex of such a cluster algebra does not depend on the choice of…

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.

In this text, we investigate webs which can be associated to cluster algebras from the point of view of the abelian functional equations these webs carry, focusing on the polylogarithmic ones. We introduce a general notion of webs whose…

We generalise surface cluster algebras to the case of infinite surfaces where the surface contains finitely many accumulation points of boundary marked points. To connect different triangulations of an infinite surface, we consider infinite…

We construct a graded cluster algebra structure on the Cox ring of a smooth complex variety $Z$, depending on a base cluster structure on the ring of regular functions of an open subset $Y$ of $Z$. After considering some elementary examples…

We show that the coordinate ring of the Vinberg monoid of a simply connected semisimple complex group is an upper cluster algebra. As an application, we construct cluster structures on a large class of flat reductive monoids. After…

We survey some recent constructions of cluster algebra structures on coordinate rings of unipotent subgroups and unipotent cells of Kac-Moody groups. We also review a quantized version of these results.

We characterize mutation-finite cluster algebras of rank at least 3 using positive semi-definite quadratic forms. In particular, we associate with every unpunctured bordered surface a positive semi-definite quadratic space $V$, and with…

For any cluster algebra whose underlying combinatorial data can be encoded by a bordered surface with marked points, we construct a geometric realization in terms of suitable decorated Teichmueller space of the surface. On the geometric…

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…

We introduce new objects, called $(G,c)$-bands, associated with a simple simply-connected algebraic group $G$, and a Coxeter element $c$ in its Weyl group. We show that bands of a given type are the $K$-points of an infinite dimensional…

The proposal is to use clusters, graphs and networks as models in order to analyse the Web structure. Clusters, graphs and networks provide knowledge representation and organization. Clusters were generated by co-site analysis. The sample…

Using vertex algebra techniques, we determine a set of generators for the cohomology ring of the Hilbert schemes of points on an arbitrary smooth projective surface over the field of complex numbers.

We report on the computation of invariants, covariants, and contravariants of cubic surfaces. All algorithms are implemented in the computer algebra system magma.

We suggest a new, alternative algebraic method for computation of geometrical quantities by means of the embedding of local loops into Lie groups.

This is a survey article on some connections between cluster algebras and link invariants, written for the Notices of the AMS.

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…

We introduce a category of cluster algebras with fixed initial seeds. This category has countable coproducts, which can be constructed combinatorially, but no products. We characterise isomorphisms and monomorphisms in this category and…

Given a certain triangulation of a punctured surface with boundary, we construct a new triangulated surface without punctures which covers it. This new surface is naturally equipped with an action of a group of order two, and its quotient…