Related papers: Bases for cluster algebras from surfaces
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…
A cluster algebra is unistructural if the set of its cluster variables determines its clusters and seeds. It is conjectured that all cluster algebras are unistructural. In this paper, we show that any cluster algebra arising from a…
We generalize the construction of the bangle, band and the bracelet bases for cluster algebras from orbifolds to the case where there is only one marked point on the boundary.
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.
We give combinatorial formulas for the Laurent expansion of any cluster variable in any cluster algebra coming from a triangulated surface (with or without punctures), with respect to an arbitrary seed. Moreover, we work in the generality…
In this note we explain how to obtain cluster algebras from triangulations of (punctured) discs following the approach of S. Fomin, M. Shapiro and D. Thurston. Furthermore, we give a description of m-cluster categories via diagonals (arcs)…
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…
We study cluster algebras that are associated to unpunctured surfaces with coefficients arising from boundary arcs. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of…
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 study quantum cluster algebras from unpunctured surfaces with arbitrary coefficients and quantization. We first give a new proof of the Laurent expansion formulas for commutative cluster algebras from unpunctured surfaces, we then give…
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…
We give a uniform geometric realization for the cluster algebra of an arbitrary finite type with principal coefficients at an arbitrary acyclic seed. This algebra is realized as the coordinate ring of a certain reduced double Bruhat cell in…
We introduce a new class of finite dimensional gentle algebras, the surface algebras, which are constructed from an unpunctured Riemann surface with boundary and marked points by introducing cuts in internal triangles of an arbitrary…
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…
We study cluster algebras with principal coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of perfect…
The cluster category is a triangulated category introduced for its combinatorial similarities with cluster algebras. We prove that a cluster algebra A of finite type can be realized as a Hall algebra, called the exceptional Hall algebra, of…
This is a first step guide to the theory of cluster algebras. We especially focus on basic notions, techniques, and results concerning seeds, cluster patterns, and cluster algebras.
We construct and study cluster algebra structures in rings of invariants of the special linear group action on collections of three-dimensional vectors, covectors, and matrices. The construction uses Kuperberg's calculus of webs on marked…
In an attempt to create an algebraic framework for dual canonical bases and total positivity in semisimple groups, we initiate the study of a new class of commutative algebras.
This is an introduction to cluster algebras and their common triangular bases. These bases are Kazhdan-Lusztig-type and serve as the canonical bases of cluster algebras from the representation-theoretic point of view. We review seeds…