Related papers: Crystal bases and categorifications
We introduce a new class of graded rings extending the class of generalized Weyl algebras. These rings are orders in crossed products of the most general type, and we introduce their basic structure theory. We provide an extensive list of…
This is a survey article on classical groups (over arbitrary division rings) and their geometries.
We construct quantized versions of generic bases in quantum cluster algebras of finite and affine types. Under the specialization of $q$ and coefficients to 1, these bases are generic bases of finite and affine cluster algebras.
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 describe the upper seminormal crystal structure for the $\mu$-supported $\delta$-vectors for any quiver with potential with reachable frozen vertices, or equivalently for the tropical points of the corresponding cluster $\mc{X}$-variety.…
We give an introduction to the theory of cluster categories and cluster tilted algebras. We include some background on the theory of cluster algebras, and discuss the interplay with cluster categories and cluster tilted algebras.
This is a survey of what is known and/or conjectured about the prime and primitive spectra of quantum algebras, of quantized coordinate rings in particular. The topological structure of these spectra, their relations to classical affine…
We give a brief introduction to (upper) cluster algebras and their quantization using examples. Then we present several important families of bases for these algebras using topological models. We also discuss tropical properties of these…
The history of the canonical basis and crystal basis of a quantized enveloping algebra and its representations is presented
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.
The cluster multiplication formulas for a generalized quantum cluster algebra of Kronecker type are explicitly given. Furthermore, a positive bar-invariant $\mathbb{Z}[q^{\pm\frac{1}{2}}]$-basis of this algebra is constructed.
We give a crystal structure on the set of all irreducible components of Lagrangian subvarieties of quiver varieties. One con show that, as a crystal, it is isomorphic to the crystal base of an irreducible highest weight representation of a…
Category Theory provides us with a clear notion of what is an internal structure. This will allow us to focus our attention on a certain type of relationship between context and structure.
This is a survey article on some connections between cluster algebras and link invariants, written for the Notices of the AMS.
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…
A major direction in the theory of cluster algebras is to construct (quantum) cluster algebra structures on the (quantized) coordinate rings of various families of varieties arising in Lie theory. We prove that all algebras in a very large…
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.
Cluster algebras were introduced by S. Fomin and A. Zelevinsky in math.RT/0104151; their study continued in math.RA/0208229, math.RT/0305434. This is a family of commutative rings designed to serve as an algebraic framework for the theory…
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…
Let $\g$ be an affine Kac-Moody Lie algebra. Let $\U^+$ be the positive part of the Drinfeld-Jimbo quantum enveloping algebra associated to $\g$. We construct a basis of $\U^+$ which is related to the Kashiwara-Lusztig global crystal basis…