Related papers: Crystal bases and categorifications

We consider, for each exchange matrix B, a category of geometric cluster algebras over B and coefficient specializations between the cluster algebras. The category also depends on an underlying ring R, usually the integers, rationals, or…

This paper studies cluster algebras locally, by identifying a special class of localizations which are themselves cluster algebras. A `locally acyclic cluster algebra' is a cluster algebra which admits a finite cover (in a geometric sense)…

In this paper, we give a quantum cluster algebra structure on the deformed Grothendieck ring of $\CC_{n}$, where $\CC_{n}$ is a full subcategory of finite dimensional representations of $U_q(\widehat{sl_{2}})$ defined in section II.

We survey several generalizations of the Weyl algebra including generalized Weyl algebras, twisted generalized Weyl algebras, quantized Weyl algebras, and Bell-Rogalski algebras. Attention is paid to ring-theoretic properties,…

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…

A lot of recent activity has been directed towards various constructions of "natural" bases in cluster algebras. We develop a new approach to this problem which is close in spirit to Lusztig's construction of a canonical basis, and the…

This paper is a survey of knot theory and invariants of knots and links from the point of view of categories of diagrams. The topics range from foundations of knot theory to virtual knot theory and topological quantum field theory.

We show how the theory of canonical bases in modified universal enveloping algebras can be used to develop the theory of Chevalley groups over any commutative ring with 1.

This is a survey on coarse geometry with an emphasis on coarse homology theories.

Crystal base of the level 0 part of the modified quantum affine algebra $\widetilde U_q(\widehat{sl_2})_0$ is given by path. Weyl group actions, extremal vectors and crystal structure of all irreducible components are described explicitly.

Elementary review article on quantum cryptography.

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.

The present survey results from the will to reconcile two approaches to quantum probabilities: one rather physical and coming directly from quantum mechanics, the other more algebraic. The second leading idea is to provide a unified picture…

The theory of quantum symmetric pairs as developed by the second author is based on coideal subalgebras of the quantized universal enveloping algebra for a semisimple Lie algebra. This paper investigates the center of these coideal…

We demonstrate how one can see quantization of geometry, and quantum algebraic structure in supersymmetric gauge theory.

This paper extends the study of group algebras of finite groups in which the socle of the center is an ideal. We provide a detailed analysis of the structure of these groups. In a particular case, we reach a complete characterization of the…

Inspired by recent work of Geiss-Leclerc-Schroer, we use Hom-finite cluster categories to give a good candidate set for a basis of (upper) cluster algebras with coefficients arising from quivers. This set consists of generic values taken by…

The classification of Grassmannian cluster algebras resembles that of regular polygonal tilings. We conjecture that this resemblance may indicate a deeper connection between these seemingly unrelated structures.

In this paper is discussed description of some algebraic structures in quantum theory by using formal recursive constructions with "complex Poincar\'e group" ISO(4,C).

An algebraic structure underlying the quantity calculus is proposed consisting in an algebraic fiber bundle, that is, a base structure which is a free Abelian group together with fibers which are one dimensional vector spaces, all of them…