Quantum cluster algebras

Abstract

Cluster algebras form an axiomatically defined class of commutative rings designed to serve as an algebraic framework for the theory of total positivity and canonical bases in semisimple groups and their quantum analogs. In this paper we introduce and study quantum deformations of cluster algebras.

Quantum cluster variables via Serre polynomials

“…Let R be an integral domain (i.e. an integral commutative ring) and let v be an invertible element in R. We shall define generalized quantum cluster algebras over (R, v), in analogy with quantum cluster algebras as defined in [BZ05]. In fact, in the present paper, we shall only be interested in the following cases:…”

“…Consider the seeds (Λ(t), B(t), X(t)) and (Λ(t ), B(t ), X(t )) associated to t and t respectively. By Section 6 of [BZ05], every quantum cluster variable X j (t ), 1 ≤ j ≤ n, has an expansion of the form X j (t ) = d∈Z m J d X(t)(d), where the coefficients J d belong to Z[q ± 1 2 ] and can also be written as subtraction-free rational expressions in q 1 2 . Then this property also holds for X(t )(e ), and we can write…”

“…For this, we adopt the idea of the proof of Theorem 6.1 of [BZ05]. Fix two vertices t and t of T n and two quantum cluster monomials X(t)(e) and X(t )(e ) as in Definition 2.1.6 such that they have the same image under ev 1 .…”

“…They are commutative subalgebras of Laurent polynomial rings. Later, quantum cluster algebras were introduced in [BZ05]. Therefore, since its very beginning, the theory of (commutative and quantum) cluster algebras is closely related to topics in quantum groups and canonical bases.…”

T-Systems, Networks and Dimers

“…Via the projection π we may decompose this intersection into N +1 2 hexagons, which are the projection of bottom half cubes, ready for undergoing the cube 2 The reader should not confuse this term with its meaning in the previous section: the notion of flatness is relative to the geometry of the underlying cubic lattice here, and the "flat" surface is perpendicular to the direction (1, 1, 1). The term "flat" is used with this meaning throughout this section.…”