Quantum cluster algebras

By Arkady Berenstein, Andrei Zelevinsky. 2004

384 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/19510d6542e179f2928e673675f8a70f85e6e5f4

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In Algebra and Number Theory

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In Statistical and Nonlinear Physics

Citations in the corpus

Positivity for cluster algebras (2013, positive): “positivity has been shown in [17], building on [4], [16], [25], [26], using monoidal categorifications of quantum cluster algebras and perverse sheaves over graded quiver varieties.”
On a Quantum Analog of the Caldero–Chapoton Formula (2010, methodology): “The quantum cluster algebra Aq(Q) is of finite type and we have the following formulas for the cluster variables in terms of the initial cluster (X1, X2): X3 := X[1] = X (−1,1) +X X4 := X[1,2] = X (−1,0) +X +X X5 := X[1,2,1] = X (1,−1) +X X6 := X[1,2,1,2] = X (1,0) = X1 X7 := X[1,2,1,2,1] = X (0,1) ”
Introduction to Cluster Algebras (2018, background): “Rather than giving precise definitions, we indicate here the intuition motivating the definition and refer the reader to [4] for more details.”
The quantum dilogarithm and representations of quantum cluster varieties (2007, methodology): “for i ∈ I − I0. Now eca = Id implies ecaq = Id by [ BZq ], the latter implies by Lemma 5.23 that the cluster transformation ecdq of the space D × q,|ei| is the identity.”
Monoidal categorification of cluster algebras (2014, methodology): “By the Laurent phenomenon ([2]), we can write xX = ∑”
Cluster algebras and derived categories (2012, neutral): “…turned out that the combinatorics of cluster algebras also appear in many other subjects, for example in • Poisson geometry [60] [61] [62] [63] [10] . . . ; • discrete dynamical systems [27] [47] [74] [79] [84] [81] [91] . . . ; • higher Teichmüller spaces [35] [36] [37] [38] [39] . . . ; •…”
A Quantum Cluster Algebra of Kronecker Type and the Dual Canonical Basis (2010, background): “The quantized enveloping algebra Uq(g) is a Q(q)-algebra generated by Ei, (i = 1, 2), Fi, (i = 1, 2), Ki,K −1 i , (i = 1, 2), subject to the following relations KiKj = KjKi, (i 6= j) (11) KiK −1 i = K −1 i Ki = 1, (i = 1, 2) (12) KiEjK −1 i = q Ej , (1 ≤ i, j ≤ 2) (13) KiFjK −1 i = q Fj , (1 ≤ i, j ”
R-Twisting and 4d/2d Correspondences (2010, background): “For us it is especially important to consider the quantum version of these algebras [40, 50–53].”
Quiver Grassmannians and degenerate flag varieties (2011, background): “In recent papers, authors noticed that also the Poincaré polynomials of quiver Grassmannians play an important role in the study of quantum cluster algebras [Qin 2010; Berenstein and Zelevinsky 2005].”
Weak separation and plabic graphs (2011, background): “Let Cq(G(k, n)) be the quantization of the coordinate ring of G(k, n) introduced in [1]; since pI and pJ are in a common cluster, they quasi-commute in this ring.”