On the cluster nature and quantization of geometric \(R\)-matrices

DOI10.4171/PRIMS/55-1-2zbMath1442.13070arXiv1607.00722OpenAlexW2963663079MaRDI QIDQ1731795

Thomas Lam, Rei Inoue, Pavlo Pylyavskyy

Publication date: 14 March 2019

Published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1607.00722

zbMATH Keywords

cluster algebraquantum cluster algebrageometric \(R\)-matrix

Mathematics Subject Classification ID

Quantum groups (quantized enveloping algebras) and related deformations (17B37) Groups and algebras in quantum theory and relations with integrable systems (81R12) Yang-Baxter equations (16T25) Cluster algebras (13F60)

Related Items

Plabic R-matrices, Discrete dynamics in cluster integrable systems from geometric \(R\)-matrix transformations, Invariants of Weyl group action and \(q\)-characters of quantum affine algebras, Kirillov-Reshetikhin crystals \(B^{1, s}\) for \(\widehat{\mathfrak{s}\mathfrak{l}}_n\) using Nakajima monomials, Polygon recutting as a cluster integrable system, Cluster realization of Weyl groups and \(q\)-characters of quantum affine algebras, Cluster realizations of Weyl groups and higher Teichmüller theory, Solution of tetrahedron equation and cluster algebras, Tensor diagrams and cluster combinatorics at punctures, Geometric lifting of the integrable cellular automata with periodic boundary conditions, Differential equations for the closed geometric crystal chains

Cites Work

• Unnamed Item
• Classical and quantum dilogarithm identities
• Total positivity in loop groups. I: Whirls and curls
• The quantum dilogarithm and representations of quantum cluster varieties
• Discrete dynamical systems with \(W(A^{(1)}_{m-1} \times A^{(1)}_{n-1})\) symmetry
• Rigged configurations and cylindric loop Schur functions
• Cluster algebras. III: Upper bounds and double Bruhat cells.
• Parametrizations of canonical bases and totally positive matrices
• Crystals and total positivity on orientable surfaces
• Periodicities of T-systems and Y-systems, dilogarithm identities, and cluster algebras. I: Type \(B_r\)
• Positivity for cluster algebras
• Quantum cluster algebras.
• Cluster algebras I: Foundations
• Tropical R maps and affine geometric crystals
• Cluster algebras IV: Coefficients
• Cluster ensembles, quantization and the dilogarithm
• AFFINE CRYSTALS AND VERTEX MODELS
• Geometric Crystals and Set-Theoretical Solutions to the Quantum Yang-Baxter Equation
• Periodicities in cluster algebras and dilogarithm identities