Relativistic Classical Integrable Tops and Quantum R-matrices

Abstract

We describe classical top-like integrable systems arising from the quantum exchange relations and corresponding Sklyanin algebras. The Lax operator is expressed in terms of the quantum non-dynamical $R$-matrix even at the classical level, where the Planck constant plays the role of the relativistic deformation parameter in the sense of Ruijsenaars and Schneider (RS). The integrable systems (relativistic tops) are described as multidimensional Euler tops, and the inertia tensors are written in terms of the quantum and classical $R$-matrices. A particular case of ${\mathrm{gl}}_N$ system is gauge equivalent to the $N$-particle RS model while a generic top is related to the spin generalization of the RS model. The simple relation between quantum $R$-matrices and classical Lax operators is exploited in two ways. In the elliptic case we use the Belavin's quantum $R$-matrix to describe the relativistic classical tops. Also by the passage to the noncommutative torus we study the large $N$ limit corresponding to the relativistic version of the nonlocal 2d elliptic hydrodynamics. Conversely, in the rational case we obtain a new ${\mathrm{gl}}_N$ quantum rational non-dynamical $R$-matrix via the relativistic top, which we get in a different way -- using the factorized form of the RS Lax operator and the classical Symplectic Hecke (gauge) transformation. In particular case of ${\mathrm{gl}}_2$ the quantum rational $R$-matrix is 11-vertex. It was previously found by Cherednik. At last, we describe the integrable spin chains and Gaudin models related to the obtained $R$-matrix.

36 pages, minor corrections

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Cited by in corpus (36)

• Planck Constant as Spectral Parameter in Integrable Systems and KZB Equations
• Classical integrable systems and soliton equations related to eleven-vertex R-matrix
• Quantum Baxter-Belavin R-matrices and multidimensional Lax pairs for Painleve VI
• Trigonometric integrable tops from solutions of associative Yang-Baxter equation
• Trigonometric version of quantum-classical duality in integrable systems
• Noncommutative extensions of elliptic integrable Euler-Arnold tops and Painleve VI equation
• Field analogue of the Ruijsenaars-Schneider model
• Generalized model of interacting integrable tops
• Field theory generalizations of two-body Calogero-Moser models in the form of Landau-Lifshitz equations
• On R-matrix valued Lax pairs for Calogero-Moser models
• On factorized Lax pairs for classical many-body integrable systems
• Higher rank 1+1 integrable Landau-Lifshitz field theories from associative Yang-Baxter equation
• Lax representation of the hyperbolic van Diejen dynamics with two coupling parameters
• Relativistic elliptic matrix tops and finite Fourier transformations
• Anisotropic spin generalization of elliptic Macdonald-Ruijsenaars operators and R-matrix identities
• GL(NM) quantum dynamical $R$-matrix based on solution of the associative Yang-Baxter equation
• Integrable System of Generalized Relativistic Interacting Tops
• Lax equations for relativistic $\mathrm{GL}(NM,\mathbb{C})$ Gaudin models on elliptic curve
• Relativistic interacting integrable elliptic tops
• Gauge equivalence between 1+1 rational Calogero-Moser field theory and higher rank Landau-Lifshitz equation
• Higher order analogues of unitarity condition for quantum R-matrices
• Associative Yang-Baxter equation for quantum (semi-)dynamical R-matrices
• Characteristic determinant and Manakov triple for the double elliptic integrable system
• Gauge equivalence of 1+1 Calogero-Moser-Sutherland field theory and higher rank trigonometric Landau-Lifshitz model
• Self-duality and scattering map for the hyperbolic van Diejen systems with two coupling parameters (with an appendix by S. Ruijsenaars)
• On the classical r-matrix structure of the rational BC(n) Ruijsenaars-Schneider-van Diejen system
• Higher rank generalization of 11-vertex rational R-matrix: IRF-Vertex relations and associative Yang-Baxter equation
• Supersymmetric generalization of q-deformed long-range spin chains of Haldane-Shastry type and trigonometric GL(N|M) solution of associative Yang-Baxter equation
• Asymmetric 6-vertex model and classical Ruijsenaars-Schneider system of particles
• Classical integrable spin chains of Landau-Lifshitz type from R-matrix identities
• Integrable many-body systems of Calogero-Ruijsenaars type
• On R-matrix identities related to elliptic anisotropic spin Ruijsenaars-Macdonald operators
• Multi-pole extension for elliptic models of interacting integrable tops
• Elliptic Ruijsenaars-Toda and elliptic Toda chains: classical r-matrix structure and relation to XYZ chain
• Nonstandard representations of type C affine Hecke algebra from K-operators
• Large N limit of spectral duality in classical integrable systems