Rational Top and its Classical R-matrix

Abstract

We construct a rational integrable system (the rational top) on a coadjoint orbit of ${\mathrm{SL}}_N$ Lie group. It is described by the Lax operator with spectral parameter and classical non-dynamical skew-symmetric $r$-matrix. In the case of the orbit of minimal dimension the model is gauge equivalent to the rational Calogero-Moser (CM) system. To obtain the results we represent the Lax operator of the CM model in two different factorized forms -- without spectral parameter (related to spinless case) and another one with the spectral parameter. The latter gives rise to the rational top while the first one is related to generalized Cremmer-Gervais $r$-matrices. The gauge transformation relating the rational top and CM model provides a classical rational version of the IRF-Vertex correspondence. From a geometrical point of view it describes the modification of $\mathrm{SL}(N,\mathbb{C})$-bundles over degenerated elliptic curve. In view of Symplectic Hecke Correspondence the rational top is related to the rational spin CM model. Possible applications and generalizations of the suggested construction are discussed. In particular, the obtained $r$-matrix defines a class of KZB equations.

19 pages

References in corpus (1)

• Classification of Isomonodromy Problems on Elliptic Curves

Cited by in corpus (19)

• 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
• Generalized model of interacting integrable tops
• Field theory generalizations of two-body Calogero-Moser models in the form of Landau-Lifshitz equations
• 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 interacting integrable elliptic tops
• Integrable System of Generalized Relativistic Interacting Tops
• Associative Yang-Baxter equation for quantum (semi-)dynamical R-matrices
• Gauge equivalence between 1+1 rational Calogero-Moser field theory and higher rank Landau-Lifshitz equation
• 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
• Asymmetric 6-vertex model and classical Ruijsenaars-Schneider system of particles
• Higher rank generalization of 11-vertex rational R-matrix: IRF-Vertex relations and associative Yang-Baxter equation
• Integrable many-body systems of Calogero-Ruijsenaars type