Trigonometric integrable tops from solutions of associative Yang-Baxter equation

Abstract

We consider a special class of quantum non-dynamical $R$-matrices in the fundamental representation of ${\mathrm{GL}}_N$ with spectral parameter given by trigonometric solutions of the associative Yang-Baxter equation. In the simplest case $N=2$ these are the well-known 6-vertex $R$-matrix and its 7-vertex deformation. The $R$-matrices are used for construction of the classical relativistic integrable tops of the Euler-Arnold type. Namely, we describe the Lax pairs with spectral parameter, the inertia tensors and the Poisson structures. The latter are given by the linear Poisson-Lie brackets for the non-relativistic models, and by the classical Sklyanin type algebras in the relativistic cases. In some particular cases the tops are gauge equivalent to the Calogero-Moser-Sutherland or trigonometric Ruijsenaars-Schneider models.

arXiv comment22 pages, minor corrections

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Cited by in corpus (15)

• Field analogue of the Ruijsenaars-Schneider model
• Infinite-dimensional Lie bialgebras via affinization of Novikov bialgebras and Koszul duality
• Generalized model of interacting integrable tops
• Field theory generalizations of two-body Calogero-Moser models in the form of Landau-Lifshitz equations
• Higher rank 1+1 integrable Landau-Lifshitz field theories from associative Yang-Baxter equation
• Anisotropic spin generalization of elliptic Macdonald-Ruijsenaars operators and R-matrix identities
• Elliptic generalization of integrable q-deformed anisotropic Haldane-Shastry long-range spin chain
• Relativistic interacting integrable elliptic tops
• Lax equations for relativistic $\mathrm{GL}(NM,\mathbb{C})$ Gaudin models on elliptic curve
• 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
• Classical integrable spin chains of Landau-Lifshitz type from R-matrix identities
• Supersymmetric generalization of q-deformed long-range spin chains of Haldane-Shastry type and trigonometric GL(N|M) solution of associative Yang-Baxter equation
• On R-matrix identities related to elliptic anisotropic spin Ruijsenaars-Macdonald operators