On factorized Lax pairs for classical many-body integrable systems

Abstract

In this paper we study factorization formulae for the Lax matrices of the classical Ruijsenaars-Schneider and Calogero-Moser models. We review the already known results and discuss their possible origins. The first origin comes from the IRF-Vertex relations and the properties of the intertwining matrices. The second origin is based on the Schlesinger transformations generated by modifications of underlying vector bundles. We show that both approaches provide explicit formulae for $M$-matrices of the integrable systems in terms of the intertwining matrices (and/or modification matrices). In the end we discuss the Calogero-Moser models related to classical root systems. The factorization formulae are proposed for a number of special cases.

42 pages, minor changes

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• Field analogue of the Ruijsenaars-Schneider model
• Dualities in quantum integrable many-body systems and integrable probabilities -- I
• Lax equations for relativistic $\mathrm{GL}(NM,\mathbb{C})$ Gaudin models on elliptic curve
• Non-ultralocal classical r-matrix structure for 1+1 field analogue of elliptic Calogero-Moser model
• Characteristic determinant and Manakov triple for the double elliptic integrable system
• Quantum-classical duality for Gaudin magnets with boundary
• Higher rank generalization of 11-vertex rational R-matrix: IRF-Vertex relations and associative Yang-Baxter equation
• Quantum-classical correspondence for gl(1|1) supersymmetric Gaudin magnet with boundary
• Elliptic Ruijsenaars-Toda and elliptic Toda chains: classical r-matrix structure and relation to XYZ chain