Classical R-Matrices and the Feigin-Odesskii Algebra via Hamiltonian and Poisson Reductions

Abstract

We present a formula for a classical $r$-matrix of an integrable system obtained by Hamiltonian reduction of some free field theories using pure gauge symmetries. The framework of the reduction is restricted only by the assumption that the respective gauge transformations are Lie group ones. Our formula is in terms of Dirac brackets, and some new observations on these brackets are made. We apply our method to derive a classical $r$-matrix for the elliptic Calogero-Moser system with spin starting from the Higgs bundle over an elliptic curve with marked points. In the paper we also derive a classical Feigin-Odesskii algebra by a Poisson reduction of some modification of the Higgs bundle over an elliptic curve. This allows us to include integrable lattice models in a Hitchin type construction.

arXiv comment27 pages LaTex

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• Solutions of the quantum dynamical Yang-Baxter equation and dynamical quantum groups
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• Multiplicative Hitchin Systems and Supersymmetric Gauge Theory
• 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
• R-Matrix Structure of Hitchin System in Tyurin Parameterization
• Three lectures on classical integrable systems and gauge field theories
• On the r-matrix structure of the hyperbolic BC(n) Sutherland model
• On the classical r-matrix structure of the rational BC(n) Ruijsenaars-Schneider-van Diejen system
• Geometry of Higgs bundles over elliptic curves related to automorphisms of simple Lie algebras, Calogero-Moser systems and KZB equations