Action-angle map and duality for the open Toda lattice in the perspective of Hamiltonian reduction

Abstract

An alternative derivation of the known action-angle map of the standard open Toda lattice is presented based on its identification as the natural map between two gauge slices in the relevant symplectic reduction of the cotangent bundle of GL(n, R). This then permits to interpret Ruijsenaars' action-angle duality for the Toda system in the same group-theoretic framework which was established previously for Calogero type systems.

Duality between the trigonometricBCnSutherland system and a completed rational Ruijsenaars–Schneider–van Diejen system

“…Specifically, we will reduce the cotangent bundle T * U(2n) with respect to the symmetry group G + × G + , where G + ∼ = U(n) × U(n) is the fix-point subgroup of an involution of U(2n). This enlarges the range of the reduction approach to action-angle dualities [11,12,18], which realizes [5,6,7,8,9] the following scenario. Pick a higher dimensional symplectic manifold (P, Ω) equipped with two Abelian Poisson algebras Q 1 and Q 2 formed by invariants under a symmetry group acting on P .…”

Global Description of Action-Angle Duality for a Poisson–Lie Deformation of the Trigonometric $$\varvec{\mathrm {BC}_n}$$ BC n Sutherland System

“…It was realized by Gorsky and his collaborators [14,15,21], and explored in detail by others [5][6][7][8][9][10][11][12][13]24,25], that dual pairs of integrable many-body systems can be derived by Hamiltonian reduction utilizing the following mechanism. Suppose that we have a higher dimensional 'master phase space' M that admits a symmetry group G, and two distinguished independent Abelian Poisson algebras H 1 and H 2 formed by G-invariant, smooth functions on M. Then we can apply Hamiltonian reduction to M and obtain a reduced phase space M red equipped with two Abelian Poisson algebras H 1 red and H 2 red that descend, respectively, from H 1 and H 2 .…”

Aspects of the inverse problem for the Toda chain

“…Its classical integrability has been established in [7,16]. Explicit formulae for the inverse action-angle map have been obtained first by Ruijssenaars [30] and recently rederived by Fehér [6] by means of a much simpler setting. Further, Olshanetsky and Perelomov [28] constructed the quantum integrals of motion inductively whereas Kostant [20] identified eigenfunctions of the open chain with Whittaker functions for GL(N, R).…”