Bi-Hamiltonian Structure of Spin Sutherland Models: The Holomorphic Case

L Fehér

doi:10.1007/s00023-021-01084-7

Abstract

We construct a bi-Hamiltonian structure for the holomorphic spin Sutherland hierarchy based on collective spin variables. The construction relies on Poisson reduction of a bi-Hamiltonian structure on the holomorphic cotangent bundle of mathrm{GL}(n,mathbb {C}), which itself arises from the canonical symplectic structure and the Poisson structure of the Heisenberg double of the standard mathrm{GL}(n,mathbb {C}) Poisson–Lie group. The previously obtained bi-Hamiltonian structures of the hyperbolic and trigonometric real forms are recovered on real slices of the holomorphic spin Sutherland model.

Highlights

The Hamiltonians tr(Lk) generate an integrable system on M, which reduces to the spin Sutherland system by keeping only the observables that are invariant under simultaneous conjugations of g and L by arbitrary elements of GL(n, C)
The bi-Hamiltonian structure was found by interpreting this hierarchy as the Poisson reduction of a bi-Hamiltonian hierarchy on the holomorphic cotangent bundle T ∗GL(n, C), described by Theorems 2.1, 2.2 and Proposition 2.4
We reproduced our previous results on real forms of the system [12,13] by considering real slices of the holomorphic reduced phase space

Summary

Introduction

The theory of integrable systems is an interesting field of mathematics motivated by influential examples of exactly solvable models of theoretical physics. We may refer to (1.1) as the holomorphic spin Sutherland hierarchy It is known (see, for example, [21]) that the holomorphic spin Sutherland hierarchy is a reduction of a natural integrable system on the cotangent bundle M := T ∗GL(n, C) equipped with its canonical symplectic form. The Hamiltonians tr(Lk) generate an integrable system on M, which reduces to the spin Sutherland system by keeping only the observables that are invariant under simultaneous conjugations of g and L by arbitrary elements of GL(n, C). We summarize the main results once more and highlight a few open problems

Bi-Hamiltonian Hierarchy on the Cotangent Bundle

The Reduced Bi-Hamiltonian Hierarchy

Recovering the Real Forms

Conclusion