Lax operator algebras and Hamiltonian integrable hierarchies

TL;DR

This paper develops a new framework using Lax operator algebras to construct and analyze integrable hierarchies on Riemann surfaces, extending previous approaches and deriving classical systems like Calogero-Moser.

Contribution

It introduces Lax operator algebras for complex Lie algebras on Riemann surfaces and constructs Hamiltonian integrable hierarchies, generalizing prior methods.

Findings

Constructed hierarchies of commuting Lax flows

Proved these flows are Hamiltonian with respect to a symplectic structure

Derived classical integrable systems such as elliptic Calogero-Moser models

Abstract

We consider the theory of Lax equations in complex simple and reductive classical Lie algebras with the spectral parameter on a Riemann surface of finite genus. Our approach is based on the new objects -- the Lax operator algebras, and develops the approach of I.Krichever treating the $\gl(n)$ case. For every Lax operator considered as the mapping sending a point of the cotangent bundle on the space of extended Tyrin data to an element of the corresponding Lax operator algebra we construct the hierarchy of mutually commuting flows given by Lax equations and prove that those are Hamiltonian with respect to the Krichever-Phong symplectic structure. The corresponding Hamiltonians give integrable finite-dimensional Hitchin-type systems. For example we derive elliptic $A_n$, $C_n$, $D_n$ Calogero-Moser systems in frame of our approach.