Complete integrability from Poisson-Nijenhuis structures on compact hermitian symmetric spaces

TL;DR

This paper explores Poisson-Nijenhuis systems on compact hermitian symmetric spaces, revealing their spectrum, integrability, and connection to Gelfand-Tsetlin variables, with a focus on Grassmannians.

Contribution

It introduces a new class of integrable models on symmetric spaces using Poisson-Nijenhuis structures, linking eigenvalues to action variables and providing explicit formulas for connections.

Findings

Eigenvalues of Nijenhuis tensor are Gelfand-Tsetlin variables in Grassmannians

Models are integrable with respect to both Poisson structures

Complete integrability is established via nested subalgebras

Abstract

We study a class of Poisson-Nijenhuis systems defined on compact hermitian symmetric spaces, where the Nijenhuis tensor is defined as the composition of Kirillov-Konstant-Souriau symplectic form with the so called Bruhat-Poisson structure. We determine its spectrum. In the case of Grassmannians the eigenvalues are the Gelfand-Tsetlin variables. We introduce the abelian algebra of collective hamiltonians defined by a chain of nested subalgebras and prove complete integrability. By construction, these models are integrable with respect to both Poisson structures. The eigenvalues of the Nijenhuis tensor are a choice of action variables. Our proof relies on an explicit formula for the contravariant connection defined on vector bundles that are Poisson with respect to the Bruhat-Poisson structure.