Invariant Poisson-Nijenhuis structures on Lie groups and classification

TL;DR

This paper classifies invariant Poisson-Nijenhuis structures on Lie groups and their Lie algebras, linking them to solutions of the classical Yang-Baxter equation and applying results to dynamical systems with Lie group symmetries.

Contribution

It introduces the concept of r-n structures as infinitesimal counterparts to invariant Poisson-Nijenhuis structures and classifies all such structures with invertible r on four-dimensional symplectic Lie algebras.

Findings

Classification of r-matrices and r-n structures on 4D symplectic Lie algebras.

Establishment of a correspondence between r-n structures and solutions to the classical Yang-Baxter equation.

Application to dynamical systems with Lie group symmetries.

Abstract

We study right-invariant (resp., left-invariant) Poisson-Nijenhuis structures on a Lie group $G$ and introduce their infinitesimal counterpart, the so-called r-n structures on the corresponding Lie algebra $\mathfrak g$. We show that $r$-$n$ structures can be used to find compatible solutions of the classical Yang-Baxter equation. Conversely, two compatible r-matrices from which one is invertible determine an $r$-$n$ structure. We classify, up to a natural equivalence, all $r$-matrices and all $r$-$n$ structures with invertible $r$ on four-dimensional symplectic real Lie algebras. The result is applied to show that a number of dynamical systems which can be constructed by $r$-matrices on a phase space whose symmetry group is Lie group $G$, can be specifically determined.