Symmetry and quaternionic integrable systems

TL;DR

This paper explores the symmetries of hyperhamiltonian systems on hyperkahler manifolds, showing how quaternionic oscillators relate to integrability and symmetry characterization, especially in Euclidean spaces.

Contribution

It establishes a connection between quaternionic oscillators and integrable hyperhamiltonian systems, highlighting symmetry roles in their classification.

Findings

Symmetries characterize integrability in hyperhamiltonian systems.

Quaternionic oscillators serve as models for integrable hyperhamiltonian dynamics.

Euclidean case provides a clear symmetry-integrability correspondence.

Abstract

Given a hyperkahler manifold M, the hyperkahler structure defines a triple of symplectic structures on M; with these, a triple of Hamiltonians defines a so called hyperhamiltonian dynamical system on M. These systems are integrable when can be mapped to a system of quaternionic oscillators. We discuss the symmetry of integrable hyperhamiltonian systems, i.e. quaternionic oscillators; and conversely how these symmetries characterize, at least in the Euclidean case, integrable hyperhamiltonian systems.