Exchanging role of the phase space and symmetry group of integrable Hamiltonian systems related to Lie bialgebras of bi-symplectic types

TL;DR

This paper introduces a novel method to interchange the roles of phase space and symmetry group in integrable Hamiltonian systems based on Lie bialgebras, revealing new relations between integrals of motion.

Contribution

It constructs integrable systems with bi-symplectic Lie bialgebras and develops transformations that swap phase space and symmetry group roles, providing new insights into their structure.

Findings

New transformations exchanging phase space and symmetry group roles

Relations between integrals of motion in swapped systems

Examples of four-dimensional bi-symplectic Lie bialgebras

Abstract

We construct integrable Hamiltonian systems with Lie bialgebras $({\bf g} , {\bf \tilde{g}})$ of the bi-symplectic type for which the Poisson-Lie groups ${\bf G}$ play the role of the phase spaces, and their dual Lie groups ${\bf {\tilde {G}}}$ play the role of the symmetry groups of the systems. We give the new transformations to exchange the role of phase spaces and symmetry groups and obtain the relations between integrals of motions of these integrable systems. Finally, we give some examples of real four-dimensional Lie bialgebras of bi-symplectic type.