Completely Integrable Systems Connected with Lie Algebras

Abstract

In a recent paper Ballersteros and Ragnisco (1998) have proposed a new method of constructing integrable Hamiltonian systems. A new class of integrable systems may be devised using the following sequence: $\mathcal{A} \to \Lambda \to C \to \tilde \Lambda \to \{ .,\;.\} _{\tilde \Lambda } \to (A,\;\vartriangle )$, where A is a Lie algebra ${\text{(R}}^{\text{3}} ,[.,\;.]),\;\Lambda$ is a Lie–Poisson structure on R ₃, C is a Casimir for $\Lambda ,\;\{ .,\;.\} _{\tilde \Lambda }$ is a reduced Poisson bracket and (A, ▵) is a bialgebra. We study the relation between a Lie-Poisson stucture Λ and a reduced Poisson bracket $\{ ,\;\} _{\tilde \Lambda }$, which is a key element in using the Lie algebra A to constructing this sequence. New examples of Lie algebras and their related integrable Hamiltonian systems are given.