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Completely Integrable Systems Connected with Lie Algebras

Authors:	Stanisław P. Kasperczuk

Affiliation:	(1) Institute of Physics, Pedagogical University, Plac Słowiański 6, 65-069 Zielona Góra, Poland

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. This revised version was published online in July 2006 with corrections to the Cover Date.

Keywords:	Lie algebra Poisson manifold Casimir function Poisson bialgebra integrable system
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