On a trivial family of noncommutative integrable systems (Q352472)

Because integrable systems are rare, it is always interesting and valuable to find new one. In this paper, the author shows how some new integrable systems can be constructed from old ones. The main idea is the following. Suppose we have a manifold \(M\) and a Hamiltonian function \(H\) on \(M\) with a canonical Poisson bivector \(P\) defined on the phase space \(M\).NEWLINENEWLINE A second Poisson bivector \(P'\) is given by the Lie derivative of \(P\) along the Liouville vector field \(Y= AdH\), where \(A\) is a 2-tensor field acting on the differential of the Hamiltonian. So \(P'= L_YP\). \(P'\) is a trivial deformation because it is simultaneously a 2-coboundary and a 2-cocycle in the Poisson-Lichnerowicz cohomology defined by \(P\).NEWLINENEWLINE The author then exhibits 2-tensor fields \(A\) associated with well-known integrable systems (including the Toda lattice, the relativistic Toda lattice, and the rational Calogero-Moser model). Then he proves the noncommutative integrability of a generalization of the rational Calogero-Moser system with three-particle interaction.