On the Lie-Poisson structure of the nonlinearized discrete eigenvalue problem (Q2738151)

It is well-known that the Lie-Poisson structure associated with a Lie algebra constitutes a generalized Hamiltonian structure on a Poisson manifold. In the paper under review a discrete version of the generalized finite-dimensional integrable Hamiltonian system is constructed. First a Lie group homomorphism from \(\text{SL}(2,\mathbb R)\) into \(\text{SL}(3,\mathbb R)\) is introduced by means of the adjoint action of \(\text{SL}(2, \mathbb R)\) onto its Lie algebra. Next a \(3\times 3\) discrete matrix eigenvalue problem, which has the same isospectral evolution equation as the \(2\times 2\) matrix eigenvalue problem, is given. Under a constraint between potentials and eigenfunctions a Poisson map, which is the nonlinearization of the \(3\times 3\) matrix spectral problem, is obtained and the integrability of this map is investigated. Finally a reduction on the symplectic foliation induced by the corresponding Lie-Poisson structure is established, and the resulting integrable symplectic map is exactly the nonlinearized \(2\times 2\) eigenvalue problem.