Poisson structures and degenerations of integrable systems related to y (gl2)

The Toda lattice is an important dynamical system studied in the theory of integrable systems. It is known that the Toda lattice is related to other integrable systems: the DST system and the XXX model. We will generalize these three systems by attaching a Hamiltonian system to a nonincreasing sequence ▁k=(k_0,k_1,...,k_N) such that k_i −k_(i+1)≤2. The Toda system corresponds to the constant sequence k_i = k, the DST system to k_i=k_(i+1)+1, and the XXX system to k_i=k_(i+1)+2. We will express the variables in all these systems in terms of τ-functions, and use this to give the relation between the 2 × 2 and N × N Lax matrix descriptions of the systems. We show that all these systems are completely integrable, giving explicit action-angle variables. In the past, these three systems were studied using independent sets of variables. Since we can express any system corresponding to such k using the same set of variables (τ-functions), we prove that all these systems are in fact isomorphic to the Toda system, and hence to each other. This seems not to have been known before. Sklyanin showed that the Toda lattice, the DST system, and the XXX model are related by degenerations. We will use deformed τ-functions to deform the systems attached to sequences k as above. The deformed systems will correspond to two sequences ▁k,▁s. Then we define explicit degenerations of these deformed systems, generalizing Sklyanin’s results to our deformed systems.LimitedAuthor requested closed access (OA after 2yrs) in Vireo ETD syste