The Poisson structure on the coordinate ring of discrete Lax operator and Toda lattice equation (Q1342752)

A new Poisson structure on the coordinate ring of the discrete Lax operator is constructed. On this basis a new Hamiltonian formalism is obtained. The equation under consideration is a completely integrable system of the form \({\partial L\over \partial t} = [A,L]\), where \(L= \sum_{i\in Z} E_{i,i+1} + \sum_{i<j} L_{i,j} E_{i,j}\), \(A= (A_{i,j})_{i,j \in Z}\), \(A_{i,j}\) is a polynomial of \(L_{k,l}\), \(E_{i,j}\) is an \((i,j)\) matrix unit. The calculus for the Poisson structure applied to the (one-dimensional) Toda lattice equation \[ \partial^2_t u(s)= \exp \bigl(u(s+1) - u(s)\bigr) - \exp \bigl(u(s) - u(s-1) \bigr). \] The Poisson structure of the finite rank Lax operators is also discussed.