The Hamiltonian structures of the two-dimensional Toda lattice and \(R\)-matrices (Q812958)

The Lax representation of the two-dimensional Toda hierarchy is introduced and basic facts about the classical \(R\)-matrix theory applied to integrable systems are recalled. An algebra of difference operators naturally associated with two-dimensional Toda hierarchy is defined. Then the \(R\)-matrix is introduced and it is shown that it comes from a nontrivial splitting of the algebra. It is checked that the skew-symmetric part of the \(R\)-matrix satisfies the modified Young-Baxter equation and thus defined three compatible Poisson structures. By a Dirac reduction to an affine space for the two-dimensional Toda lattice hierarchy, three compatible Poisson structures are obtained. The coordinate form of the Poisson brackets on the space of dependent variable is derived. At last, the tri-Hamiltonian formulation of the two-dimensional Toda flows is suggested.