A trihamiltonian extension of the Toda lattice

A new Poisson structure is defined on a subspace of the Kupershmidt algebra, isomorphic to the space HH of n×nn×n Hermitian matrices. The new Poisson structure is of Lie–Poisson type with respect to the standard Lie bracket of HH. This Poisson structure (together with two already known ones, obtained through a rr-matrix technique) allows to construct an extension of the periodic Toda lattice with nn particles that fits in a trihamiltonian recurrence scheme. Some explicit examples of the construction and of the first integrals found in this way are given.