A trihamiltonian extension of the Toda lattice

A new Poisson structure is defined on a subspace of the Kupershmidt algebra, isomorphic to the space H of n × n Hermitian matrices. The new Poisson structure is of Lie-Poisson type with respect to the standard Lie bracket of H. This Poisson structure (together with two already known ones, obtained through a r-matrix technique) allows to construct an extension of the periodic Toda lattice with n 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.