A trihamiltonian extension of the Toda lattice

The authors consider a tri-Hamiltonian formulation giving rise not to a single chain of recurrences but a two-dimensional scheme labelled by two indices. An application of this new notion is considered in the case of the periodic Toda lattice. A new (linear) Poisson structure is defined on a subspace of the Kupershmidt algebra isomorphic to the space of Hermitian matrices. This new Poisson structure together with the two well-known linear and quadratic Toda brackets gives rise to a tri-Hamiltonian extension of the periodic lattice which fits the two-dimensional scheme mentioned above. Some explicit examples of the construction are given.