Some remarks on the bi-Hamiltonian structure of integral and discrete evolution equations

From the MR review by W.Oevel: "The authors investigate the relations between two abstract algebraic structures leading to a set of integrable evolution equations admitting a bi-Hamiltonian structure. The first structure, recently introduced by the authors, consists of an associative algebra with unit, endowed with a linear map solving the Yang-Baxter equation. A(1,1)-tensor field with vanishing Nijenhuis torsion (i.e., a hereditary recursion operator) can be built from these structures, leading to a bi-Hamiltonian scheme based on the Lie-Kirillov bracket. By suitable realisations of the abstract algebra integrable equations such as the ILW hierarchy in one and two spatial dimensions can be described. The second structure, recently introduced by Ragnisco and Santini, also consists of an abstract associative algebra, now endowed with an algebra homomorphism. Again, an abstract hereditary recursion operator can be defined in terms of the homomorphism, leading to another bi-Hamiltonian scheme. It is shown that both constructions lead to the same Nijenhuis operator by defining a suitable solution of the Yang-Baxter equation from the algebra homomorphism. Nevertheless, only for a special choice of parameters involved will the related Poisson brackets coincide. By a suitable realisation of the abstract scheme the infinite and the finite nonperiodic Toda lattices are considered. In "physical variables'' the abstract structure leads to a bi-Hamiltonian scheme with a hereditary operator. In "Flaschka variables'' the two basic Poisson brackets can be reduced, whereas the abstract hereditary operator becomes singular on the submanifold under consideration."