R-matrix theory, formal Casimirs and the periodic Toda lattice

From the MR review by Malcolm Adams: "This paper gives a very thorough discussion of the tri-Hamiltonian structure for the periodic Toda lattice. The approach is from the nonunitary R-matrix viewpoint along with the theory of reduction of Poisson structures. The R-matrix theory is presented in an appropriately general algebraic setting so that it can be applied in the various algebraic settings for the periodic Toda lattice. In particular, the equivalence of two standard Lax pair formulations in loop algebras is demonstrated by showing each of their equivalences to the formal differential difference scheme of B. A. Kupershmidt [Astérisque123 (1985)]. The paper is nicely written, giving brief reviews of the necessary R-matrix theory, Poisson reduction theory, and algebraic settings for the periodic Toda lattice. The paper also gives a particularly nice dicussion of the relations of alternate systems of integrals in the multi-Hamiltonian setting by means of generalized Schur polynomials."